Extension and embedding theorems for Campanato spaces on $C^{0,\gamma}$ domains

TL;DR

This paper studies Campanato spaces on $C^{0,eta}$ domains, establishing embedding theorems and extension results that preserve the space parameters, thus advancing understanding of function regularity in anisotropic settings.

Contribution

It introduces extension and embedding theorems for Campanato spaces on $C^{0,eta}$ domains with anisotropic metrics, preserving exponents during extension.

Findings

Established Campanato embedding theorem for $C^{0,eta}$ domains.

Proved functions in these spaces can be extended to Euclidean space without losing properties.

Analyzed anisotropic metric effects on function space embeddings.

Abstract

We consider Campanato spaces with exponents $\lambda , p$ on domains of class $C^{0,\gamma}$ in the N-dimensional Euclidean space endowed with a natural anisotropic metric depending on $\gamma$. We discuss several results including the appropriate Campanato's embedding theorem and we prove that functions of those spaces can be extended to the whole of the Euclidean space without deterioration of the exponents $\lambda, p$.

Equations

$$\underset{B}{\sup }{\left(\frac{1}{|B\cap \Omega {|}^{\lambda }}{\int }_{B\cap \Omega }|f(y)-\text{vbox}-{\int }_{B\cap \Omega }f(z)dz{|}^{p}dy\right)}^{\frac{1}{p}}$$

$${\delta }_{\gamma }(x,y)=\max \{|\bar{x}-\bar{y}{|}^{\gamma },|x_N-y_N|\},$$

$$N_{\gamma }=\frac{N-1}{\gamma }+1.$$

$$Tf(\bar{x},x_N)=f(\bar{x},2|\bar{x}{|}^{\gamma }-x_N),\mathrm{if}{x}_N>|\bar{x}{|}^{\gamma }.$$

$$B_{\gamma }(x,r)=\{y\in {\mathbb{R}}^N:{\delta }_{\gamma }(x,y)<r\}.$$

$$f_{0}(x)=\left\{\begin{array}[]{ll}f(x),&\ {\rm if}\ x\in\Omega,\\ 0,&\ {\rm if}\ x\in\mathbb{R}^{N}\setminus\Omega.\end{array}\right.$$

$$\Vert f{\Vert }_{L_{p,\gamma }^{\lambda }(\Omega )}:={\left(\underset{x\in \Omega }{\sup }\underset{r\in \left]0,{\delta }_{\gamma }(\Omega )\right]}{\sup }\frac{1}{|B_{\gamma }(x,r)\cap \Omega {|}^{\lambda }}{\int }_{B_{\gamma }(x,r)\cap \Omega }|f(y){|}^{p}dy\right)}^{\frac{1}{p}}$$

$$|f|_{{\mathcal{L}}^{\lambda}_{p,\gamma}(\Omega)}:=\sup_{x\in\Omega}\,\sup_{r\in\left]0,\delta_{\gamma}(\Omega)\right]}\!\!\left(\frac{1}{|B_{\gamma}(x,r)\cap\Omega|^{\lambda}}\int_{B_{\gamma}(x,r)\cap\Omega}\biggl{|}f(y)-\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\!\int_{B_{\gamma}(x,r)\cap\Omega}f(z)dz\biggr{|}^{p}dy\right)^{\frac{1}{p}}\,.$$

$$L_{p,\gamma }^{\lambda }(\Omega )=\{f:\Omega \to \mathbb{R}:f_0\in L_{loc}^p({\mathbb{R}}^N),\mathrm{and}\Vert f{\Vert }_{L_{p,\gamma }^{\lambda }(\Omega )}<\infty \}.$$

$${\mathcal{L}}_{p,\gamma }^{\lambda }(\Omega )=\{f:\Omega \to \mathbb{R}:f_0\in L_{loc}^p({\mathbb{R}}^N),\mathrm{and}|f{|}_{{\mathcal{L}}_{p,\gamma }^{\lambda }(\Omega )}<\infty \}.$$

$$\Vert f{\Vert }_{{\mathcal{L}}_{p,\gamma }^{\lambda }(\Omega )}:=\Vert f{\Vert }_{L^p(\Omega )}+|f{|}_{{\mathcal{L}}_{p,\gamma }^{\lambda }(\Omega )},$$

$$BMO(\Omega )=\left\{f\in L_{loc}^1(\Omega ):|f{|}_{BMO(\Omega )}\ne \infty \right\}$$

$$\underset{Q\subset \Omega }{\sup }\text{vbox}-{\int }_Q\left|f(y)-\text{vbox}-{\int }_{Q}f(z)dz\right|dy$$

$$BMO(\Omega )={\mathcal{L}}_{1,1}^1(\Omega ).$$

$$\frac{1}{|Q\cap \Omega |}{\int }_{Q\cap \Omega }|f-f_{Q\cap \Omega }|=\frac{1}{2r}{\int }_{-r}^r|x_1|d{x}_1=\frac{r}{2}$$

$$|B_{\gamma }(x,r)\cap \Omega |\ge c{r}^{N_{\gamma }},$$

$$\Vert f{\Vert }_{C^{0,\alpha }(\overline{\Omega },{\delta }_{\gamma })}=\underset{x\in \overline{\Omega }}{\sup }|f(x)|+\underset{x,y\in \overline{\Omega }}{\sup }\frac{|f(x)-f(y)|}{{\delta }_{\gamma }^{\alpha }(x,y)}.$$

$$\text{vbox}-{\int }_{B_{\gamma }(x,r)\cap \Omega }|f(y)-f_{B_{\gamma }(x,r)\cap \Omega }|dy\le {\left(\text{vbox}-{\int }_{B_{\gamma }(x,r)\cap \Omega }|f(y)-f_{B_{\gamma }(x,r)\cap \Omega }{|}^{p}dy\right)}^{\frac{1}{p}},$$

$$\left|\left\{x\in B_{\gamma }:\left|f(x)-f_{B_{\gamma }}\right|>t\right\}\right|\le c_1{e}^{-\frac{c_{2}t}{a}}|B_{\gamma }|$$

$$\begin{array}{rl}& \frac{1}{|B_{\gamma }|}{\int }_{B_{\gamma }}|f(y)-f_{B_{\gamma }}{|}^{p}dy\\ & =\frac{p}{|B_{\gamma }|}{\int }_0^{\infty }{t}^{p-1}|\left\{y\in B_{\gamma }:|f(y)-f_{B_{\gamma }}|>t\right\}|dt\\ & \le c_{1}p{\int }_0^{\infty }{t}^{p-1}{e}^{-\frac{c_{2}t}{a}}dt=\frac{c_1}{c_2^p}\Gamma (p+1)|f{|}_{{\mathcal{L}}_{1,\gamