Local and semilocal Poincar\'e inequalities on metric spaces

TL;DR

This paper investigates local versions of doubling and Poincaré inequalities in metric spaces, showing they self-improve to semilocal forms in proper connected spaces and exploring their implications for geometric and analytical properties.

Contribution

It demonstrates that local assumptions on metric spaces lead to significant self-improvement and various properties, advancing understanding of local geometric and analytical structures.

Findings

Local assumptions self-improve to semilocal in proper connected spaces

Local properties like quasiconvexity and Poincaré inequalities hold under local assumptions

Many properties of p-harmonic functions can be proved locally, except Liouville theorem

Abstract

We consider several local versions of the doubling condition and Poincar\'e inequalities on metric spaces. Our first result is that in proper connected spaces, the weakest local assumptions self-improve to semilocal ones, i.e. holding within every ball. We then study various geometrical and analytical consequences of such local assumptions, such as local quasiconvexity, self-improvement of Poincar\'e inequalities, existence of Lebesgue points, density of Lipschitz functions and quasicontinuity of Sobolev functions. It turns out that local versions of these properties hold under local assumptions, even though they are not always straightforward. We also conclude that many qualitative, as well as quantitative, properties of p-harmonic functions on metric spaces can be proved in various forms under such local assumptions, with the main exception being the Liouville theorem, which fails without global assumptions.

Equations

$$\mathop{\mathchoice{\kern 1.28333pt\vbox{\hrule width=3.50002pt}\kern-4.78336pt}{\kern 1.16664pt\vbox{\hrule width=3.50002pt}\kern-4.66666pt}{\kern 0.91664pt\vbox{\hrule width=2.75pt}\kern-3.66664pt}{\kern 0.74997pt\vbox{\hrule width=2.25pt}\kern-2.99998pt}}\mathopen{}\int_{B}|u-u_{B}|\,d\mu\leq Cr_{B}\biggl{(}\mathop{\mathchoice{\kern 1.28333pt\vbox{\hrule width=3.50002pt}\kern-4.78336pt}{\kern 1.16664pt\vbox{\hrule width=3.50002pt}\kern-4.66666pt}{\kern 0.91664pt\vbox{\hrule width=2.75pt}\kern-3.66664pt}{\kern 0.74997pt\vbox{\hrule width=2.25pt}\kern-2.99998pt}}\mathopen{}\int_{\lambda B}g^{p}\,d\mu\biggr{)}^{1/p},$$

$$|f(\gamma (0))-f(\gamma (l_{\gamma }))|\le {\int }_{\gamma }gds,$$

$$\|f\|_{N^{1,p}(X)}=\biggl{(}\int_{X}|f|^{p}\,d\mu+\inf_{g}\int_{X}g^{p}\,d\mu\biggr{)}^{1/p},$$

$$N^{1,p}(X)=\{f:\Vert f{\Vert }_{N^{1,p}(X)}<\infty \}.$$

$$C_p(E)=\underset{u}{\inf }\Vert u{\Vert }_{N^{1,p}(X)}^p,$$

$$\mu (2B)\le \mu (B(x_i,16r^{\prime }))\le C_{\mu }\mu (B(x_i,8r^{\prime }))\le C_{\mu }^4\mu (B(x_i,r^{\prime })),$$

$$N\underset{i}{\min }\mu (B(x_i,r^{\prime }))\le \sum _{i=1}^N\mu (B(x_i,r^{\prime }))\le \mu (2B)\le C_{\mu }^4\underset{i}{\min }\mu (B(x_i,r^{\prime })),$$

$$\mu\bigl{(}\bigl{(}\tfrac{2}{3}-\delta\bigr{)}B_{0}\bigr{)}\leq\mu\bigl{(}B\bigl{(}x_{i},\tfrac{2}{3}r_{0}\bigr{)}\bigr{)}=\mu\bigl{(}B(x_{i},16r^{\prime}))\leq C_{\mu}^{4}\mu(B(x_{i},r^{\prime})).$$

$$N\underset{i}{\min }\mu (B(x_i,r^{\prime }))$$

$$\displaystyle\leq\frac{C_{\mu}^{4}\mu(B_{0})}{\mu\bigl{(}\bigl{(}\tfrac{2}{3}-\delta\bigr{)}B_{0}\bigr{)}}\min_{i}\mu(B(x_{i},r^{\prime}))\leq M\min_{i}\mu(B(x_{i},r^{\prime})),$$

$$B^{\prime}\subset\bigcup_{i=1}^{N}B(x_{i},2r^{\prime})\subset\bigcup_{i=1}^{N}B\bigl{(}x_{i},\tfrac{1}{2}r\bigr{)}.$$

$$d(x,y)=\{\begin{array}{ll}|x_2-y_2|,& \text{if }j=k\text{ or }j=0\text{ or }k=0,\\ 1,& \text{if }j\ne k,j,k\ge 1.\end{array}$$

$$\mu (B(x,r))\ge \underset{j}{\min }\mu (B_j^{\prime })>0.$$

$$M_{\Omega ,B_0}^{\ast }f(x):=\underset{B}{\sup }\text{vbox}\text{hrule}width=3.50002pt{\int }_{B}fd\mu ,x\in \Omega ,$$

$$\mu (E_{\tau })\le \frac{C}{\tau }{\int }_{E_{\tau }}|f|d\mu ,\text{where }{E}_{\tau }=\{x\in \Omega :M_{\Omega ,B_0}^{\ast }f(x)>\tau \}.$$

$${\int }_{\Omega }(M_{\Omega ,B_0}^{\ast }f{)}^{t}d\mu \le C_t{\int }_{\Omega }|f{|}^{t}d\mu .$$

$$\biggl{(}\mathop{\mathchoice{\kern 1.28333pt\vbox{\hrule width=3.50002pt}\kern-4.78336pt}{\kern 1.16664pt\vbox{\hrule width=3.50002pt}\kern-4.66666pt}{\kern 0.91664pt\vbox{\hrule width=2.75pt}\kern-3.66664pt}{\kern 0.74997pt\vbox{\hrule width=2.25pt}\kern-2.99998pt}}\mathopen{}\int_{B}|u-u_{B}|^{q}\,d\mu\biggr{)}^{1/q}\leq Cr_{B}\biggl{(}\mathop{\mathchoice{\kern 1.28333pt\vbox{\hrule width=3.50002pt}\kern-4.78336pt}{\kern 1.16664pt\vbox{\hrule width=3.50002pt}\kern-4.66666pt}{\kern 0.91664pt\vbox{\hrule width=2.75pt}\kern-3.66664pt}{\kern 0.74997pt\vbox{\hrule width=2.25pt}\kern-2.99998pt}}\mathopen{}\int_{\lambda B}g^{p}\,d\mu\biggr{)}^{1/p}.$$

$$X=\mathbf{R}^{2}\setminus\bigl{\{}(x,y)\in\mathbf{R}^{2}:0<y<\sqrt{1-x^{2}}-e^{-1/|x|}\bigr{\}},$$

$$C_{0}=\bigl{\{}(x,y)\in\mathbf{R}^{2}:0<\sqrt{1-x^{2}}-e^{-1/|x|}\leq y<\sqrt{1-x^{2}}\bigr{\}},$$

$$X=({\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle B(0,2)}\kern 0.0pt}$\hss}{B(0,2)}}{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle B(0,2)}\kern 0.0pt}$\hss}{B(0,2)}}{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle B(0,2)}\kern 0.0pt}$\hss}{B(0,2)}}{\hbox to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle B(0,2)}\kern 0.0pt}$\hss}{B(0,2)}}}\setminus B(0,1))\cup\{x=(x_{1},x_{2}):0<|x|\leq 2\text{ and }x_{1}x_{2}\geq 0\}\subset\mathbf{R}^{2}.$$

$$r^{\prime }=\underset{j}{\min }{r}_j,C=\underset{j}{\max }{C}_j\text{and}\lambda =\underset{j}{\max }{\lambda }_j,$$

$${\rho }_{\epsilon }(z)=\inf \sum _{i=1}^{m}d(x_{i-1},x_i),$$

$$|{\rho }_{\epsilon }(y)-({\rho }_{\epsilon }{)}_{B_0}|$$

$$\le \sum _{j=0}^{\infty }\text{vbox}\text{hrule}width=3.50002pt{\int }_{B_{j+1}}|{\rho }_{\epsilon }-({\rho }_{\epsilon }{)}_{B_j}|d\mu$$

$$\le \sum _{j=0}^{\infty }{C}_{\mu }^3\text{vbox}\text{hrule}width=3.50002pt{\int }_{B_j}|{\rho }_{\epsilon }-({\rho }_{\epsilon }{)}_{B_j}|d\mu$$

$$\displaystyle\leq C_{\mu}^{3}C_{\rm PI}\sum_{j=0}^{\infty}r_{B_{j}}\biggl{(}\mathop{\mathchoice{\kern 1.28333pt\vbox{\hrule width=3.50002pt}\kern-4.78336pt}{\kern 1.16664pt\vbox{\hrule width=3.50002pt}\kern-4.66666pt}{\kern 0.91664pt\vbox{\hrule width=2.75pt}\kern-3.66664pt}{\kern 0.74997pt\vbox{\hrule width=2.25pt}\kern-2.99998pt}}\mathopen{}\int_{\lambda B_{j}}1^{p}\,d\mu\biggr{)}^{1/p}$$

$$=\Lambda d(x,y).$$

$${\rho }_{\epsilon }(y)=|{\rho }_{\epsilon }(y)-{\rho }_{\epsilon }(x)|\le 2\Lambda d(x,y)<10\Lambda d(x,y).$$

$$\sum _{i=1}^{M_n}d(x_{i-1}^n,x_i^n)\le 3\Lambda d(x,y).$$

$$x={\hat{x}}_0^n,{\hat{x}}_1^n,.\text{nonscript}.\text{nonscript}.,{\hat{x}}_{m_n}^n=y,\text{satisfying }{\epsilon }_n\le d({\hat{x}}_{i-1}^n,{\hat{x}}_i^n)\le 3{\epsilon }_n,$$

$$0<\mathop{\mathchoice{\kern 1.28333pt\vbox{\hrule width=3.50002pt}\kern-4.78336pt}{\kern 1.16664pt\vbox{\hrule width=3.50002pt}\kern-4.66666pt}{\kern 0.91664pt\vbox{\hrule width=2.75pt}\kern-3.66664pt}{\kern 0.74997pt\vbox{\hrule width=2.25pt}\kern-2.99998pt}}\mathopen{}\int_{B_{0}}|u-u_{B_{0}}|\,d\mu\leq Cr_{B_{0}}\biggl{(}\mathop{\mathchoice{\kern 1.28333pt\vbox{\hrule width=3.50002pt}\kern-4.78336pt}{\kern 1.16664pt\vbox{\hrule width=3.50002pt}\kern-4.66666pt}{\kern 0.91664pt\vbox{\hrule width=2.75pt}\kern-3.66664pt}{\kern 0.74997pt\vbox{\hrule width=2.25pt}\kern-2.99998pt}}\mathopen{}\int_{\lambda B_{0}}g^{p}\,d\mu\biggr{)}^{1/p}=0,$$

$$\Vert u-u_A{\Vert }_{L^q(A)}\le Q\text{and}\Vert u-u_E{\Vert }_{L^q(E)}\le Q.$$

$$\Vert u-u_{A\cup E}{\Vert }_{L^q(A\cup E)}\le 4(1+{\theta }^{-1/q})Q.$$

Full text

Local and semilocal Poincaré inequalities

on metric spaces

Anders Björn and Jana Björn

  

Anders Björn

*Department of Mathematics, Linköping University,

SE-581 83 Linköping, Sweden; anders.bjorn@liu.se

Jana Björn

Department of Mathematics, Linköping University,

SE-581 83 Linköping, Sweden; jana.bjorn@liu.se *

(To appear in J. Math. Pures Appl.)

Abstract. We consider several local versions of the doubling condition and Poincaré inequalities on metric measure spaces. Our first result is that in proper connected spaces, the weakest local assumptions self-improve to semilocal ones, i.e. holding within every ball.

We then study various geometrical and analytical consequences of such local assumptions, such as local quasiconvexity, self-improvement of Poincaré inequalities, existence of Lebesgue points, density of Lipschitz functions and quasicontinuity of Sobolev functions. It turns out that local versions of these properties hold under local assumptions, even though they are not always straightforward.

We also conclude that many qualitative, as well as quantitative, properties of $p\mspace{1.0mu}$-harmonic functions on metric spaces can be proved in various forms under such local assumptions, with the main exception being the Liouville theorem, which fails without global assumptions.

Résumé. Nous considérons plusieurs versions locales des conditions de doublement et des inégalités de Poincaré dans des espaces métriques mesurés. Notre premier résultat stipule que dans un espace propre connexe, les hypothèses locales les plus faibles s’améliorent en semi-locales, c.à.d. elles sont valables dans chaque boule.

Nous étudions ensuite certaines conséquences géométriques et analytiques de telles hypothèses locales tel que la quasi-convexité locale, l’auto amélioration des inégalités de Poincaré, l’existence des points Lebesgue, la densité des fonctions Lipschitz et la quasi-continuité des fonctions Sobolev. Il s’avère que les versions locales de ces propriétés restent valables sous les hypothèses locales même si elles ne sont pas toujours immédiates.

Nous concluons également que sous telles hypothéses locales, plusieurs propriétés qualitatives, ainsi que quantitatives, des fonctions p-harmoniques sur des espaces métriques peuvent être prouvées sous diverses formes, l’exception principale étant le théorème de Liouville qui échoue sans hypothéses globales.

Key words and phrases: capacity, density of Lipschitz functions, Lebesgue point, local doubling, metric measure space, Newtonian space, nonlinear potential theory, $p\mspace{1.0mu}$-harmonic function, Poincaré inequality, quasicontinuity, quasiminimizer, semilocal doubling, Sobolev space.

Mathematics Subject Classification (2010): Primary: 31E05; Secondary: 30L99, 31C45, 35J60, 46E35

1 Introduction

In the last two decades, an extensive part of first-order analysis, such as the Sobolev space theory and nonlinear potential theory for $p\mspace{1.0mu}$-harmonic functions, has been developed on metric spaces. Standard assumptions are very often that $\mu$ is a doubling measure supporting a $p\mspace{1.0mu}$-Poincaré inequality and that the space $X$ is complete. The doubling condition controls changes in scales, while the Poincaré inequality guarantees that functions are controlled by their so-called upper gradients. Both of these conditions play a vital role in many proofs. These assumptions are usually imposed globally on the whole space.

In this paper we study how these conditions can be relaxed and replaced by similar local or semilocal assumptions, while retaining most of the important consequences. We assume throughout the paper that $1\leq p<\infty$ and that $X=(X,d,\mu)$ is a metric space equipped with a metric $d$ and a positive complete Borel measure $\mu$ such that $0<\mu(B)<\infty$ for all balls $B\subset X$.

Definition 1.1**.**

The measure $\mu$ is locally doubling if for every $x\in X$ there are $r,C>0$ (depending on $x$) such that $\mu(2B)\leq C\mu(B)$ for all balls $B\subset B(x,r)$. If such a $C$ exists for all $x\in X$ and all $r>0$, then $\mu$ is semilocally doubling. (Here and below, $\lambda B$ stands for the ball concentric with $B$ and with $\lambda$-times the radius.)

The measure $\mu$ supports a local $p\mspace{1.0mu}$-Poincaré inequality, $p\geq 1$, if for every $x\in X$ there are $r,C>0$ and $\lambda\geq 1$ such that for all balls $B\subset B(x,r)$, all integrable functions $u$ on $\lambda B$ and all upper gradients $g$ of $u$,

[TABLE]

where $u_{B}:=\mathop{\mathchoice{\kern 1.28333pt\vbox{\hrule width=3.50002pt}\kern-4.78336pt}{\kern 1.16664pt\vbox{\hrule width=3.50002pt}\kern-4.66666pt}{\kern 0.91664pt\vbox{\hrule width=2.75pt}\kern-3.66664pt}{\kern 0.74997pt\vbox{\hrule width=2.25pt}\kern-2.99998pt}}\mathopen{}\int_{B}u\,d\mu:=\int_{B}u\,d\mu/\mu(B)$. If such $C$ and $\lambda$ exist for all $x$ and all $r$, then $X$ supports a semilocal $p\mspace{1.0mu}$-Poincaré inequality.

It may be worth comparing this with the definition of local function spaces. A function $f\in L^{1}_{\rm loc}(X)$ if for every $x\in X$ there is $r>0$ such $f\in L^{1}(B(x,r))$. If $X$ is proper (i.e. all closed bounded sets are compact), then it is a well-known (and useful) fact that this is equivalent to requiring that $f\in L^{1}_{\rm loc}(B)$ for every ball $B$ in $X$. It turns out that a similar fact is true for the doubling property. Note that if $\mu$ is globally doubling, then $X$ is proper if and only if it is complete.

Proposition 1.2**.**

If $X$ is proper and $\mu$ is locally doubling, then $\mu$ is semilocally doubling.

This is perhaps not very surprising, and essentially just needs a standard compactness argument to be shown. That a similar result is true also for Poincaré inequalities is far less obvious, and requires several pages to prove. Poincaré inequalities are intimately related to connectivity properties, and thus the precise statement is as follows. The assumptions of properness and connectedness cannot be dropped.

Theorem 1.3**.**

If $X$ is proper and connected and $\mu$ is locally doubling and supports a local $p\mspace{1.0mu}$-Poincaré inequality, then it supports a semilocal $p\mspace{1.0mu}$-Poincaré inequality.

As already mentioned, in much of the metric space literature on first-order analysis it is assumed that $\mu$ is globally doubling and supports a global $p\mspace{1.0mu}$-Poincaré inequality, while it is folklore that much of the theory holds under local assumptions. For instance, the assumptions are global in the monographs Hajłasz–Koskela [23], Björn–Björn [6] and Heinonen–Koskela–Shanmugalingam–Tyson [28]. In [23] and [6] it is mentioned that Riemannian manifolds with Ricci curvature bounded from below support semilocal assumptions with the implicit implication that most of the developed theory holds under these weaker assumptions.

There are some papers requiring local assumptions, but they often take different forms from paper to paper. Sometimes it is assumed that the constants involved are uniform, something we do not assume (but for Section 6). Such assumptions (of different kinds) are e.g. assumed in Cheeger [18], Danielli–Garofalo–Marola [19], Garofalo–Marola [21] and Holopainen–Shanmugalingam [29]. The requirements are in all cases more restrictive than our local assumptions, and those in [18] are more restrictive than our semilocal assumptions.

Once we have established Proposition 1.2 and Theorem 1.3 (which we do in Sections 3 and 4), we take a look at which useful consequences of global doubling and Poincaré inequalities can be obtained already under (semi)local assumptions, with and without properness. We study the self-improvement of Poincaré inequalities, Lebesgue points, density of Lipschitz and locally Lipschitz functions, quasicontinuity and $p\mspace{1.0mu}$-harmonic functions under (semi)local assumptions.

In Section 5 we concentrate on the self-improvement of Poincaré inequalities and prove two results: one improving the norm on the left-hand side of (1.1) and the other one the norm on the right-hand side. In Section 6 we see how uniformly local assumptions give slightly stronger self-improvement conclusions. Under global assumptions these important results are due to Hajłasz–Koskela [22], [23] and Keith–Zhong [32], respectively. In particular, the latter result can be localized in the following way.

Theorem 1.4**.**

If $X$ is locally compact and $\mu$ is locally doubling and supports a local $p\mspace{1.0mu}$-Poincaré inequality, both with uniform constants $C$ and $\lambda$ and with $p>1$, then $X$ supports a local $q$-Poincaré inequality for some $q<p$ with new uniform constants $C$ and $\lambda$.

Neither in the assumptions nor in the conclusion do we assume uniformity in the radius $r$ of the balls $B(x,r)$ involved. It is worth noting that the corresponding result with global assumptions and a global conclusion fails in locally compact spaces, due to a counterexample by Koskela [39]. In complete spaces it holds by Keith–Zhong [32].

In Section 7, we turn to Lebesgue points and show that Sobolev (Newtonian) functions have Lebesgue points outside a set of zero $p\mspace{1.0mu}$-capacity. Traditionally, as well as in metric spaces, such results are shown using the density of continuous functions. Here we avoid using this property and instead exploit the Newtonian theory in a different and novel way, which may be of interest also under global assumptions.

In the next section we consider the density of Lipschitz and locally Lipschitz functions in the Sobolev (Newtonian) space $N^{1,p}(X)$. There are two existing results under global assumptions in the literature, one assuming doubling and a Poincaré inequality, due to Shanmugalingam [42], and the other more recent one assuming $p>1$, completeness and doubling but no Poincaré inequality, due to Ambrosio–Colombo–Di Marino [3] and Ambrosio–Gigli–Savaré [4]. We extend both results to local assumptions, and combine them. Among other results, we obtain the following “local-to-global” density result.

Theorem 1.5**.**

If $X$ is proper and connected and $\mu$ is locally doubling and supports a local $p\mspace{1.0mu}$-Poincaré inequality then Lipschitz functions with compact support are dense in $N^{1,p}(X)$.

In Section 9 we look at consequences of the obtained density results, primarily quasicontinuity and various properties of the Sobolev capacity ${C_{p}}$. We end the paper with a discussion on how much of the nonlinear potential theory for $p\mspace{1.0mu}$-harmonic functions, developed e.g. in the book [6], holds under local assumptions, and explain that indeed most of the results therein extend to this setting, with the main exception being the Liouville theorem, which actually fails without global assumptions. As most of the results in [6] are either local or semilocal (e.g. on bounded domains) this is not so surprising, and indeed this has already been hinted upon in the literature, as mentioned above.

The importance of distinguishing between local and global assumptions is certainly more apparent when discussing global properties, such as the Dirichlet problem on unbounded domains (as in Hansevi [24], [25]) or existence of global singular functions (as in Holopainen–Shanmugalingam [29]). We hope that the theory developed in this paper will provide a suitable foundation for such studies.

Acknowledgement. We would like to thank the anonymous referee for encouraging us to add more details to some of the proofs. The authors were supported by the Swedish Research Council, grants 621-2011-3139, 621-2014-3974 and 2016-03424.

2 Upper gradients and Newtonian spaces

We assume throughout the paper that $1\leq p<\infty$ and that $X=(X,d,\mu)$ is a metric space equipped with a metric $d$ and a positive complete Borel measure $\mu$ such that $0<\mu(B)<\infty$ for all balls $B\subset X$. It follows that $X$ is separable and Lindelöf. Proofs of the results in this section can be found in the monographs Björn–Björn [6] and Heinonen–Koskela–Shanmugalingam–Tyson [28].

A curve is a continuous mapping from an interval, and a rectifiable curve is a curve with finite length. We will only consider curves which are compact and rectifiable, and thus each curve can be parameterized by its arc length $ds$. A property is said to hold for $p\mspace{1.0mu}$-almost every curve if it fails only for a curve family $\Gamma$ with zero $p\mspace{1.0mu}$-modulus, i.e. there exists $0\leq\rho\in L^{p}(X)$ such that $\int_{\gamma}\rho\,ds=\infty$ for every rectifiable curve $\gamma\in\Gamma$.

Following Heinonen–Koskela [27], we introduce upper gradients as follows (they called them very weak gradients).

Definition 2.1**.**

A Borel function $g:X\to[0,\infty]$ is an upper gradient of a function $f:X\to{\overline{\mathbf{R}}}:=[-\infty,\infty]$ if for all nonconstant rectifiable curves $\gamma\colon[0,l_{\gamma}]\to X$,

[TABLE]

where the left-hand side is considered to be $\infty$ whenever at least one of the terms therein is infinite. If $g:X\to[0,\infty]$ is measurable and (2.1) holds for $p\mspace{1.0mu}$-almost every nonconstant rectifiable curve, then $g$ is a $p\mspace{1.0mu}$-weak upper gradient of $f$.

The $p\mspace{1.0mu}$-weak upper gradients were introduced in Koskela–MacManus [40]. It was also shown therein that if $g\in L^{p}_{\rm loc}(X)$ is a $p\mspace{1.0mu}$-weak upper gradient of $f$, then one can find a sequence $\{g_{j}\}_{j=1}^{\infty}$ of upper gradients of $f$ such that $\|g_{j}-g\|_{L^{p}(X)}\to 0$. If $f$ has an upper gradient in $L^{p}_{\rm loc}(X)$, then it has an a.e. unique minimal $p\mspace{1.0mu}$-weak upper gradient $g_{f}\in L^{p}_{\rm loc}(X)$ in the sense that for every $p\mspace{1.0mu}$-weak upper gradient $g\in L^{p}_{\rm loc}(X)$ of $f$ we have $g_{f}\leq g$ a.e., see Shanmugalingam [43]. Following Shanmugalingam [42], we define a version of Sobolev spaces on the metric space $X$.

Definition 2.2**.**

For a measurable function $f:X\to{\overline{\mathbf{R}}}$, let

[TABLE]

where the infimum is taken over all upper gradients $g$ of $f$. The Newtonian space on $X$ is

[TABLE]

The quotient space $N^{1,p}(X)/{\sim}$, where $f\sim h$ if and only if $\|f-h\|_{N^{1,p}(X)}=0$, is a Banach space and a lattice, see Shanmugalingam [42]. In this paper we assume that functions in $N^{1,p}(X)$ are defined everywhere (with values in ${\overline{\mathbf{R}}}$), not just up to an equivalence class in the corresponding function space. This is important for upper gradients to make sense.

For a measurable set $E\subset X$, the Newtonian space $N^{1,p}(E)$ is defined by considering $(E,d|_{E},\mu|_{E})$ as a metric space in its own right. We say that $f\in N^{1,p}_{\rm loc}(E)$ if for every $x\in E$ there exists a ball $B_{x}\ni x$ such that $f\in N^{1,p}(B_{x}\cap E)$. If $f,h\in N^{1,p}_{\rm loc}(X)$, then $g_{f}=g_{h}$ a.e. in $\{x\in X:f(x)=h(x)\}$, in particular for $c\in\mathbf{R}$ we have $g_{\min\{f,c\}}=g_{f}\chi_{\{f<c\}}$ a.e.

Definition 2.3**.**

The (Sobolev) capacity of a set $E$ is the number

[TABLE]

where the infimum is taken over all $u\in N^{1,p}(X)$ such that $u=1$ on $E$.

We say that a property holds quasieverywhere (q.e.) if the set of points for which the property does not hold has capacity zero. The capacity is the correct gauge for distinguishing between two Newtonian functions. If $u\in N^{1,p}(X)$, then $u\sim v$ if and only if $u=v$ q.e. Moreover, if $u,v\in N^{1,p}_{\rm loc}(X)$ and $u=v$ a.e., then $u=v$ q.e.

We let $B=B(x,r)=\{y\in X:d(x,y)<r\}$ denote the ball with centre $x$ and radius $r$, and let $\lambda B=B(x,\lambda r)$. We assume throughout the paper that balls are open. In metric spaces it can happen that balls with different centres and/or radii denote the same set. We will however make the convention that a ball $B$ comes with a predetermined centre and radius $r_{B}$. Note that it can happen that $B(x_{0},r_{0})\subset B(x_{1},r_{1})$ even when $r_{0}>r_{1}$. In disconnected spaces this can happen also when $r_{0}>2r_{1}$. If $X$ is connected, then $r_{0}>2r_{1}$ is possible only when $B(x_{0},r_{0})=B(x_{1},r_{1})=X$.

3 Local doubling

One can think of several different possibilities for local assumptions. We will make them precise below. In this section we concentrate on the doubling property and then consider Poincaré inequalities in the next section.

Definition 3.1**.**

The measure $\mu$ is doubling within $B(x_{0},r_{0})$ if there is $C>0$ (depending on $x_{0}$ and $r_{0}$) such that $\mu(2B)\leq C\mu(B)$ for all balls $B\subset B(x_{0},r_{0})$.

We say that $\mu$ is locally doubling (on $X$) if for every $x_{0}\in X$ there is $r_{0}>0$ (depending on $x_{0}$) such that $\mu$ is doubling within $B(x_{0},r_{0})$.

If $\mu$ is doubling within every ball $B(x_{0},r_{0})$ then it is semilocally doubling (on $X$), and if moreover $C$ is independent of $x_{0}$ and $r_{0}$, then $\mu$ is globally doubling (on $X$).

Note that when saying that $\mu$ is doubling within $B(x_{0},r_{0})$ this is (implicitly) done with respect to $X$ as the balls are all with respect to $X$, and moreover $2B$ does not have to be a subset of $B(x_{0},r_{0})$. This is not the same as saying that $\mu$ is globally doubling on $B(x_{0},r_{0})$, which refers to balls with respect to $B(x_{0},r_{0})$.

If $\mu$ is locally doubling on $X$ and $\Omega\subset X$ is open, then $\mu$ is also locally doubling on $\Omega$. This hereditary property fails for semilocal and global doubling, see Remark 3.3 below.

An even weaker property is that $\mu$ is pointwise doubling at $x_{0}\in X$ if there are $C,r_{0}>0$ such that $\mu(B(x_{0},2r))\leq C\mu(B(x_{0},r))$ for $0<r<r_{0}$. Requiring such a pointwise assumption and a similar pointwise Poincaré inequality at every $x_{0}\in X$ is too weak for most results. See however, Björn–Björn–Lehrbäck [9], [10] for capacity estimates using such pointwise assumptions.

Definition 3.2**.**

The space $X$ is globally doubling if there is a constant $N$ such that every ball $B(x,r)$ can be covered by at most $N$ balls with radii $\frac{1}{2}r$.

The space $X$ is locally doubling if for every $x_{0}\in X$ there is $r_{0}>0$ such that $B(x_{0},r_{0})$ is globally doubling. Moreover, $X$ is semilocally doubling if every ball $B\subset X$ is globally doubling.

Remark 3.3**.**

Let $B$ be a ball. If $\mu$ is globally doubling then it does not follow that $\mu|_{B}$ is globally doubling on $B$, see Example 4.3. On the other hand if the space $X$ is globally doubling then so is $B$ (as a metric space). This is why Definition 3.2 differs from Definition 3.1 in that it considers balls with respect to $B$ which are not necessarily balls with respect to $X$. It is possible to give an equivalent definition of (semi)local doubling of $X$ more in the spirit of Definition 3.1, which only uses balls with respect to $X$, but such a definition is more technical to state and hence we prefer our Definition 3.2.

It is rather immediate that every subset of a globally doubling metric space is itself globally doubling, and hence the same hereditary property also holds for (semi)local doubling. It is also easy to see that every bounded set in a semilocally doubling metric space is totally bounded. See Heinonen [26, Section 10.13] for more on doubling metric spaces.

If $\mu$ is (semi)locally resp. globally doubling, then so is $X$ by the following result.

Proposition 3.4**.**

Assume that $\mu$ is doubling within $B_{0}=B(x_{0},r_{0})$ in the sense of Definition 3.1. Then $\delta B_{0}$ is globally doubling for every $\delta<\tfrac{2}{3}$, with $N$ depending only on $\delta$ and the doubling constant within $B_{0}$.

Example 3.5 below shows that the constant $\tfrac{2}{3}$ is optimal and that it can even happen that $\tfrac{2}{3}B_{0}$ is not totally bounded.

• Proof.

Let $B^{\prime}=B(x,r)\cap\delta B_{0}$ be an arbitrary ball with respect to $\delta B_{0}$ for some $\delta<\tfrac{2}{3}$. Then $x\in\delta B_{0}$ and we may assume that $r\leq 2\delta r_{0}$. Let $B=B(x,r)$ and r^{\prime}=\min\bigl{\{}\tfrac{1}{4}r,\tfrac{1}{24}r_{0}\bigr{\}}. Assume that $x_{i}\in B^{\prime}$, $i=1,\mathinner{.}\nonscript\!.\nonscript\!\mathinner{.},N$, are such that $d(x_{i},x_{j})\geq 2r^{\prime}$ if $i\neq j$. Then $B(x_{i},r^{\prime})$ are pairwise disjoint and $B(x_{i},8r^{\prime})\subset B_{0}$. We shall show that there is a bound for $N$. Let $C_{\mu}$ be the doubling constant for $\mu$ within $B_{0}$.

If $r^{\prime}=\tfrac{1}{4}r$, then

[TABLE]

and hence

[TABLE]

which yields that $N\leq C_{\mu}^{4}$. On the other hand, if $r^{\prime}=\tfrac{1}{24}r_{0}$ then as in (3.1),

[TABLE]

Therefore

[TABLE]

where $M$ only depends on $C_{\mu}$ and $\delta$. Hence $N\leq M$.

We can thus find a maximal pairwise disjoint collection of $\{B(x_{i},r^{\prime})\}_{i=1}^{N}$ with $N\leq\max\{M,C_{\mu}^{4}\}$ elements. As the collection is maximal we see that

[TABLE]

Hence $\delta B_{0}$ is globally doubling. ∎

Example 3.5**.**

Let $I_{0}=\{0\}\times[-1,0]\subset\mathbf{R}^{2}$ and I_{j}=\{j\}\times\bigl{[}\tfrac{2}{3}-\tfrac{1}{3}\cdot 2^{-j},1\bigr{]}\subset\mathbf{R}^{2}, $j=1,2,\mathinner{.}\nonscript\!.\nonscript\!\mathinner{.}$, be vertical linear segments in the plane. Equip each $I_{j}$ with a multiple of the 1-dimensional Hausdorff measure so that $\mu(I_{j})=2^{-j}$, $j=0,1,\mathinner{.}\nonscript\!.\nonscript\!\mathinner{.}$. Let $X=\bigcup_{j=0}^{\infty}I_{j}$, equipped with $\mu$ and the metric $d$ so that for $x=(j,x_{2})\in I_{j}$ and $y=(k,y_{2})\in I_{k}$,

[TABLE]

Let $x_{0}=(0,0)$. Then it is easily verified that $\mu$ is doubling within $B_{0}=B(x_{0},1)$. However, $\tfrac{2}{3}B_{0}$ is not totally bounded, and thus not globally doubling.

This example also shows that the next two results are sharp. More precisely, $X$ is bounded and complete but noncompact and thus not proper. Both $X$ and $\mu$ are locally doubling but neither is semilocally doubling.

The most common global assumptions are that $X$ is complete and supports a global $p\mspace{1.0mu}$-Poincaré inequality and that $\mu$ is globally doubling. It then follows that $X$ is proper and connected (and even quasiconvex). Under local assumptions these properties need to be imposed separately. Connectedness is strongly related to Poincaré inequalities, which we discuss in the next section. Properness always implies completeness and the proof of [6, Proposition 3.1] also shows the following equivalence.

Lemma 3.6**.**

Assume that $X$ is semilocally doubling. Then $X$ is proper if and only if $X$ is complete.

If $X$ is only locally doubling and complete, then it is locally compact but not necessarily proper as the following example shows.

Example 3.7**.**

Let $X=\mathbf{R}^{2}$ equipped with the Gaussian measure $Ce^{-x_{1}^{2}-x_{2}^{2}}\,dx$ but with the distance $d(x,y)=\arctan|x-y|$. Then $X$ is bounded and complete, but not proper (as $X$ is a closed bounded noncompact set). At the same time, $\mu$ clearly is locally doubling and supports a local $1$-Poincaré inequality.

In proper spaces, local and semilocal doubling are equivalent, as we shall now see.

Proposition 3.8**.**

Assume that $X$ is proper. If $X$ resp. $\mu$ is locally doubling, then it is semilocally doubling.

• Proof.

Let $B_{0}$ be a ball. If $X$ is locally doubling, we can for each x\in{\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle B}\kern 0.0pt}\hss}{B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle B}\kern 0.0pt}\hss}{B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle B}\kern 0.0pt}\hss}{B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle B}\kern 0.0pt}\hss}{B}}}_{0} find a globally doubling ball $B_{x}\ni x$. Since $X$ is proper, {\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle B}\kern 0.0pt}\hss}{B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle B}\kern 0.0pt}\hss}{B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle B}\kern 0.0pt}\hss}{B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle B}\kern 0.0pt}\hss}{B}}}_{0} is compact and thus we can find a finite set $\{x_{i}\}_{i=1}^{N}$ such that $B_{0}\subset\bigcup_{i=1}^{N}B_{x_{i}}$. It is easily seen that a finite union of globally doubling sets is globally doubling, and hence $B_{0}$ is globally doubling.

Now assume instead that $\mu$ is locally doubling and $B_{0}=B(x_{0},r_{0})$. By enlarging $r_{0}$, if necessary, we may assume that either $r_{0}=\operatorname{dist}(x_{0},X\setminus B_{0})$ or $B_{0}=X$. Since {\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle B}\kern 0.0pt}\hss}{B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle B}\kern 0.0pt}\hss}{B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle B}\kern 0.0pt}\hss}{B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle B}\kern 0.0pt}\hss}{B}}}_{0} is compact (as $X$ is proper), it can be covered by finitely many balls $B_{j}=B(x_{j},r_{j})$ such that $\mu$ is doubling within each ball $2B_{j}$. Let $r^{\prime}=\min_{j}r_{j}$. Again by compactness, we can cover {\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle B}\kern 0.0pt}\hss}{B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle B}\kern 0.0pt}\hss}{B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle B}\kern 0.0pt}\hss}{B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle B}\kern 0.0pt}\hss}{B}}}_{0} by finitely many balls $B^{\prime}_{j}$ with radii $r^{\prime}/2$. Let $B(x,r)\subset B_{0}$ be arbitrary and find $j$ and $j^{\prime}$ such that $x\in B_{j}$ and $x\in B^{\prime}_{j^{\prime}}$.

If $r\leq r^{\prime}$ then $B(x,r)\subset 2B_{j}$ and hence the conclusion of the proposition holds for $B(x,r)$ with constant $\max_{j}C_{j}$. On the other hand, if $r>r^{\prime}$ then $x\in B^{\prime}_{j^{\prime}}$ and hence $B^{\prime}_{j^{\prime}}\subset B(x,r)$, which yields the lower bound

[TABLE]

Since $B(x,2r)\subset B(x_{0},5r_{0})$, we also have a uniform upper bound for $\mu(B(x,2r))$, which proves that $\mu$ is semilocally doubling. ∎

We will need the following local maximal function estimate.

Proposition 3.9**.**

Assume that $\mu$ is doubling within the ball $B_{0}$ in the sense of Definition 3.1, and let $\Omega\subset B_{0}$ be open. For $f\in L^{1}(\Omega)$, define the noncentred local maximal function

[TABLE]

where the supremum is taken over all balls $B$ such that $x\in B\subset\Omega$ and $\frac{5}{2}B\subset B_{0}$. Then

[TABLE]

Moreover, if $t>1$, then

[TABLE]

The constant $5$ in the factor $\frac{5}{2}$ above and in the proof below comes from the $5$-covering lemma. It is well known that also the $(3+\varepsilon)$-covering lemma holds, for every $\varepsilon>0$, see [6, Remark 1.8, Example 1.9 and p. 36]. Thus the factor $\frac{5}{2}$ can be replaced by any factor $>\frac{3}{2}$, which would also make it possible to decrease some other constants in this paper. For simplicity we have chosen to just rely on the $5$-covering lemma, as is common practice in analysis on metric spaces.

• Proof.

Since $\mu$ is doubling within the ball $B_{0}$, it is true that $\mu(5B)\leq C\mu(B)$ for every ball $B$ used in (3.2). Thus, the proof of Lemma 3.12 in [6] directly applies also here showing the first estimate (3.3). The second estimate then follows just as in the proof of Theorem 3.13 in [6], with $X$ therein replaced by $\Omega$. ∎

We end the section by noting the following consequence of local doubling, which will be useful later.

Theorem 3.10**.**

(The Lebesgue differentiation theorem)* Assume that $\mu$ is locally doubling. If $f\in L^{1}_{\rm loc}(X)$, then a.e. point is a Lebesgue point for $f$.*

• Proof.

As $X$ is Lindelöf, we can cover $X$ by balls $\{B_{j}\}_{j=1}^{\infty}$ such that $f\in L^{1}(10B_{j})$ and $\mu$ is doubling within each $10B_{j}$. By the proof of Theorem 1.6 in Heinonen [26], the Vitali covering theorem holds in each $B_{j}$. It then follows from Remark 1.13 in [26] that the Lebesgue differentiation theorem holds within each $B_{j}$, and hence in $X$, as the union is countable. ∎

4 Local Poincaré inequalities

In this section we study local aspects of the Poincaré inequality similarly to the doubling property in Section 3. It will turn out that connectivity plays an important role here.

Definition 4.1**.**

Let $1\leq q<\infty$. We say that the $(q,p)$-Poincaré inequality holds within $B(x_{0},r_{0})$ if there are constants $C>0$ and $\lambda\geq 1$ (depending on $x_{0}$ and $r_{0}$) such that for all balls $B\subset B(x_{0},r_{0})$, all integrable functions $u$ on $\lambda B$, and all upper gradients $g$ of $u$,

[TABLE]

We also say that $X$ (or $\mu$) supports a local $(q,p)$-Poincaré inequality (on $X$) if for every $x_{0}\in X$ there is $r_{0}$ (depending on $x_{0}$) such that the $(q,p)$-Poincaré inequality holds within $B(x_{0},r_{0})$.

If the $(q,p)$-Poincaré inequality holds within every ball $B(x_{0},r_{0})$ then $X$ supports a semilocal $(q,p)$-Poincaré inequality, and if moreover $C$ and $\lambda$ are independent of $x_{0}$ and $r_{0}$, then $X$ supports a global $(q,p)$-Poincaré inequality.

If $q=1$ we usually just write $p\mspace{1.0mu}$-Poincaré inequality.

The inequality (4.1) can equivalently be required for all integrable functions $u$ on $\lambda B$, and all $p\mspace{1.0mu}$-weak upper gradients $g$ of $u$; see [6, Proposition 4.13] for other equivalent formulations.

As in the case of the doubling condition, local Poincaré inequalities are inherited by open subsets, i.e. if $\Omega\subset X$ is open and $X$ supports a local $(q,p)$-Poincaré inequality, then so does $\Omega$. This hereditary property fails for semilocal and global Poincaré inequalities.

Remark 4.2**.**

When defining (semi)local doubling in Definition 3.1 it is primarily a matter of taste (except for the constant $\tfrac{2}{3}$ in Proposition 3.4) whether the condition is required for $B\subset B(x_{0},r_{0})$ or for $B\subset 2B\subset B(x_{0},r_{0})$, and the same is true for $B$ and $\lambda B$ in the local Poincaré inequalities in Definition 4.1. However, for semilocal Poincaré inequalities it is vital that the condition is for all $B\subset B_{0}$, rather than for all $B\subset\lambda B\subset B_{0}$, which is a weaker requirement since $\lambda$ is allowed to depend on $B_{0}$. (Consider e.g. $X=(-\infty,0]\cup[1,\infty)$ with the Euclidean metric and Lebesgue measure. Then for $x_{0}=0$, $r_{0}\geq 2$ and $\lambda:=r_{0}$, the requirement $\lambda B\subset B(x_{0},r_{0})$ implies that $r_{B}\leq 1$ and hence (4.1) holds for all such balls, while it clearly fails for $B(0,2)\subset B(x_{0},r_{0})$.)

Example 4.3**.**

If a Poincaré inequality holds within $B_{0}=B(x_{0},r_{0})$, then it does not mean that $B_{0}$ (or {\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle B}\kern 0.0pt}\hss}{B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle B}\kern 0.0pt}\hss}{B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle B}\kern 0.0pt}\hss}{B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle B}\kern 0.0pt}\hss}{B}}}_{0}) itself (as a metric space) supports a global Poincaré inequality. Similarly, if $\mu$ is doubling within $B_{0}$, then it does not follow that $\mu|_{B_{0}}$ (or \mu|_{{\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle B}\kern 0.0pt}\hss}{B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle B}\kern 0.0pt}\hss}{B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle B}\kern 0.0pt}\hss}{B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle B}\kern 0.0pt}\hss}{B}}}_{0}}) is doubling. To see this, let

[TABLE]

i.e. $X$ is $\mathbf{R}^{2}$ with the open unit upper half-disc removed and the curved cusp of exponential type at $(0,1)$,

[TABLE]

inserted back into the hole. Then $\operatorname{int}X$ is a uniform domain and hence the Lebesgue measure, restricted to it, is globally doubling and supports a global $1$-Poincaré inequality, by Theorem 4.4 in Björn–Shanmugalingam [16] (or [6, Theorem A.21]). By Aikawa–Shanmugalingam [2, Proposition 7.1], the same is true for $X$ itself.

Now, if $x_{0}$ is the origin and we let $B_{0}=B(x_{0},1)$, then $B_{0}\cap X$ and {\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle B_{0}\cap X}\kern 0.0pt}\hss}{B_{0}\cap X}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle B_{0}\cap X}\kern 0.0pt}\hss}{B_{0}\cap X}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle B_{0}\cap X}\kern 0.0pt}\hss}{B_{0}\cap X}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle B_{0}\cap X}\kern 0.0pt}\hss}{B_{0}\cap X}}} have the cusps $C_{0}$ and {\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle C}\kern 0.0pt}\hss}{C}}}_{0} in the vicinity of the point $(0,1)$. Hence the Lebesgue measure restricted to these sets is not doubling near or at this point.

Moreover, $C_{0}$ is disconnected at $(0,1)$ and {\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle C}\kern 0.0pt}\hss}{C}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle C}\kern 0.0pt}\hss}{C}}}_{0} is essentially disconnected at the point $(0,1)$ (which has zero capacity with respect to {\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle B_{0}\cap X}\kern 0.0pt}\hss}{B_{0}\cap X}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle B_{0}\cap X}\kern 0.0pt}\hss}{B_{0}\cap X}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle B_{0}\cap X}\kern 0.0pt}\hss}{B_{0}\cap X}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle B_{0}\cap X}\kern 0.0pt}\hss}{B_{0}\cap X}}} for all $p\geq 1$), so neither $B_{0}\cap X$ nor {\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle B_{0}\cap X}\kern 0.0pt}\hss}{B_{0}\cap X}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle B_{0}\cap X}\kern 0.0pt}\hss}{B_{0}\cap X}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle B_{0}\cap X}\kern 0.0pt}\hss}{B_{0}\cap X}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle B_{0}\cap X}\kern 0.0pt}\hss}{B_{0}\cap X}}} supports any global or semilocal Poincaré inequalities. For {\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle B_{0}\cap X}\kern 0.0pt}\hss}{B_{0}\cap X}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle B_{0}\cap X}\kern 0.0pt}\hss}{B_{0}\cap X}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle B_{0}\cap X}\kern 0.0pt}\hss}{B_{0}\cap X}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle B_{0}\cap X}\kern 0.0pt}\hss}{B_{0}\cap X}}} even the local doubling and all local Poincaré inequalities fail at $(0,1)$.

Propositions 1.2 and 3.8 about (semi)local doubling are rather straightforward. A bit more surprising, perhaps, is that Theorem 1.3 is true for Poincaré inequalities. We will obtain the following more general version of Theorem 1.3.

Theorem 4.4**.**

If $X$ is proper and connected and $\mu$ is locally doubling and supports a local $(q,p)$-Poincaré inequality, then it supports a semilocal $(q,p)$-Poincaré inequality.

The proof of Theorem 4.4 will be split into a number of lemmas, some of which may be of independent interest. It will be concluded at the end of this section. But first, we discuss the assumptions in Theorem 4.4 as well as some consequences of local Poincaré inequalities. To start with, it is easily verified that if $X$ supports a semilocal $p\mspace{1.0mu}$-Poincaré inequality then it is connected, cf. the proof of [6, Proposition 4.2]. The following example shows that this conclusion fails if we replace the semilocal assumption with a local one, even if $X$ is proper.

Example 4.5**.**

Let $X$ be the union of two disjoint closed balls in $\mathbf{R}^{n}$, which is proper and such that the Lebesgue measure is globally doubling and supports a local $1$-Poincaré inequality on $X$. However $X$ is not connected.

The above example also shows that the connectedness assumption in Theorem 4.4 cannot be dropped, while the following example shows that neither can the properness assumption.

Example 4.6**.**

Let

[TABLE]

Then $X$ is connected and the Lebesgue measure is globally doubling on $X$ and supports a local $1$-Poincaré inequality. However, $X$ is neither proper nor supports any semilocal Poincaré inequality.

The following is a partial result on the way to proving Theorem 4.4.

Lemma 4.7**.**

If $X$ is proper and supports a local $(q,p)$-Poincaré inequality, then for every ball $B_{0}$ there exist $r^{\prime},C>0$ and $\lambda\geq 1$ such that the $(q,p)$-Poincaré inequality (4.1) holds for all balls $B=B(x,r)\subset B_{0}$ with $r\leq r^{\prime}$.

• Proof.

As in the proof of Proposition 3.8, the compact set {\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle B}\kern 0.0pt}\hss}{B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle B}\kern 0.0pt}\hss}{B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle B}\kern 0.0pt}\hss}{B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle B}\kern 0.0pt}\hss}{B}}}_{0} can be covered by finitely many balls $B_{j}=B(x_{j},r_{j})$ so that the $(q,p)$-Poincaré inequality holds within each $2B_{j}$, with constants $C_{j}$ and $\lambda_{j}$. Letting

[TABLE]

together with the fact that $B(x,r)\subset 2B_{j}$ for some $j$, concludes the proof. ∎

We shall now see which connectivity properties follow from the local Poincaré inequality.

Proposition 4.8**.**

Assume that $X$ is locally compact and that $\mu$ is locally doubling and supports a local $p\mspace{1.0mu}$-Poincaré inequality. Then $X$ is locally quasiconvex (and thus locally rectifiably pathconnected), i.e. for every $x_{0}\in X$ there are $r_{0},L>0$ (depending on $x_{0}$) such that every pair of points $x,y\in B(x_{0},r_{0})$ can be connected by a curve of length at most $Ld(x,y)$.

If $X$ is moreover proper and connected, then it is semilocally quasiconvex, i.e. the above connectivity property holds in every ball $B(x_{0},r_{0})$ (with $L$ depending on it).

Local quasiconvexity and its behaviour under various transformations of $X$ were considered by Buckley–Herron–Xie [17], cf. Section 2 and Proposition 4.2 therein. Note that if $X$ is connected and locally (rectifiably) pathconnected, then it is (rectifiably) pathconnected, as the (rectifiably) pathconnected components must be open. In particular, local quasiconvexity and connectedness imply that $X$ is rectifiably pathconnected, but not necessarily quasiconvex.

Later on it will be important to have the following more precise version of the above connectivity result, which will also be used to deduce Proposition 4.8.

Lemma 4.9**.**

Let $x,y\in X$ and assume that the $p\mspace{1.0mu}$-Poincaré inequality and the doubling property for $\mu$ hold (with constants $C_{\rm PI}$ and $C_{\mu}$) within $B_{0}=B(x,2d(x,y))$ in the sense of Definitions 3.1 and 4.1. Let $\Lambda=3C_{\mu}^{3}C_{\rm PI}$. If the ball {\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle\Lambda B}\kern 0.0pt}\hss}{\Lambda B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle\Lambda B}\kern 0.0pt}\hss}{\Lambda B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle\Lambda B}\kern 0.0pt}\hss}{\Lambda B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle\Lambda B}\kern 0.0pt}\hss}{\Lambda B}}}_{0} is compact then $x$ and $y$ can be connected by a curve in {\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle\Lambda B}\kern 0.0pt}\hss}{\Lambda B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle\Lambda B}\kern 0.0pt}\hss}{\Lambda B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle\Lambda B}\kern 0.0pt}\hss}{\Lambda B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle\Lambda B}\kern 0.0pt}\hss}{\Lambda B}}}_{0}, of length at most $Ld(x,y)$, where $L=9\Lambda$.

Note that, as in [6, Theorem 4.32], the constants $\Lambda$ and $L$ are independent of the dilation constant $\lambda$ in the $p\mspace{1.0mu}$-Poincaré inequality.

• Proof.

Let $\lambda$ be the dilation constant in the $p\mspace{1.0mu}$-Poincaré inequality within $B_{0}$. Following Semmes’s chaining argument, define for $\varepsilon>0$ and $z\in\lambda B_{0}$,

[TABLE]

where the infimum is taken over all collections $\{x_{i}\}_{i=0}^{m}\subset X$ such that $x_{0}=x$, $x_{m}=z$ and $d(x_{i-1},x_{i})<\varepsilon$ for all $i=1,2,\mathinner{.}\nonscript\!.\nonscript\!\mathinner{.},m$. Should there be no such chain, we let $\rho_{\varepsilon}(z)=10\Lambda d(x,y)$. Then it is easily verified that $\rho_{\varepsilon}$ is locally 1-Lipschitz and has 1 as an upper gradient.

Since the $p\mspace{1.0mu}$-Poincaré inequality and the doubling property for $\mu$ hold for all the balls $B_{0}=B(x,2d(x,y))$ and $B_{j}=B(y,2^{-j}d(x,y))\subset B_{0}$, with $j=1,2,\mathinner{.}\nonscript\!.\nonscript\!\mathinner{.}$, a standard telescoping argument shows that

[TABLE]

A similar estimate with $\rho_{\varepsilon}(x)=0$ then yields that

[TABLE]

In particular, for each $\varepsilon_{n}=3\cdot 2^{-n}\Lambda d(x,y)$, $n=1,2,\mathinner{.}\nonscript\!.\nonscript\!\mathinner{.}$, there is a chain $x=x_{0}^{n},x_{1}^{n},\mathinner{.}\nonscript\!.\nonscript\!\mathinner{.},x_{M_{n}}^{n}=y$ such that $d(x_{i-1},x_{i})\leq\varepsilon_{n}$ for all $i$ and

[TABLE]

Moreover, as $d(x,x_{i}^{n})\leq 2\Lambda d(x,y)=\Lambda r_{B_{0}}$ or $d(y,x_{i}^{n})\leq\Lambda d(x,y)$, we conclude that all $x_{i}^{n}$ belong to the compact set {\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle\Lambda B}\kern 0.0pt}\hss}{\Lambda B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle\Lambda B}\kern 0.0pt}\hss}{\Lambda B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle\Lambda B}\kern 0.0pt}\hss}{\Lambda B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle\Lambda B}\kern 0.0pt}\hss}{\Lambda B}}}_{0}.

Using [6, Lemma 4.34], we can find a subchain

[TABLE]

which implies that $m_{n}\leq 3\Lambda d(x,y)/\varepsilon_{n}=2^{n}$. Letting $S_{n}=\{2^{-n}i:i=0,1,\mathinner{.}\nonscript\!.\nonscript\!\mathinner{.},2^{n}\}$, the function \gamma_{n}:S_{n}\to{\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle\Lambda B}\kern 0.0pt}\hss}{\Lambda B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle\Lambda B}\kern 0.0pt}\hss}{\Lambda B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle\Lambda B}\kern 0.0pt}\hss}{\Lambda B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle\Lambda B}\kern 0.0pt}\hss}{\Lambda B}}}_{0}, defined by $\gamma_{n}(2^{-n}i)={\hat{x}}^{n}_{\min\{i,m_{n}\}}$, is easily shown to be $9\Lambda d(x,y)$-Lipschitz.

Using a diagonal argument, we can from the sequence $\{\gamma_{n}\}_{n=1}^{\infty}$ choose a subsequence, which converges on $S=\bigcup_{n=1}^{\infty}S_{n}$. Since each $\gamma_{n}$ is $9\Lambda d(x,y)$-Lipschitz, so is the limiting function \gamma:S\to{\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle\Lambda B}\kern 0.0pt}\hss}{\Lambda B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle\Lambda B}\kern 0.0pt}\hss}{\Lambda B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle\Lambda B}\kern 0.0pt}\hss}{\Lambda B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle\Lambda B}\kern 0.0pt}\hss}{\Lambda B}}}_{0}. Finally, $\gamma$ extends to a $9\Lambda d(x,y)$-Lipschitz function on $[0,1]$, which after reparameterization provides us with the desired curve. ∎

• Proof of Proposition 4.8.

Let $x_{0}\in X$ and find a ball $B_{0}^{\prime}$ centred at $x_{0}$ so that {{\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle B_{0}^{\prime}}\kern 0.0pt}\hss}{B_{0}^{\prime}}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle B_{0}^{\prime}}\kern 0.0pt}\hss}{B_{0}^{\prime}}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle B_{0}^{\prime}}\kern 0.0pt}\hss}{B_{0}^{\prime}}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle B_{0}^{\prime}}\kern 0.0pt}\hss}{B_{0}^{\prime}}}}} is compact and the $p\mspace{1.0mu}$-Poincaré inequality and the doubling property for $\mu$ hold within $B_{0}^{\prime}$, with constants $C_{\rm PI}$ and $C_{\mu}$. Let $\Lambda=3C_{\mu}^{3}C_{\rm PI}$ and $B_{0}=\Lambda^{-1}B_{0}^{\prime}$. Now, if $x,y\in\tfrac{1}{5}B_{0}$ then $B(x,2d(x,y))\subset B_{0}$ and {\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle\Lambda B}\kern 0.0pt}\hss}{\Lambda B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle\Lambda B}\kern 0.0pt}\hss}{\Lambda B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle\Lambda B}\kern 0.0pt}\hss}{\Lambda B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle\Lambda B}\kern 0.0pt}\hss}{\Lambda B}}}_{0} is compact. Hence Lemma 4.9 shows the existence of a connecting curve of length at most $9\Lambda d(x,y)$. Thus $X$ is locally quasiconvex, which proves the first part of the proposition.

Now assume that $X$ is in addition proper and connected. Let $B_{0}=B(x_{0},r_{0})$ be arbitrary. By Proposition 3.8, $\mu$ is semilocally doubling. Furthermore, Lemma 4.7 implies that for some $0<r^{\prime}\leq r_{0}$, the $p\mspace{1.0mu}$-Poincaré inequality (1.1) holds for all $B=B(x,r)\subset 5B_{0}$ with $0<r\leq 2r^{\prime}$.

Since $\overline{5\Lambda B}_{0}$ is compact, by the properness of $X$, Lemma 4.9 yields that every pair of points x,y\in{\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle B}\kern 0.0pt}\hss}{B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle B}\kern 0.0pt}\hss}{B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle B}\kern 0.0pt}\hss}{B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle B}\kern 0.0pt}\hss}{B}}}_{0} with $d(x,y)\leq r^{\prime}$, can be connected by a curve of length at most $L^{\prime}d(x,y)$, where $L^{\prime}$ depends only on the doubling and Poincaré constants provided for $5B_{0}$ by Proposition 1.2 and Lemma 4.7.

By compactness, {\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle B}\kern 0.0pt}\hss}{B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle B}\kern 0.0pt}\hss}{B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle B}\kern 0.0pt}\hss}{B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle B}\kern 0.0pt}\hss}{B}}}_{0} can be covered by finitely many balls $B(x_{k},r^{\prime})$ with x_{k}\in{\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle B}\kern 0.0pt}\hss}{B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle B}\kern 0.0pt}\hss}{B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle B}\kern 0.0pt}\hss}{B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle B}\kern 0.0pt}\hss}{B}}}_{0}, $k=1,2,\mathinner{.}\nonscript\!.\nonscript\!\mathinner{.},N$. As $X$ is connected and locally quasiconvex, it is rectifiably connected. Hence, there is for each pair $x_{j},x_{k}$, $j\neq k$, a rectifiable curve $\gamma_{j,k}\subset X$ connecting $x_{j}$ to $x_{k}$. Since there are only finitely many such pairs, it follows that $M:=\sup_{j,k}l_{\gamma_{j,k}}<\infty$.

Finally, let $x,y\in B_{0}$ be arbitrary. If $d(x,y)\leq r^{\prime}$, then we already know that $x$ and $y$ can be connected by a curve of length at most $L^{\prime}d(x,y)$. If $d(x,y)>r^{\prime}$, then $x$ and $y$ can be connected to some $x_{j}$ and $x_{k}$, respectively, by curves of lengths at most $L^{\prime}r^{\prime}$. Adding $\gamma_{j,k}$ to these curves produces a curve from $x$ to $y$ of length at most $2L^{\prime}r^{\prime}+M<(2L^{\prime}+M/r^{\prime})d(x,y)$. ∎

The local quasiconvexity proved in Proposition 4.8 can be further bootstrapped by the following result.

Lemma 4.10**.**

Let $B_{0}$ be a ball such that $\mu$ is locally doubling and supports a local $p\mspace{1.0mu}$-Poincaré inequality on $B_{0}$ (as the underlying space). Assume that the $p\mspace{1.0mu}$-Poincaré inequality (1.1) holds for $B_{0}$ (in place of $B$) with dilation constant $\lambda\geq 1$ and that $\lambda B_{0}$ is locally compact.

Then $B_{0}$ is rectifiably connected within $\lambda B_{0}$, i.e. any two points $x,y\in B_{0}$ can be connected by a rectifiable curve lying within $\lambda B_{0}$.

• Proof.

Divide $\lambda B_{0}$ into its rectifiable components, i.e. $x,y\in\lambda B_{0}$ belong to the same rectifiable component if there is a rectifiable curve within $\lambda B_{0}$ from $x$ to $y$. For each $x\in\lambda B_{0}$, let $G_{x}$ denote the rectifiable component containing $x$, which is measurable by Järvenpää–Järvenpää–Rogovin–Rogovin–Shanmugalingam [30, Corollary 1.9 and Remark 3.1], since $\lambda B_{0}$ is locally compact. The local assumptions on $B_{0}$, together with Proposition 4.8, imply that the sets $G_{x}\cap B_{0}$ are open for all $x$.

Let $G$ be such a rectifiable component intersecting $B_{0}$. We shall show that $B_{0}\subset G$. Assume on the contrary that $B_{0}\setminus G\neq\varnothing$. Since all $G_{x}\cap B_{0}$ are open, so is $B_{0}\setminus G=\bigcup_{x\notin G}(G_{x}\cap B_{0})$. Let $u=\chi_{G}$, which has $g\equiv 0$ as an upper gradient in the open set $\lambda B_{0}$ as there are no rectifiable curves in $\lambda B_{0}$ from $G$ to $\lambda B_{0}\setminus G$. Since both $B_{0}\cap G$ and $B_{0}\setminus G$ are nonempty and open, they both have positive measure. Hence, by the $p\mspace{1.0mu}$-Poincaré inequality for $B_{0}$,

[TABLE]

a contradiction. ∎

The following lemma makes it possible to lift the Poincaré inequality from small to large sets.

Lemma 4.11**.**

Let $1\leq q<\infty$ and $A,E\subset X$ be such that $\mu(A\cap E)\geq\theta\mu(E)$ for some $\theta>0$. Also assume that for some $Q\geq 0$ and a measurable function $u$,

[TABLE]

Then

[TABLE]

• Proof.

To start with, we have by the triangle inequality,

[TABLE]

This yields

[TABLE]

Finally, Lemma 4.17 in [6] allows us to replace $u_{A}$ on the left-hand side by $u_{A\cup E}$, at the cost of an extra factor 2 on the right-hand side. ∎

We are now ready to conclude the proof of the semilocal $(q,p)$-Poincaré inequality.

• Proof of Theorem 4.4.

Let $B_{0}=B(x_{0},r_{0})$ be fixed. As $X$ is proper and connected, Proposition 4.8 shows that there is $\sigma\geq 1$ such that every pair of points in {\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle B}\kern 0.0pt}\hss}{B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle B}\kern 0.0pt}\hss}{B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle B}\kern 0.0pt}\hss}{B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle B}\kern 0.0pt}\hss}{B}}}_{0} can be connected by a rectifiable curve within $\sigma B_{0}$. By Lemma 4.7, there exist $r^{\prime},C>0$ and $\lambda\geq 1$ such that the $(q,p)$-Poincaré inequality (4.1) holds for every ball $B$ with radius $r_{B}\leq r^{\prime}$ and centre in $\overline{\sigma B}_{0}$. By decreasing $r^{\prime}$, if necessary, we may assume that $\lambda r^{\prime}\leq r_{0}$.

Next, let $B\subset B_{0}$ be an arbitrary ball. If $r_{B}\leq r^{\prime}$, then the $(q,p)$-Poincaré inequality (4.1) holds for $B$. Assume therefore that $r_{B}>r^{\prime}$. By compactness, {\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle B}\kern 0.0pt}\hss}{B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle B}\kern 0.0pt}\hss}{B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle B}\kern 0.0pt}\hss}{B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle B}\kern 0.0pt}\hss}{B}}}_{0} can be covered by finitely many balls $B^{\prime}_{j}$, $j=1,2,\mathinner{.}\nonscript\!.\nonscript\!\mathinner{.},M$, with radius $r^{\prime}$ and centres x_{j}\in{\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle B}\kern 0.0pt}\hss}{B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle B}\kern 0.0pt}\hss}{B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle B}\kern 0.0pt}\hss}{B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle B}\kern 0.0pt}\hss}{B}}}_{0}. Proposition 4.8 provides us for each $j$ with a rectifiable curve $\gamma_{j}$ in $\sigma B_{0}$ connecting $x_{j}$ to $x_{j+1}$. Following this curve, we define balls

[TABLE]

Note that $\tfrac{1}{2}B_{j,k+1}\subset B_{j,k}$. Add all these balls to the chain $\{B^{\prime}_{j}\}_{j=1}^{M}$, in between $B^{\prime}_{j}$ and $B^{\prime}_{j+1}$, and renumber the sequence as $B_{1},B_{2},\mathinner{.}\nonscript\!.\nonscript\!\mathinner{.},B_{N}$. Note that all the balls $B_{j}$ have the same radius $r^{\prime}$ and that $\tfrac{1}{2}B_{j+1}\subset B_{j}\cap B_{j+1}$.

Next, let $A_{j}=\bigcup_{i=1}^{j}B_{i}$. Note that $B_{0}\subset A_{N}$. Then $A_{j}\cap B_{j+1}\supset\tfrac{1}{2}B_{j+1}$ and the semilocal doubling property of $\mu$, provided by Proposition 1.2, implies that for some $\theta>0$ (depending on $2\sigma B_{0}$),

[TABLE]

Since all the balls $B_{j}$ have radius $r^{\prime}$, the $(q,p)$-Poincaré inequality (4.1) holds for them, and we have

[TABLE]

where we have used the semilocal doubling property, together with the fact that $\lambda B_{j}\subset 2\sigma B_{0}$ and that $r^{\prime}$ and $r_{0}$ are fixed. Lemma 4.11 with $A=A_{1}$ and $E=B_{2}$ now yields

[TABLE]

Another application of Lemma 4.11 with $A=A_{2}$ and $E=B_{3}$ then gives

[TABLE]

Continuing in this way along the whole sequence $\{B_{j}\}_{j=1}^{N}$, we can conclude after finitely many iterations that

[TABLE]

Let $\lambda^{\prime}=3\sigma r_{0}/r^{\prime}$. Since $r_{B}\geq r^{\prime}$, the measures of the balls $B\subset 2\sigma B_{0}\subset\lambda^{\prime}B$ are all comparable, due to the semilocal doubling property of $\mu$, and thus

[TABLE]

Finally, [6, Lemma 4.17] allows us to replace $u_{A_{N}}$ by $u_{B}$ on the left-hand side (at the cost of an extra factor 2 on the right-hand side), which completes the proof. ∎

We will also need the following lemma, which shows the reverse doubling condition under suitable local assumptions.

Lemma 4.12**.**

Assume that the doubling property for $\mu$ holds within $B_{0}$ in the sense of Definition 3.1, with doubling constant $C_{\mu}$, and that the $p\mspace{1.0mu}$-Poincaré inequality (1.1) holds for $B_{0}$ (in place of $B$). Let $B\subset 2B\subset B_{0}$ be a ball with $r_{B}<\frac{2}{3}\operatorname{diam}B_{0}$. Then there is $\theta<1$, only depending on $C_{\mu}$, such that

[TABLE]

• Proof.

Assume that $B=B(x,r)$. If there were no $y$ such that $d(x,y)=\frac{3}{4}r$, then zero would be an upper gradient of $\chi_{\frac{3}{4}B}$, which would violate the $p\mspace{1.0mu}$-Poincaré inequality for the ball $B_{0}$, as in (4.3), since $\operatorname{diam}\overline{\tfrac{3}{4}B}\leq\tfrac{3}{2}r<\operatorname{diam}B_{0}$ and thus $B_{0}\setminus\overline{\tfrac{3}{4}B}\neq\varnothing$. Hence there is $y$ such that $d(x,y)=\frac{3}{4}r$.

As $B(y,r)\subset 2B\subset B_{0}$, we get from the doubling property within $B_{0}$ that

[TABLE]

and thus

[TABLE]

5 Better Poincaré inequalities

There are two types of better Poincaré inequalities. The first type of result is the Sobolev–Poincaré inequality, due to Hajłasz–Koskela [22], [23], which strengthens the left-hand side of the inequality. The arguments in [22] and [23] also show how, in sufficiently nice spaces such as $\mathbf{R}^{n}$, the dilation constant $\lambda>1$ in the Poincaré inequality can be improved to $\lambda=1$. These results were originally proved under global doubling and Poincaré assumptions, but since all the considerations in the proof are of local nature, they can also be obtained under (semi)local assumptions in the following form.

Theorem 5.1**.**

(Local Sobolev–Poincaré inequality)* Let $B_{0}$ be a ball such that the $p\mspace{1.0mu}$-Poincaré inequality (with dilation constant $\lambda$) and the doubling property for $\mu$ hold within $B_{0}$ in the sense of Definitions 3.1 and 4.1. Assume that the dimension condition*

[TABLE]

holds for some $C_{0},s>0$ and all balls $X\neq B^{\prime}\subset B\subset B_{0}$.

Then there exists $C$, depending only on $p$, the doubling constant and both constants in the $p\mspace{1.0mu}$-Poincaré inequality within $B_{0}$, such that for all balls $B$ with $5\lambda B\subset B_{0}$, all integrable functions $u$ on $2\lambda B$, and all $p\mspace{1.0mu}$-weak upper gradients $g$ of $u$,

[TABLE]

where $q=p^{*}:=sp/(s-p)>p$ if $p<s$ while $q<\infty$ is arbitrary when $p\geq s$ (in which case $C$ depends also on $q$).

If $L$ is a local quasiconvexity constant for $B_{0}$, in the sense that every pair of points $x,y\in B_{0}$ can be connected (in $X$) by a curve of length at most $Ld(x,y)$, then (5.2) holds for all balls $B$ with $\tfrac{5}{2}LB\subset B_{0}$, and the dilation constant $2\lambda$ in (5.2) can be replaced by $L$.

*In particular, if $\mu$ is *(semi)*locally doubling and supports a *(semi)*local $p\mspace{1.0mu}$-Poincaré inequality then $X$ supports a *(semi)local $(p,p)$-Poincaré inequality.

• Proof.

It suffices to consider the case $s>p$ since when $s\leq p$, one can instead assume the dimension condition (5.1) with $s$ replaced by $p+\varepsilon$ for arbitrarily small $\varepsilon>0$.

We will follow the arguments in the proof of [6, Theorem 4.39] and prove (5.2) under the assumption of local $L$-quasiconvexity, i.e. with $LB$ in the right-hand side. To obtain (5.2) with $2\lambda B$ in the right-hand side, replace the balls in the chain below by the balls $B^{0,0}:=B(x,2r)$, $B^{i,0}:=B(x^{\prime},2^{-(i+1)}r)$, $i=1,2,\mathinner{.}\nonscript\!.\nonscript\!\mathinner{.}$, and $\widehat{B}^{i}:=\lambda B^{i,0}$. Note that $\frac{5}{2}\widehat{B}^{i}\subset B_{0}$, $i=0,1,\mathinner{.}\nonscript\!.\nonscript\!\mathinner{.}$.

Let B\bigl{(}x,\tfrac{5}{2}Lr\bigr{)}\subset B_{0} be arbitrary. We can assume that $Lr\leq\operatorname{diam}B_{0}$. Let $\rho_{0}=Lr/2\lambda$ and $\rho_{i}=2^{-i}\rho_{0}$, $i=1,2,\mathinner{.}\nonscript\!.\nonscript\!\mathinner{.}$ . For $x^{\prime}\in B(x,r)$, consider an $L$-quasiconvex curve $\gamma$ from $x$ to $x^{\prime}$, i.e. $\gamma(0)=x$ and $\gamma(l_{\gamma})=x^{\prime}$, where $l_{\gamma}\leq Ld(x,x^{\prime})$ is the length of $\gamma$. Find the smallest integer $i^{\prime}\geq 0$ such that $2\lambda\rho_{i^{\prime}}\leq L(r-d(x,x^{\prime}))$. For each $i=0,1,\mathinner{.}\nonscript\!.\nonscript\!\mathinner{.},i^{\prime}-1$, consider all the integers $j\geq 0$, such that

[TABLE]

and let $x_{i,j}=\gamma(t_{i,j})$. There are at most $\lambda$ such $x_{i,j}$’s for each $i$. Similarly, for $i=i^{\prime}$ there are at most $2\lambda$ integers $j\geq 0$ and points $x_{i,j}=\gamma(t_{i,j})$ such that $t_{i,j}<l_{\gamma}$. For $i>i^{\prime}$, let $j=0$ and $x_{i,j}=x^{\prime}$.

It is now easily verified that $d(x,x_{i,j})+\lambda\rho_{i}<Lr$ and hence

[TABLE]

Ordering the balls $B^{i,j}$ lexicographically, we obtain a chain of balls from $x$ to $x^{\prime}$, with substantial overlaps. Assuming that $x^{\prime}$ is a Lebesgue point of $u$, a standard telescoping argument using the $p\mspace{1.0mu}$-Poincaré inequality for each $B^{i,j}\subset B_{0}$, as in (Proof.), then yields the estimate

[TABLE]

where the sum is taken over all balls $B$ in the chain. Note that, because of the doubling property within $B_{0}$, the balls $\lambda B$ and $B(x^{\prime},\lambda r_{B})$ have comparable measures. The dimension condition (5.1) therefore yields

[TABLE]

where the summations in $\Sigma^{\prime}$ and $\Sigma^{\prime\prime}$ are over $B$ with $r_{B}>\rho_{i_{0}}$ and $r_{B}\leq\rho_{i_{0}}$, respectively (and $i_{0}\geq 0$ will be chosen later). Since $\lambda\rho_{i}\leq\lambda\rho_{0}=\frac{1}{2}Lr\leq\frac{1}{2}\operatorname{diam}B_{0}$, Lemma 4.12 implies that there exists $\theta\in(0,1)$, independent of $x^{\prime}$ and $i$, such that

[TABLE]

and hence, as $1/s-1/p<0$,

[TABLE]

Similarly,

[TABLE]

and hence, as $1/s>0$,

[TABLE]

where $M(x^{\prime}):=M^{*}_{B(x,Lr),B_{0}}g_{u}^{p}(x^{\prime})$ is the noncentred local maximal function given by (3.2). Here we have also used that both $x^{\prime}$ and $\lambda B^{i,j}$ are contained in $\widehat{B}^{i}:=B(x_{i,0},2\lambda\rho_{i})\subset B(x,Lr)$, and \tfrac{5}{2}\widehat{B}^{i}\subset B\bigl{(}x,\tfrac{5}{2}Lr\bigr{)}\subset B_{0}, $i\geq 0$. Choosing $i_{0}\geq 0$ so that

[TABLE]

we can conclude that

[TABLE]

which gives a lower bound for $M(x^{\prime})$. Proposition 3.9 then yields the level set estimate

[TABLE]

which in turn implies (5.2) with $B(x,Lr)$ in the right-hand side, by [6, Lemma 4.25].

To conclude the statement of the theorem under local assumptions, let $x_{0}\in X$ be arbitrary and find $r_{0}>0$ so that the local assumptions hold within $B(x_{0},r_{0})$. Then choose a radius $0<r_{0}^{\prime}\leq(11\lambda)^{-1}r_{0}$ so that $B_{0}^{\prime}:=B(x_{0},r_{0}^{\prime})\neq X$ and $\operatorname{dist}(x_{0},X\setminus B_{0}^{\prime})=r_{0}^{\prime}$. For $B=B(x,r)\subset B_{0}^{\prime}$ it then follows that $r_{B}\leq 2r_{0}^{\prime}$ and hence $5\lambda B\subset B(x_{0},r_{0})$. The already proved first part of the theorem then implies that (5.2) holds for $B$.

Under semilocal assumptions, let $B_{0}^{\prime}:=B(x_{0},r_{0}^{\prime})\neq X$ be arbitrary and such that $\operatorname{dist}(x_{0},X\setminus B_{0}^{\prime})=r_{0}^{\prime}$. (If $B_{0}^{\prime}=X$, the proof is similar.) Then $r_{B}\leq 2r_{0}^{\prime}$ whenever $B=B(x,r)\subset B_{0}^{\prime}$, and hence $2B\subset 5B_{0}^{\prime}$. Note that above, when proving (5.2) with $2\lambda B$ on the right-hand side, the $p\mspace{1.0mu}$-Poincaré inequality is only used to balls within $2B$ (to obtain (5.3)), while (5.1) and the doubling property are used for balls within $5\lambda B$, where $\lambda$ is the dilation constant in the $p\mspace{1.0mu}$-Poincaré inequality within $2B$. Thus, to obtain (5.2) for $B\subset B_{0}^{\prime}$ we need to apply the $p\mspace{1.0mu}$-Poincaré inequality with $\lambda$ and $C_{\rm PI}$ determined by $5B_{0}^{\prime}$, followed by (5.1) and the doubling property with constants determined by $11\lambda B_{0}^{\prime}$. This can be done because of the semilocal assumptions, since the doubling property for $\mu$ within $11\lambda B_{0}^{\prime}$ implies (5.1) within $11\lambda B_{0}^{\prime}$ for some $s>0$. ∎

Remark 5.2**.**

There is a converse relation between $s$ and $q$ in Theorem 5.1 as well, namely if the $(q,p)$-Poincaré inequality (5.2) holds for all balls $B$ with $5\lambda B\subset B_{0}$, and $\mu$ is doubling within $B_{0}$, then (5.1) holds with $s=qp/(q-p)$ for all balls $X\neq B^{\prime}\subset B$ with $15\lambda B\subset B_{0}$; this follows from the proof of [6, Proposition 4.20]. Note that the formulas $s(q)$ and $q(s)$ are inverse functions of each other, if $p<s$. In particular, if we let

[TABLE]

then

[TABLE]

Note that there need not exist an optimal $s$ (for a given $B_{0}$ in Theorem 5.1), i.e. the set of values of $s$ for which (5.1) holds may be an open interval, cf. Example 3.1 in Björn–Björn–Lehrbäck [9]. Similarly there need not be an optimal $q$.

The second type of self-improvement for Poincaré inequalities is the open-ended property due to Keith–Zhong [32, Theorem 1.0.1] which strengthens the right-hand side of the inequality, see also Heinonen–Koskela–Shanmugalingam–Tyson [28, Theorem 12.3.9], Eriksson-Bique [20] and Kinnunen–Lehrbäck–Vähäkangas–Zhong [35]. A careful analysis of the proof (in [32] or [28]) shows that all the balls considered therein lie within a constant dilation of the ball in the Poincaré inequality under consideration. This makes it possible to prove the following local version.

Theorem 5.3**.**

Assume that $p>1$ and let $B_{0}=B(x_{0},r_{0})$ be a ball such that {\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle B}\kern 0.0pt}\hss}{B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle B}\kern 0.0pt}\hss}{B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle B}\kern 0.0pt}\hss}{B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle B}\kern 0.0pt}\hss}{B}}}_{0} is compact and the $p\mspace{1.0mu}$-Poincaré inequality and the doubling property for $\mu$ hold within $B_{0}$ in the sense of Definitions 3.1 and 4.1.

Then there exist constants $C$, $\lambda$ and $q<p$, depending only on $p$, the doubling constant and both constants in the $p\mspace{1.0mu}$-Poincaré inequality within $B_{0}$, such that for all balls $B$ with $\lambda B\subset B_{0}$, all integrable functions $u$ on $\lambda B$, and all $q$-weak upper gradients $g$ of $u$,

[TABLE]

In the proof below, we will use the inner metric which is defined by

[TABLE]

where the infimum is taken over all curves $\gamma$ connecting $x$ and $y$. If there are no such curves then $d_{\rm in}(x,y)=\infty$. As $X$ may be disconnected, this is not always a metric, but we will nevertheless still use the name “inner metric”. Balls with respect to $d_{\rm in}$, defined in the obvious way, will be denoted by $B_{\rm in}$.

• Proof.

Let $\lambda^{\prime}$ be the dilation constant in the $p\mspace{1.0mu}$-Poincaré inequality within $B_{0}$. Lemma 4.9 shows that there exist $\Lambda\geq 1$ and $L=9\Lambda$ such that

[TABLE]

whenever $B(x,2\Lambda d(x,y))\subset B_{0}$. It follows that if $2\Lambda B(x,r)\subset B_{0}$, then

[TABLE]

We will now explain how the arguments in the proof of [6, Theorem 4.39] can be used to show that for every inner ball $B_{\rm in}=B_{\rm in}(x,r)$ such that $B(x,2r)\subset B_{0}$, the following inner $p\mspace{1.0mu}$-Poincaré inequality with dilation constant $1$ holds:

[TABLE]

(Since we only assume a local $p\mspace{1.0mu}$-Poincaré inequality with respect to ordinary balls, Theorem 4.39 in [6] cannot be applied directly and care has to be taken when comparing ordinary and inner balls.)

More precisely, let $\rho_{0}=r/2\lambda^{\prime}L$ and $\rho_{i}=2^{-i}\rho_{0}$, $i=1,2,\mathinner{.}\nonscript\!.\nonscript\!\mathinner{.}$ . For $x^{\prime}\in B_{\rm in}(x,r)$, consider a $d_{\rm in}$-geodesic $\gamma$ from $x$ to $x^{\prime}$, i.e. $\gamma(0)=x$ and $\gamma(d^{\prime})=x^{\prime}$, where $d^{\prime}=d_{\rm in}(x,x^{\prime})<r$ is the length of $\gamma$. Such a geodesic exists by Ascoli’s theorem and the compactness of {\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle B}\kern 0.0pt}\hss}{B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle B}\kern 0.0pt}\hss}{B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle B}\kern 0.0pt}\hss}{B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle B}\kern 0.0pt}\hss}{B}}}_{0}. Find the smallest integer $i^{\prime}\geq 0$ such that $2\lambda^{\prime}\rho_{i^{\prime}}\leq r-d^{\prime}$. For each $i=0,1,\mathinner{.}\nonscript\!.\nonscript\!\mathinner{.},i^{\prime}-1$, consider all the integers $j\geq 0$, such that

[TABLE]

and let $x_{i,j}=\gamma(t_{i,j})$. There are at most $\lambda^{\prime}L$ such $x_{i,j}$’s for each $i$. Similarly, for $i=i^{\prime}$, there are at most $2\lambda^{\prime}L$ integers $j\geq 0$ and points $x_{i,j}=\gamma(t_{i,j})$ such that $t_{i,j}<d^{\prime}$. For $i>i^{\prime}$, let $j=0$ and $x_{i,j}=x^{\prime}$.

It is now easily verified that $d_{\rm in}(x,x_{i,j})+\lambda^{\prime}L\rho_{i}<r$ and hence, with $B^{i,j}=B(x_{i,j},\rho_{i})$,

[TABLE]

so (5.5) implies that

[TABLE]

For later reference, let $y_{i}=\gamma((1-2^{-(i+1)})d^{\prime})$ when $i\leq i^{\prime}$ and $y_{i}=x^{\prime}$ otherwise. Then each ball $\widehat{B}^{i,j}:=B(y_{i},(L+1)\lambda^{\prime}\rho_{i})$ contains both $\lambda^{\prime}B^{i,j}$ and $x^{\prime}$. Moreover,

[TABLE]

so $\tfrac{5}{2}\widehat{B}^{i,j}\subset B(x,2r)\subset B_{0}$.

Ordering the balls $B^{i,j}$ lexicographically, we obtain a chain of balls from $x$ to $x^{\prime}$, with substantial overlaps. Assuming that $x^{\prime}$ is a Lebesgue point of $u$, a standard telescoping argument using the $p\mspace{1.0mu}$-Poincaré inequality for each $B^{i,j}\subset B_{0}$, as in (Proof.), then yields the estimate

[TABLE]

where the sum is taken over all balls $B$ in the above chain. We can now estimate the measure of the set

[TABLE]

as follows. Writing $t=C_{2}t\sum_{B}r_{B}/r$ and comparing with (5.7), we can for every Lebesgue point $x^{\prime}\in E_{t}$ find some ball $B_{x^{\prime}}:=B^{i,j}$ from the corresponding chain so that

[TABLE]

By the above, the corresponding ball $\widehat{B}_{x^{\prime}}:=\widehat{B}^{i,j}$ contains both $\lambda^{\prime}B_{x^{\prime}}$ and $x^{\prime}$. Hence, using the 5-covering lemma we can from the collection $\widehat{B}_{x^{\prime}}$, where $x^{\prime}\in E_{t}$ are Lebesgue points of $u$, extract a countable pairwise disjoint subcollection $\widehat{B}_{x^{\prime}_{k}}$, $k=1,2,\mathinner{.}\nonscript\!.\nonscript\!\mathinner{.}$, such that

[TABLE]

where in the last inequality we have used that $\tfrac{5}{2}\widehat{B}_{x^{\prime}_{k}}\subset B_{0}$, so that the doubling condition can be applied. Note also that the measures of $\lambda^{\prime}B_{x^{\prime}_{k}}$ and $\widehat{B}_{x^{\prime}_{k}}$ are comparable. Estimating the balls in the last sum using (5.8), together with the fact that the balls $\lambda^{\prime}B_{x^{\prime}_{k}}\subset\widehat{B}_{x^{\prime}_{k}}\cap B_{\rm in}(x,r)$ are disjoint, now yields the level set estimate

[TABLE]

which in turn implies (5.6), by [6, Lemma 4.25].

Next, still with respect to $d_{\rm in}$ and within $B_{0}$, the proof in [28, Theorem 12.3.9] (or Keith–Zhong [32]), which is written for geodesic spaces, can be applied to show that there exists $q<p$ such that for every inner ball $B_{\rm in}=B_{\rm in}(x,r)$ with $1280B(x,r)\subset B_{0}$,

[TABLE]

(Here it is also used that if $B_{\rm in}(x^{\prime},r^{\prime})\subset 256B_{\rm in}$ then $B_{\rm in}(x^{\prime},r^{\prime})=B_{\rm in}(x^{\prime},r^{\prime\prime})$ for some $r^{\prime\prime}\leq 512r$, and hence

[TABLE]

so (5.6) holds for every such inner ball $B_{\rm in}(x^{\prime},r^{\prime})\subset 256B_{\rm in}$ and can be used in the arguments leading to [28, Theorem 12.3.9].) Now, by (5.5),

[TABLE]

all with comparable measures (depending on $L$). Hence, (5.9) yields (5.4) with $\lambda=1280L$ and $B$ replaced by $B(x,r/L)$. ∎

In Heinonen–Koskela–Shanmugalingam–Tyson [28, Proposition 12.3.10] it is explained how (under global assumptions) the properness of $X$ in Keith–Zhong [32] can be relaxed to local compactness, at the price that the resulting $q$-Poincaré inequality only holds for $u\in N^{1,p}(\lambda B)$, which is however enough for many applications. A counterexample by Koskela [39] shows that one cannot deduce a standard $q$-Poincaré inequality in this case. A similar improvement can be proved under local assumptions and in this case we do conclude a standard local $q$-Poincaré inequality, even though $q$ may vary from ball to ball. Under semiuniformly local assumptions there is even a fixed $q<p$, see Theorem 1.4 whose proof is given in Section 6 below.

Theorem 5.4**.**

If $X$ is locally compact and supports a local $p\mspace{1.0mu}$-Poincaré inequality, $p>1$, and $\mu$ is locally doubling, then for every $x_{0}\in X$ there is a ball $B_{0}^{\prime}\ni x_{0}$, and $q<p$, such that a $q$-Poincaré inequality holds within $B_{0}^{\prime}$ in the sense of Definition 4.1.

If $X$ is in addition proper and connected, then the conclusion is semilocal, i.e. it holds for all balls $B_{0}^{\prime}\subset X$.

Note that for a semilocal conclusion it is not enough to assume that $X$ is locally compact and that the doubling and Poincaré assumptions are semilocal. This is another consequence of the counterexample in Koskela [39].

• Proof.

Let $x_{0}\in X$ be arbitrary and find $r_{0}>0$ so that {\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle B(x_{0},r_{0})}\kern 0.0pt}\hss}{B(x_{0},r_{0})}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle B(x_{0},r_{0})}\kern 0.0pt}\hss}{B(x_{0},r_{0})}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle B(x_{0},r_{0})}\kern 0.0pt}\hss}{B(x_{0},r_{0})}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle B(x_{0},r_{0})}\kern 0.0pt}\hss}{B(x_{0},r_{0})}}} is compact and the local assumptions hold within $B(x_{0},r_{0})$. Let $\lambda$ be given by Theorem 5.3. Then choose a radius $0<r_{0}^{\prime}\leq(3\lambda)^{-1}r_{0}$ so that $B_{0}^{\prime}:=B(x_{0},r_{0}^{\prime})\neq X$ and $\operatorname{dist}(x_{0},X\setminus B_{0}^{\prime})=r_{0}^{\prime}$. For $B\subset B_{0}^{\prime}$ it then follows that $r_{B}\leq 2r_{0}^{\prime}$ and hence $\lambda B\subset B(x_{0},r_{0})$. The first statement then follows from Theorem 5.3.

If $X$ is in addition proper and connected, then let $B_{0}^{\prime}:=B(x_{0},r_{0}^{\prime})$ be arbitrary and assume that $\operatorname{dist}(x_{0},X\setminus B_{0}^{\prime})=r_{0}^{\prime}$ (the proof is similar for $B_{0}^{\prime}=X$). Since $\lambda$ in Theorem 5.3 depends on $B_{0}$, we cannot directly obtain a semilocal conclusion from it.

Instead, let $L$ be provided by Lemma 4.9 with $B_{0}$ replaced by $5B_{0}^{\prime}$. Then for every ball $B(x,r)\subset B_{0}^{\prime}$, we have $r\leq 2r_{0}^{\prime}$ and hence $B(x,1280Lr)\subset 2561LB_{0}^{\prime}=:B_{0}$. Because the $p\mspace{1.0mu}$-Poincaré inequality and the doubling property for $\mu$ hold within $B_{0}$ (by Proposition 1.2 and Theorem 1.3), the proof of Theorem 5.3 (with constants dictated by $B_{0}$) yields (5.9). In particular, as $B(x,1280Lr)\subset B_{0}$, the inner $q$-Poincaré inequality (5.9) holds with $B_{\rm in}$ replaced by $B_{\rm in}(x,Lr)$ and with constants $q<p$ and $C^{\prime\prime}>0$ depending on $B_{0}$.

Now, $B(x,2r)\subset 5B_{0}^{\prime}$ (as $r\leq 2r_{0}^{\prime}$) and hence, since $X$ is proper, Lemma 4.9 implies that

[TABLE]

all with comparable measures (by the semilocal doubling property of $\mu$ provided by Proposition 1.2). We can therefore conclude that (5.4) holds for every $B(x,r)\subset B_{0}^{\prime}$ with $\lambda=256L$. ∎

6 Semiuniformly local assumptions

A possible strengthening of our local assumptions is to also require uniformity in the constants and/or the radii.

Definition 6.1**.**

The measure $\mu$ is semiuniformly locally doubling if there is a (uniform) constant $C$ such that for each $x\in X$ there is $r>0$ so that $\mu(2B)\leq C\mu(B)$ for all balls $B\subset B(x,r)$. If $r$ is independent of $x$, then $\mu$ is uniformly locally doubling.

Note that there is no uniformity of the radii when $\mu$ is semiuniformly locally doubling. (Semi)uniformly local Poincaré inequalities are defined similarly, with uniform constants $C$ and $\lambda$. Semiuniformly local assumptions were used by Holopainen–Shanmugalingam [29].

It may seem more natural to impose uniformly local assumptions but, as we shall see, semiuniformly local assumptions are sometimes enough. The semiuniformly local assumptions also have the advantage that they are inherited by open subsets, and in particular are satisfied on all open subsets of spaces supporting global assumptions. Moreover, any strictly positive continuous weight on $\mathbf{R}^{n}$ is semiuniformly locally doubling and supports a semiuniformly local $1$-Poincaré inequality. On the other hand, our local assumptions are more general, as seen in Example 6.3 below, and sufficient for many purposes, as demonstrated in this paper. However, under semiuniformly local assumptions the constants and exponents in the local (but not necessarily the semilocal) results in Section 5 are also uniform. We now make this more precise.

Theorem 6.2**.**

If $\mu$ is semiuniformly locally doubling and $X$ supports a local (resp. semiuniformly local) $p\mspace{1.0mu}$-Poincaré inequality, then there is $q>p$ such that $X$ supports a local (resp. semiuniformly local) $(q,p)$-Poincaré inequality.

• Proof.

This is a direct consequence of Theorem 5.1, since the exponent $q$, given by an explicit formula, only depends on the local doubling constant. ∎

• Proof of Theorem 1.4.

This follows from Theorem 5.4, since the improvement $p-q$ in the exponent depends only on the local doubling constant and the constants $p$, $C$ and $\lambda^{\prime}$ in the local $p\mspace{1.0mu}$-Poincaré inequality within $B_{0}^{\prime}$. ∎

Another consequence of semiuniformly local assumptions is obvious in Lemma 4.9: the constants $\Lambda$ and $L$ therein are uniform. It would be interesting to know for which other results it is essential to require semiuniformly local assumptions, especially when the consequences for the conclusions are not just the uniformity of constants.

Example 6.3**.**

For $k\geq 3$, let

[TABLE]

i.e. $X$ is the closed lower half-plane together with countably many “skyscraper cities” $E_{k}$ near each point $x_{k}=(k,0)$, $k\geq 3$. Note that each $E_{k}$ is self-similar with the factor $1/k$ and centre $x_{k}$.

Equip $X$ with the Euclidean distance and the $2$-dimensional Lebesgue measure $\mu$. Then every $x\in X$ has a neighbourhood $V_{x}$ (with respect to $X$) whose interior (with respect to $\mathbf{R}^{2}$) is a uniform subdomain of $\mathbf{R}^{2}$. This implies that $\mu$ is doubling and supports a 1-Poincaré inequality on $\overline{V}_{x}$, by Theorem 4.4 in Björn–Shanmugalingam [16] and Proposition 7.1 in Aikawa–Shanmugalingam [2]. Thus $\mu$ is locally doubling and supports a local 1-Poincaré inequality.

For $j,k\geq 4$, let $r_{j,k}=k^{-j}$ and $x_{j,k}=(k+r_{j,k},kr_{j,k})\in E_{k}$. Then it is easily seen that $\mu(B(x_{j,k},kr_{j,k}))$ is comparable to $k^{1-2j}$, while $\mu(B(x_{j,k},2kr_{j,k}))$ is comparable to $k^{2(1-j)}$. It follows that the local doubling constant near $x_{k}$ is at least comparable to $k$.

Similarly, the ball $B(x_{j,k},3r_{j,k})$ is disconnected and contains the point $(k-r_{j,k},kr_{j,k})$. Lemma 4.10 then implies that the local dilation constant $\lambda_{k}$ in any Poincaré inequality around the point $x_{k}$ satisfies $\lambda_{k}\geq\tfrac{1}{3}k$.

Letting $k\to\infty$ shows that $\mu$ is not semiuniformly locally doubling and does not support any semiuniformly local Poincaré inequality. On the other hand, since $X$ is proper and connected, it follows from Proposition 3.8 and Theorem 4.4 that $\mu$ is semilocally doubling and supports a semilocal 1-Poincaré inequality.

If one is satisfied with a (semi)local $p\mspace{1.0mu}$-Poincaré inequality only for $p>2$, rather than a 1-Poincaré inequality, then a simpler example can be constructed by replacing each “city” $E_{k}$ with a wedge

[TABLE]

In this case, the validity of a local $p\mspace{1.0mu}$-Poincaré inequality for $p>2$ follows from [6, Example A.23].

It is also possible to make $X$ bounded if $\mathbf{R}\times(-\infty,0]$ in its construction is replaced by $(-1,1)\times(-1,0]$ and the “cities” or “wedges” are attached at the points $(1-8k^{-1},0)$ rather than at $x_{k}$, $k\geq 8$. In this case, $X$ is not complete and does not support semilocal assumptions, because they would automatically imply global assumptions as $X$ is bounded, and thus semiuniformly local assumptions would follow as well.

7 Lebesgue points

Using the better Poincaré inequalities of Section 5 it can be shown that Newtonian functions have $L^{p}$-Lebesgue points q.e. also under local assumptions. By Hölder’s inequality every $L^{p}$-Lebesgue point is an $L^{r}$-Lebesgue point for all $1\leq r\leq p$.

Theorem 7.1**.**

Assume that $p>1$, that $\mu$ is locally doubling and that one of the following conditions is satisfied:**

$X$* is locally compact and supports a local $p\mspace{1.0mu}$-Poincaré inequality*;** 2. 2.

for every $x\in X$ there are $r>0$ and $q<p$ such that a $q$-Poincaré inequality holds within $B(x,r)$ in the sense of Definition 4.1.

If $u\in N^{1,p}_{\rm loc}(X)$, then q.e. $x$ is an $L^{p}$-Lebesgue point of $u$, i.e.

[TABLE]

On metric spaces with global assumptions such results have been obtained by Kinnunen–Latvala [34] and Björn–Björn–Parviainen [12]. Traditionally (for Sobolev functions), as well as in [34] and [12], this is shown using the density of continuous functions. Here we offer a different approach based on the fact that Newtonian functions are more precisely defined than arbitrary a.e.-representatives.

Note that, by Theorem 8.4 below, locally Lipschitz functions are dense in $N^{1,p}_{\rm loc}(X)$ under the assumptions in Theorem 7.1, even though we do not use this fact here. In case 1 it also follows from Theorem 9.1 below that the functions in $N^{1,p}_{\rm loc}(X)$ are quasicontinuous. Even though this is not known in case 2, they are still, by their Newtonian definition, more precisely defined than arbitrary a.e.-representatives, which enables us to prove the existence of Lebesgue points q.e.

• Proof of Theorem 7.1.

Theorem 5.4 shows that the assumptions 1 imply 2. Thus in both cases we can consider a ball $B_{0}$ such that $u\in N^{1,p}(B_{0})$ and a $q$-Poincaré inequality and the doubling property for $\mu$ hold within $B_{0}$, where $q<p$ depends on $B_{0}$. Theorem 5.1 shows that if $q<p$ is chosen close enough to $p$, then the Sobolev–Poincaré inequality

[TABLE]

with some $\lambda\geq 2$, holds for all balls $B$ with $\frac{5}{2}\lambda B\subset B_{0}$ and for every upper gradient $g$ of $u$.

For $x\in B_{0}$, let $r_{j}=2^{-j}$ and $v(x)=\limsup_{j\to\infty}v_{j}(x)$, where

[TABLE]

Note that $u\in L^{p}(B_{0})$ and hence, by the Lebesgue differentiation theorem (Theorem 3.10), $v=0$ a.e. in $B_{0}$. We shall show that $v\in N^{1,p}(B_{0})$ and hence $v=0$ q.e. in $B_{0}$, by [6, Proposition 1.59].

To this end, let $\gamma\colon[0,l_{\gamma}]\to B_{0}$ be a nonconstant rectifiable curve (parameterized by arc length) and $g\in L^{p}(B_{0})$ be an upper gradient of $u$. By splitting $\gamma$ into parts, if necessary, and by considering sufficiently large $j$, we can assume that $\frac{1}{2}r_{j}\leq l_{\gamma}\leq r_{j}$ and that $\tfrac{5}{2}\lambda B\subset B_{0}$, where $B:=B(x,2r_{j})$ with $x=\gamma(0)$ and $y=\gamma(l_{\gamma})$ being the endpoints of $\gamma$. Since $u\in L^{p}(B_{0})$, both $v_{j}(x)$ and $v_{j}(y)$ are finite, and hence

[TABLE]

where we have used that by the triangle inequality for the normalized $L^{p}$-norm,

[TABLE]

and similarly for \bigl{|}v_{j}(y)-|u_{B}-u(y)|\bigr{|}.

The local doubling property within $B_{0}$ and the Sobolev–Poincaré inequality (7.1) then imply that

[TABLE]

where $C^{\prime\prime\prime}$ is independent of $j$ and $g_{M}^{q}:=M^{*}_{B_{0},B_{0}}g^{q}$ is the noncentred local maximal function defined by (3.2). Glueing together all the parts of $\gamma$ and by applying [6, Lemma 1.52] we conclude that $C^{\prime\prime\prime}(g_{M}+g)$ is a $p\mspace{1.0mu}$-weak upper gradient of $v$ in $B_{0}$. Since $q<p$, the noncentred local maximal operator is bounded on $L^{p/q}(B_{0})$, by Proposition 3.9, which yields that

[TABLE]

and hence $v\in N^{1,p}(B_{0})$, as required.

As $X$ is Lindelöf there is a countable cover of $X$ by balls $B_{0}$ of the type above, and since the capacity is countably subadditive we conclude the existence of Lebesgue points q.e. ∎

Remark 7.2**.**

Since Lebesgue points are of a local nature, the proof of Theorem 7.1 can be modified so that

[TABLE]

holds for q.e. $x$ and all

[TABLE]

where

[TABLE]

cf. Remark 5.2. If $\mu$ is semiuniformly locally doubling, then ${\hat{s}}:=\sup_{x}s(x)<\infty$, and we can use ${\hat{s}}$ instead of $s(x)$ to find a common $q=q(x)>p$ so that (7.4) holds for q.e. $x$.

In fact, if $s(x)$ is “attained” then it is even possible to reach the borderline case, as in Heinonen–Koskela–Shanmugalingam–Tyson [28, Theorem 9.2.8]. More precisely, if there is $r_{0}>0$ such that

[TABLE]

then we may let $q(x)=s(x)p/(s(x)-p)$. To see this, one uses the following estimate with $q=q(x)$ and $y\in B(x,r_{0})$,

[TABLE]

where the $(q,p)$-Poincaré inequality within $B(x,r_{0})$ is provided by Theorem 5.1. The second term on the right-hand side tends to [math] q.e., as we already know that $u$ has $L^{1}$-Lebesgue points q.e., while the first term tends to [math] q.e. in $B(x,r_{0})$ by Lemma 9.2.4 in Heinonen–Koskela–Shanmugalingam–Tyson [28].

In particular, if for each $x\in X$ there is $r>0$ such that (5.1) holds for all balls $B^{\prime}\subset B\subset B(x,r)$ with the same $s>0$ (independent of $x$), then $u$ has $L^{q}$-Lebesgue points q.e., with $q=sp/(s-p)$ for $p<s$ and all $q<\infty$ for $p\geq s$, provided that the assumptions in Theorem 7.1 are fulfilled.

Remark 7.3**.**

The proof of Theorem 7.1 shows that the assumptions can be further relaxed with a somewhat weaker conclusion. Namely, if $\mu$ is locally doubling and only supports a local $p\mspace{1.0mu}$-Poincaré inequality (which need not improve to a $q$-Poincaré inequality as $X$ is not necessarily locally compact), then it can be verified that the function $v$, defined by (7.2), belongs to $N^{1,q}(B_{0})$ for every $1\leq q<p$. To see this one replaces $q$ by $p$ in (7.3), and then uses the $L^{1}$ to weak-$L^{1}$ boundedness (3.3) of the noncentred local maximal operator, which yields that $g_{M}$ belongs to weak-$L^{p}(B_{0})$ and thus to $L^{q}(B_{0})$ for all $q<p$.

An immediate consequence is that $v=0$ outside a set of zero $q$-capacity and hence, every $u\in N^{1,p}_{\rm loc}(X)$ has Lebesgue points $q$-quasieverywhere. Since it is not clear whether $\mu$ supports a local $q$-Poincaré inequality for some $q<p$, this cannot be deduced directly from the inclusion $N^{1,p}_{\rm loc}(X)\subset N^{1,q}_{\rm loc}(X)$ as we have no $q$-quasieverywhere Lebesgue point result available for functions in $N^{1,q}_{\rm loc}(X)$ under these assumptions.

8 Density of Lipschitz functions

Density of smooth functions is a useful property of Sobolev spaces, with many important consequences (which we will discuss in Section 9). In metric spaces the smoothest functions to consider are (locally) Lipschitz functions. There are two types of density results for $N^{1,p}(X)$ in the literature. The first one is due to Shanmugalingam [42] (it can also be found in [6, Theorem 5.1] and [28, Theorem 8.2.1]).

Theorem 8.1**.**

(Shanmugalingam [42, Theorem 4.1])* If $\mu$ is globally doubling and supports a global $p\mspace{1.0mu}$-Poincaré inequality, then Lipschitz functions are dense in $N^{1,p}(X)$.*

The following result was recently obtained by Ambrosio–Colombo–Di Marino [3] and Ambrosio–Gigli–Savaré [4]. In fact, it is not explicitly spelled out in either paper, but it is a direct consequence of a combination of results in the two papers. Below we explain this in some detail, see Remark 8.9. Note that by density we always mean norm-density in the $N^{1,p}$ norm, with the exception of Theorem 8.10.

Theorem 8.2**.**

(Ambrosio et al. [3], [4])*

Assume that $X$ is a complete globally doubling metric space and that $p>1$. Then Lipschitz functions are dense in $N^{1,p}(X)$.*

The main difference in these two results is that the former assumes doubling for $\mu$ (and not just for $X$) together with a Poincaré inequality, while the latter requires completeness and $p>1$. Note that even though the doubling property and the Poincaré inequality extend from $X$ to its closure {\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle X}\kern 0.0pt}\hss}{X}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle X}\kern 0.0pt}\hss}{X}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle X}\kern 0.0pt}\hss}{X}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle X}\kern 0.0pt}\hss}{X}}}, it need not be true that N^{1,p}({\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle X}\kern 0.0pt}\hss}{X}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle X}\kern 0.0pt}\hss}{X}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle X}\kern 0.0pt}\hss}{X}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle X}\kern 0.0pt}\hss}{X}}})=N^{1,p}(X), cf. [28, Lemma 8.2.3]. In other words, completeness (or at least local compactness) is not a negligible assumption. Thus both results have their pros and cons. Our aim in this section is to extend both of these results to local assumptions and to combine them into a unified result (when $p>1$).

Remark 8.3**.**

Without both completeness and a global Poincaré inequality, Lipschitz functions are not necessarily dense in $N^{1,p}(X)$, consider e.g. $X=\mathbf{R}\setminus\{0\}$ or the slit disc in $\mathbf{R}^{2}$, both of which support a local 1-Poincaré inequality. This also shows that the completeness assumption in Theorem 8.2 cannot be dropped nor replaced by local compactness.

It is therefore natural to obtain density of locally Lipschitz functions in most of our results below. It should be mentioned that there is no known example of a metric space $X$ such that locally Lipschitz functions are not dense in $N^{1,p}(X)$. (A function $u:X\to\mathbf{R}$ is locally Lipschitz if for every $x\in X$ there is $r>0$ such that $u|_{B(x,r)}$ is Lipschitz.)

The following is a local generalization of Theorem 8.1.

Theorem 8.4**.**

If $\mu$ is locally doubling and supports a local $p\mspace{1.0mu}$-Poincaré inequality, then locally Lipschitz functions are dense in $N^{1,p}_{\rm loc}(X)$.

Note that if $\Omega$ is an open subset of $X$, then the local assumptions are inherited by $\Omega$ and hence we can also directly conclude the density of locally Lipschitz functions in $N^{1,p}_{\rm loc}(\Omega$). The same is true for Theorem 8.6 below.

To prove Theorem 8.4 we will need the following lemma.

Lemma 8.5**.**

Assume that the $p\mspace{1.0mu}$-Poincaré inequality and the doubling property for $\mu$ hold within a ball $2B_{0}$ in the sense of Definitions 3.1 and 4.1. Then every $u\in N^{1,p}(2B_{0})$ can be approximated in the $N^{1,p}(B_{0})$-norm by Lipschitz functions $u_{k}$ with support in $2B_{0}$. Moreover, $\mu(\{x\in B_{0}:u(x)\neq u_{k}(x)\})\to 0$ as $k\to\infty$.

• Proof.

By Proposition 3.4, $B_{0}$ can be covered by finitely many balls $B_{j}$ with centres in $B_{0}$ and radii $r^{\prime}=r_{B_{0}}/22\lambda$, where $\lambda$ is the dilation constant in the $p\mspace{1.0mu}$-Poincaré inequality within $2B_{0}$. Let $\{\varphi_{j}\}_{j}$ be a Lipschitz partition of unity on $\bigcup_{j}B_{j}$ subordinate to $2B_{j}$, e.g. one constructed as in the proof of Theorem 8.4 below.

Let $u\in N^{1,p}(2B_{0})$ be arbitrary and let $g\in L^{p}(2B_{0})$ be an upper gradient of $u$. Noting that $\max\{\min\{u,k\},-k\}\to u$ in $N^{1,p}(2B_{0})$, as $k\to\infty$, we can assume without loss of generality that $|u|\leq 1$. Using the noncentred local maximal function in (3.2), let

[TABLE]

Proposition 3.9 then implies that

[TABLE]

For a fixed $j$ and $x\in 2B_{j}\setminus E_{t}$, we get for all $0<\tfrac{1}{2}r\leq\rho\leq r\leq 8r^{\prime}$ that

[TABLE]

since for such radii, $\tfrac{5}{2}\lambda B(x,r)\subset 22\lambda B_{j}\subset 2B_{0}$. A telescopic argument as in the proof of [6, Theorem 5.1] then shows that the limit $\bar{u}(x)=\lim_{r\to 0}u_{B(x,r)}$ exists everywhere in $2B_{j}\setminus E_{t}$ and is $C_{4}t$-Lipschitz therein. Also, by Lebesgue’s differentiation theorem (Theorem 3.10), $\bar{u}=u$ a.e. in $2B_{j}\setminus E_{t}$. Using e.g. truncated McShane extensions, $\bar{u}$ extends to a $C_{4}t$-Lipschitz function $u_{t,j}$ on $2B_{j}$ such that $|u_{t,j}|\leq 1$. Then also $u_{t}=\sum_{j}\varphi_{j}u_{t,j}$ equals $u$ a.e. in $B_{0}\setminus E_{t}$. In view of [6, Corollary 2.21], it follows that $(g+C_{4}t)\chi_{E_{t}\cap B_{0}}$ is a $p\mspace{1.0mu}$-weak upper gradient of $u-u_{t}$ and hence

[TABLE]

Since $g\in L^{p}(2B_{0})$, we conclude from (8.1) that $u_{t}\to u$ in $N^{1,p}(B_{0})$. By construction, $u_{t}$ is Lipschitz in $2B_{0}$ and $\operatorname{supp}u_{t}\subset\bigcup_{j}2B_{j}\subset 2B_{0}$. ∎

• Proof of Theorem 8.4.

Let $u\in N^{1,p}_{\rm loc}(X)$. For every $x\in X$, let $B_{x}=B(x,r_{x})$ be a ball such that $u\in N^{1,p}(B_{x})$ and such that the $p\mspace{1.0mu}$-Poincaré inequality and the doubling property for $\mu$ hold within $B_{x}$. As $X$ is Lindelöf, we can find a countable subcollection such that $X=\bigcup_{j=1}^{\infty}\tfrac{1}{4}B_{x_{j}}$. Let $B_{j}=\tfrac{1}{4}B_{x_{j}}$, $j=1,2,\mathinner{.}\nonscript\!.\nonscript\!\mathinner{.}$.

We construct a suitable Lipschitz partition of unity. For each $j$ we find $\psi_{j}\in\operatorname{Lip}(X)$ such that $\chi_{B_{j}}\leq\psi_{j}\leq\chi_{2B_{j}}$. Let inductively $\varphi_{1}=\psi_{1}$ and

[TABLE]

Then $\sum_{k=1}^{j}\varphi_{k}=1$ in $B_{j}$, and hence $\varphi_{k}\equiv 0$ in $B_{j}$ for $k>j$, so that

[TABLE]

As this holds for all $B_{j}$ we see that $\{\varphi_{j}\}_{j=1}^{\infty}$ is a partition of unity.

Since $4B_{j}=B_{x_{j}}$, the assumptions of Lemma 8.5 are satisfied for each $2B_{j}$ (in place of $B_{0}$). Hence, for every $\varepsilon>0$ and each $j$, there is $v_{j}\in\operatorname{Lip}(2B_{j})$ with

[TABLE]

where $L_{j}$ is the Lipschitz constant of $\varphi_{j}$. Then also

[TABLE]

As $v:=\sum_{j=1}^{\infty}\varphi_{j}v_{j}$ is a locally finite sum of Lipschitz functions, $v$ is locally Lipschitz. Combining this with the above estimate we conclude that

[TABLE]

We now turn to Theorem 8.2 which we generalize in the following way, also taking into account Theorem 8.1. Note that the set of points, for which 1 resp. 2 below holds, is open. Thus, if $X$ is connected, these two sets cannot be disjoint.

Theorem 8.6**.**

Let $p>1$ and assume that for every $x\in X$ there is a ball $B_{x}\ni x$ such that either

$B_{x}$* is locally compact and globally doubling*;* or* 2. 2.

the $p\mspace{1.0mu}$-Poincaré inequality and the doubling property for $\mu$ hold within $B_{x}$ in the sense of Definitions 3.1 and 4.1.

Then locally Lipschitz functions are dense in $N^{1,p}_{\rm loc}(X)$.

In particular, this holds if $p>1$ and $X$ is locally compact and locally doubling.

• Proof.

Without loss of generality we can replace the assumption 1 by

1. (a*′*).

{\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle B}\kern 0.0pt}\hss}{B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle B}\kern 0.0pt}\hss}{B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle B}\kern 0.0pt}\hss}{B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle B}\kern 0.0pt}\hss}{B}}}_{x} is compact and globally doubling.

The proof is now the same as the proof of Theorem 8.4, but with appeal to Theorem 8.2 instead of Lemma 8.5 for the balls $B_{x}$ satisfying (a*′*). ∎

So far we have deduced the density for locally Lipschitz functions. A natural question is when density can be obtained for Lipschitz functions. Note that under the assumptions in Theorems 8.4 and 8.6 it can happen that Lipschitz functions are not dense, see Remark 8.3.

Theorem 8.7**.**

Assume that $\mu$ is semilocally doubling and supports a semilocal $p\mspace{1.0mu}$-Poincaré inequality. Then Lipschitz functions with bounded support are dense in $N^{1,p}(X)$. If $X$ is also proper, then $\overline{{\operatorname{Lip}_{c}}(X)}=N^{1,p}(X)$.

${\operatorname{Lip}_{c}}(X)$ denotes the space of Lipschitz functions with compact support.

• Proof.

Let $u\in N^{1,p}(X)$ and $\varepsilon>0$. Find a sufficiently large ball $B$ with $r_{B}>1$ such that \|u\|_{N^{1,p}(X\setminus{\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle B}\kern 0.0pt}\hss}{B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle B}\kern 0.0pt}\hss}{B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle B}\kern 0.0pt}\hss}{B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle B}\kern 0.0pt}\hss}{B}}})}<\varepsilon. By Lemma 8.5 and the semilocal assumptions, there is $v\in\operatorname{Lip}(X)$ so that $\|u-v\|_{N^{1,p}(2B)}<\varepsilon$.

For $\eta(x)=(1-\operatorname{dist}(x,B))_{+}$ we then get that $v_{\varepsilon}:=v\eta\in\operatorname{Lip}(X)$ with $\operatorname{supp}v_{\varepsilon}\subset 2B$ and $u-v_{\varepsilon}=u(1-\eta)+(u-v)\eta$. Since

[TABLE]

a simple calculation yields $\|u-v_{\varepsilon}\|_{N^{1,p}(X)}<6\varepsilon$, which concludes the proof of the first part. If $X$ is also proper then $v_{\varepsilon}$ has compact support and thus $\overline{{\operatorname{Lip}_{c}}(X)}=N^{1,p}(X)$. ∎

• Proof of Theorem 1.5..

The assumptions in Theorem 8.7 are satisfied because of Proposition 1.2 and Theorem 1.3, and hence the result follows. ∎

Theorem 8.8**.**

Assume that $X$ is a complete semilocally doubling metric space and that $p>1$. Then $\overline{{\operatorname{Lip}_{c}}(X)}=N^{1,p}(X)$.

Note that, by Lemma 3.6 and Proposition 3.8, a metric space is complete and semilocally doubling if and only if it is proper and locally doubling. A natural question is if Theorem 8.8 holds when $X$ is only complete and locally doubling. The completeness in Theorem 8.8 cannot be replaced by local compactness, see Remark 8.3.

• Proof.

This is quite similar to the proof of Theorem 8.7. Let $u\in N^{1,p}(X)$ and $\varepsilon>0$. Find a sufficiently large ball $B$ with $r_{B}>1$ such that \|u\|_{N^{1,p}(X\setminus{\mathchoice{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle B}\kern 0.0pt}\hss}{B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle B}\kern 0.0pt}\hss}{B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle B}\kern 0.0pt}\hss}{B}}{\hbox to0.0pt{\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle B}\kern 0.0pt}\hss}{B}}})}<\varepsilon. By the semilocal assumptions, $\overline{2B}$ is a complete globally doubling metric space. Hence by Theorem 8.2 there is $v\in\operatorname{Lip}(\overline{2B})$ so that $\|u-v\|_{N^{1,p}(2B)}<\varepsilon$. The rest of the proof is identical to the second half of the proof of Theorem 8.7, since $X$ is proper by Lemma 3.6. ∎

Remark 8.9**.**

We will now explain how Theorem 8.2 follows from Ambrosio–Colombo–Di Marino [3] and Ambrosio–Gigli–Savaré [4]. In [4], a function $g$ is called a $p\mspace{1.0mu}$-upper gradient of a function $f$ if there is $\tilde{f}$ such that $\tilde{f}=f$ a.e. and $g$ is a $p\mspace{1.0mu}$-weak upper gradient of $\tilde{f}$ in the sense of Definition 2.1. In [4] they also define $p\mspace{1.0mu}$-weak upper gradients (different from ours), $p\mspace{1.0mu}$-relaxed upper gradients and $p\mspace{1.0mu}$-relaxed slopes. Furthermore, they show that if a function $f\in L^{p}(X)$ has a gradient $g\in L^{p}(X)$ in any of these four senses, then it has one in each of the four senses and the a.e.-minimal ones coincide. This is shown in [4, Theorem 7.4 and Section 8.1] assuming completeness of $X$.

In Ambrosio–Colombo–Di Marino [3], there is a different definition of $p\mspace{1.0mu}$-relaxed slope and the same definition of $p\mspace{1.0mu}$-weak upper gradient as in [4]. In [3, Theorem 6.1] it is shown that if $X$ is complete and $f\in L^{p}(X)$ has a gradient $g\in L^{p}(X)$ in either of these two senses, then it has one in the other and the a.e.-minimal ones coincide. So in conclusion the a.e.-minimal gradients in $L^{p}(X)$ in all fives senses exist simultaneously and then coincide, assuming that $f\in L^{p}(X)$ and that $X$ is complete. Moreover, the Sobolev space $W^{1,p}(X)$ in [3, Corollary 7.5] thus coincides with

[TABLE]

(recall that functions in $N^{1,p}(X)$ are defined pointwise everywhere).

The following density result is a special case of the equivalence result from [4, Theorem 7.4 and Section 8.1]. (Note that this is not a norm-density result as elsewhere in this section.)

Theorem 8.10**.**

Assume that $X$ is complete and $p>1$. Let $f\in N^{1,p}(X)$. Then there exist Lipschitz functions $f_{n}$ such that

[TABLE]

where

[TABLE]

is the local upper Lipschitz constant (also called upper pointwise dilation) of $f_{n}$, and $g_{f}$ is the minimal $p\mspace{1.0mu}$-weak upper gradient of $f$ (in the sense of Definition 2.1).

As explained at the bottom of p. 1 of [3], once we know reflexivity, the norm-density of Lipschitz functions follows directly using Mazur’s lemma. This reflexivity was obtained in Corollary 7.5 in [3] when $X$ is complete and globally doubling, and thus Theorem 8.2 follows.

Completeness is not assumed explicitly in [3, Corollary 7.5], but the result relies on other results in [3] (e.g. Theorem 7.4) for which completeness is assumed. And indeed, the completeness assumption cannot be dropped or replaced by assuming that $X$ is merely locally compact for the density result, since norm-density of Lipschitz functions then can fail, see Remark 8.3. The same counterexamples show that Theorem 8.10 cannot hold (in general) in locally compact spaces with Lipschitz functions $f_{n}$. Whether it can hold in locally compact spaces, if $f_{n}$ are just required to be locally Lipschitz, is not clear because the “partition of unity” technique used in the proof of Theorem 8.4 cannot be used to construct such an extension, at least not in such an easy way as here, since it would require controlling $\|{\operatorname{Lip}v-g_{f}}\|$ in terms of $\|{\operatorname{Lip}(\varphi_{j}v_{j})-g_{f_{j}}}\|$.

A slight word of warning: one may get the impression that as a consequence of Theorem 8.10 one can deduce that $g_{f}=\operatorname{Lip}f$ a.e. for Lipschitz functions. This is not so, and indeed it is not true in general, as seen by considering e.g. the von Koch snowflake curve, on which $g_{f}\equiv 0$ for all functions, because of the lack of rectifiable curves. The equality $g_{f}=\operatorname{Lip}f$ a.e. for Lipschitz functions is however true if $X$ is complete, $p>1$ and $\mu$ is globally doubling and supports a global $p\mspace{1.0mu}$-Poincaré inequality, by Theorem 6.1 in Cheeger [18].

Remark 8.11**.**

Occasionally it may be interesting to know when (locally) Lipschitz or continuous functions are dense in $N^{1,p}(X)$ even when $X\neq\operatorname{supp}\mu$, in which case our general condition that all balls have positive measure is invalid. It turns out that this happens if and only if they are dense in $N^{1,p}(\operatorname{supp}\mu)$. For Lipschitz and continuous functions this is Lemma 5.19 (e) in [6]. For locally Lipschitz functions this can be proved similarly, provided that one uses the locally Lipschitz extensions due to Luukkainen–Väisälä [41, Theorem 5.7]. (Note that the class LIP in [41] consists of locally Lipschitz functions.)

For quasicontinuity, which will be discussed in the next section, a similar equivalence is also true, by [6, Lemma 5.19 (d)].

9 Quasicontinuity and other consequences

Having established the density of continuous (or more exactly locally Lipschitz) functions we can now draw a number of qualitative conclusions about Newtonian functions and capacities.

Throughout this section, $\Omega\subset X$ is open. A function $u:\Omega\to{\overline{\mathbf{R}}}$ is quasicontinuous if for every $\varepsilon>0$ there is an open set $G$ with ${C_{p}}(G)<\varepsilon$ such that $u|_{\Omega\setminus G}$ is continuous. See the recent paper Björn–Björn–Malý [11] for several different characterizations of quasicontinuity, and in particular that one can equivalently replace the condition ${C_{p}}(G)<\varepsilon$ by ${C_{p}^{\Omega}}(G)<\varepsilon$, where ${C_{p}^{\Omega}}$ is the capacity associated with $N^{1,p}(\Omega)$ rather than with $N^{1,p}(X)$. Note also that, in the following theorem, $X$ can be replaced by any open subset of $X$. Moreover, the conditions in 1 and 2 below are inherited by open subsets.

Theorem 9.1**.**

Assume that $X$ is locally compact and that one of the following conditions is satisfied:**

$\mu$* is locally doubling and supports a local $p\mspace{1.0mu}$-Poincaré inequality,* 2. 2.

$p>1$* and $X$ is locally doubling, or more generally the conditions in Theorem 8.6 are satisfied*;** 3. 3.

continuous functions are dense in $N^{1,p}(X)$.

Then every $u\in N^{1,p}_{\rm loc}(X)$ is quasicontinuous in $X$ and hence ${C_{p}}$ is an outer capacity, i.e.

[TABLE]

Quasicontinuity has been established for Newtonian functions under various assumptions in Björn–Björn–Shanmugalingam [13], Björn–Björn–Lehrbäck [9] and Heinonen–Koskela–Shanmugalingam–Tyson [28] for open subsets. See also Shanmugalingam [42]. Assuming that $X$ is complete and that $\mu$ is globally doubling and supports a global $p\mspace{1.0mu}$-Poincaré inequality, quasicontinuity was also established for functions in $N^{1,p}(U)$ when $U$ is a quasiopen subset of $X$, by Björn–Björn–Latvala [8] (when $p>1$) and Björn–Björn–Malý [11].

For $p>1$, quasicontinuity also implies that ${C_{p}}$ is a Choquet capacity and thus, if $X$ is locally compact, that all Borel sets are capacitable, i.e.

[TABLE]

see e.g. Aikawa–Essén [1, Part 2, Section 10] together with [6, Theorems 6.4 and 6.7 (viii)]. It should be mentioned that there is no known example of a Newtonian function which is not quasicontinuous, nor of a metric space $X$ such that continuous functions are not dense in $N^{1,p}(X)$.

• Proof of Theorem 9.1.

By Theorems 8.4 and 8.6, 1 $\mathchoice{\quad\Longrightarrow\quad}{\Rightarrow}{\Rightarrow}{\Rightarrow}$ 3 and 2 $\mathchoice{\quad\Longrightarrow\quad}{\Rightarrow}{\Rightarrow}{\Rightarrow}$ 3. So assume that 3 holds. By [6, Theorem 5.21] every $u\in N^{1,p}_{\rm loc}(X)$ has a quasicontinuous representative. As $X$ is locally compact, Proposition 4.7 in Björn–Björn–Lehrbäck [9] then shows that every $u\in N^{1,p}_{\rm loc}(X)$ is quasicontinuous. The outer capacity property then follows from Björn–Björn–Shanmugalingam [13, Corollary 1.3] (or [6, Theorem 5.31]). ∎

Quasicontinuity of $N^{1,p}(X)$ gets inherited to open subsets in the following way. Recall the definition of $\widehat{N}^{1,p}(\Omega)$ in (8.2).

Proposition 9.2**.**

If every $u\in N^{1,p}(X)$ is quasicontinuous, then $N^{1,p}_{\rm loc}(\Omega)$ consists exactly of those $u\in\widehat{N}^{1,p}_{\rm loc}(\Omega)$ which are quasicontinuous, and similarly for $N^{1,p}(\Omega)$.

• Proof.

Clearly, $N^{1,p}_{\rm loc}(\Omega)\subset\widehat{N}^{1,p}_{\rm loc}(\Omega)$. Multiplying $u\in N^{1,p}_{\rm loc}(\Omega)$ by Lipschitz cut-off functions shows that for each $x\in\Omega$ there is $r_{x}>0$ such that $u$ is quasicontinuous in $B(x,r_{x})$. As $X$ is Lindelöf, a countable covering of $\Omega$ by such balls yields quasicontinuity in $\Omega$.

Conversely, by an argument due to Kilpeläinen [33], every quasicontinuous $u\in\widehat{N}^{1,p}_{\rm loc}(\Omega)$ is q.e. equal to a Newtonian function and hence itself in $N^{1,p}_{\rm loc}(\Omega)$, cf. [6, Propositions 5.22 and 5.23]. ∎

Quasicontinuity, or rather the outer capacity property following from it, provides us with a short proof of the following fact, cf. Kallunki–Shanmugalingam [31] where it was proved under stronger assumptions. A similar statement is not true if we drop the assumption that $K$ be compact. (Let e.g. $K$ be a countable dense subset of a ball in $\mathbf{R}^{n}$ and $p\leq n$.)

Proposition 9.3**.**

Assume that ${C_{p}}$ is an outer capacity and that continuous resp. (locally) Lipschitz functions are dense in $N^{1,p}(X)$. If $K\subset X$ is compact, then

[TABLE]

where the infimum is taken over all continuous resp. (locally) Lipschitz $u$ such that $u\geq 1$ on $K$.

Note that if continuous functions are dense in $N^{1,p}(X)$, then the condition that ${C_{p}}$ is an outer capacity is equivalent to requiring that all functions in $N^{1,p}(X)$ are quasicontinuous, see Theorems 5.20, 5.31 and Proposition 5.32 in [6].

• Proof.

Given $\varepsilon>0$, there exist an open set $G\supset K$ and $u\in N^{1,p}(X)$ such that $u=1$ on $G$ and

[TABLE]

Since $K$ is compact, there exists $0<\delta\leq 1$ such that $\operatorname{dist}(K,X\setminus G)>2\delta$. Let $\eta(x):=\min\{1,\operatorname{dist}(x,K)/\delta\}$ and set $\tilde{v}=u+\eta(v-u)$, where $v$ is continuous resp. (locally) Lipschitz in $X$ and such that $\|v-u\|_{N^{1,p}(X)}<\varepsilon\delta$. Then $\tilde{v}=1$ on $K$ and, as $g_{\eta(v-u)}\leq\eta g_{v-u}+|v-u|g_{\eta}$, we also have

[TABLE]

It then follows that

[TABLE]

which, by (9.1), tends to ${C_{p}}(K)$ as $\varepsilon\to 0$.

Finally, $\tilde{v}=1-\eta+\eta v$ is continuous resp. (locally) Lipschitz in $G$ (as $u\equiv 1$ therein), while $\tilde{v}=v$ is continuous resp. (locally) Lipschitz in the open set

[TABLE]

It follows that $\tilde{v}$ is continuous resp. locally Lipschitz in $X$ and, as $\operatorname{dist}(X\setminus G,\operatorname{supp}(1-\eta))>\delta$, also Lipschitz in $X$ whenever $v$ is Lipschitz. ∎

10 Conclusions for $p\mspace{1.0mu}$-harmonic functions

Nonlinear potential theory associated with $p\mspace{1.0mu}$-harmonic functions and quasiminimizers, $p>1$, has been extensively studied during the last 20 years on complete metric spaces equipped with globally doubling measures supporting a global $p\mspace{1.0mu}$-Poincaré inequality, see e.g. [6] and the references therein. It is therefore natural to see to which extent this theory can be extended to local assumptions.

In much of this theory the properness of $X$ plays an important role and even though some of the theory has already been developed on noncomplete spaces (see in particular Kinnunen–Shanmugalingam [38], Björn [15] and Björn–Marola [14]), we will in this section restrict ourselves to proper $X$. (See Björn–Björn [7, Section 6] for a similar discussion without the properness assumption.) We will also assume that $X$ is connected, that $\mu$ is locally doubling and supports a local $p\mspace{1.0mu}$-Poincaré inequality, and that $p>1$.

As we have seen, it then follows from Proposition 1.2 and Theorem 1.3 that $\mu$ is semilocally doubling and supports a semilocal $p\mspace{1.0mu}$-Poincaré inequality. The results in this paper show that most of the essential tools needed to develop the potential theory on metric spaces are available also under these assumptions.

Definition 10.1**.**

A function $u\in N^{1,p}_{\rm loc}(\Omega)$ is a $Q$-quasi(super)minimizer, $Q\geq 1$, in $\Omega$ if

[TABLE]

for all (nonnegative) $\varphi\in{\operatorname{Lip}_{c}}(\Omega)$.

If $Q=1$ in (10.1) then $u$ is a (super)minimizer. A $p\mspace{1.0mu}$-harmonic function is a continuous minimizer.

See Björn [5, Proposition 3.2] for equivalent ways of defining quasisuperminimizers; those equivalences also extend to spaces with our local assumptions. (Here Theorem 8.7 is needed.) Our first observation is that interior regularity is preserved under local assumptions. A function $u$ on $\Omega$ is lsc-regularized if $u(x)=\operatorname*{ess\,lim\,inf}_{y\to x}u(y)$ for all $x\in\Omega$.

Theorem 10.2**.**

Let $u$ be a quasi(super)minimizer in $\Omega$. Then $u$ has a representative ${\tilde{u}}$ which is continuous (resp. lsc-regularized).

Moreover, the weak Harnack inequalities for quasi(super)minimizers hold within every ball $B_{0}\subset X$ i.e. for every ball $B\subset B_{0}$ with $\Lambda B\subset\Omega$, where $\Lambda$ and the weak Harnack constants depend only on $B_{0}$.

See e.g.  Kinnunen–Martio [36], [37], Kinnunen–Shanmugalingam [38], Björn–Marola [14] and Björn–Björn [6] for formulations of the weak Harnack inequalities. There are various types of weak Harnack inequalities in these papers and under different assumptions. In [36], [37] and [38] a $q$-Poincaré inequality for some $q<p$ is assumed, which under our assumptions is provided by Theorem 5.4. Here $q$ will depend on the ball $B_{0}$.

Note that some weak Harnack inequalities in [36], [37] and [38] need to be modified, taking into account the dilation constant $\lambda$ from the $p\mspace{1.0mu}$-Poincaré inequality, see Björn–Marola [14, Section 10]. This is reflected in the constant $\Lambda\geq 1$ in Theorem 10.2 in the following way: Several of the weak Harnack inequalities in [6] contain a requirement that $50\lambda B\subset\Omega$. (The factor $50$ is not the same in all the papers.) For a fixed ball $B_{0}\subset X$, we let $C_{\rm PI}$ and $\lambda$ be the constants in the $p\mspace{1.0mu}$-Poincaré inequality (or $q$-Poincaré inequality) within $50B_{0}$, and $C_{\mu}$ be the doubling constant within $50\lambda B_{0}$. The weak Harnack inequality then holds for every ball $B\subset B_{0}$ provided that $50\lambda B\subset\Omega$ and with a constant depending only on $C_{\rm PI}$, $\lambda$, $C_{\mu}$ and $p$ (and $q$).

• Proof of Theorem 10.2.

The arguments in [36, Section 4 and Theorem 5.1], [37, Section 5] and [38] are all local, so local assumptions are enough. They do rely on a better $q$-Poincaré inequality but a suitable version is provided by Theorem 5.3, as continuity is a local property. For the lsc-regularity of quasisuperminimizers also a Lebesgue point result is needed, which is justified by Theorem 7.1.

For the weak Harnack inequalities it is explained above how the semilocal dependence on the constants is achieved. ∎

Apart from that the parameters may only be semilocal, the results in Chapters 7–14 in [6] all hold, with the exception of the Liouville theorem, which we look at in Example 10.3 below. This is because all the other results are of a local or semilocal nature, i.e. either in a bounded domain or concerning a local or semilocal property. In particular, in addition to the interior regularity in Theorem 10.2, one can prove various convergence results, minimum and maximum principles, solve the Dirichlet and the obstacle problem on bounded domains and obtain boundary regularity and resolutivity for suitable boundary data.

On the other hand, for results of a global nature, such as the Dirichlet problem on unbounded domains (as in Hansevi [24], [25]) or global singular functions (as in Holopainen–Shanmugalingam [29]), it is far from clear whether they hold under (semi)local assumptions. Thus, it is precisely when studying “global” properties that it is really interesting to know if the results hold with only local assumptions, possibly (semi)uniform ones, or if perhaps other properties of the space play a vital role. As an example of a “global” result, we will now have a look at the Liouville theorem, and show that it does not hold in the generality considered here, not even under uniformly local assumptions, cf. Section 6.

Example 10.3**.**

Let $d\mu=w\,dx$ on $\mathbf{R}$, where $\alpha\in\mathbf{R}$ and

[TABLE]

The measure $\mu$ is globally doubling and supports a uniformly local $1$-Poincaré inequality, and thus a semilocal $1$-Poincaré inequality. (And a global $p\mspace{1.0mu}$-Poincaré inequality if and only if $\alpha<p-1$.) A simple calculation shows that a function $u$ is $p\mspace{1.0mu}$-harmonic on $(\mathbf{R},\mu)$ if and only if there is a constant $c$ such that $u^{\prime}(x)=cw(x)^{1/(1-p)}$. If $\alpha>p-1$, this gives bounded nonconstant $p\mspace{1.0mu}$-harmonic functions on $(\mathbf{R},\mu)$, namely

[TABLE]

and $b,c\in\mathbf{R}$ are arbitrary constants. This shows that the Liouville theorem does not hold under semilocal assumptions, nor under uniformly local ones.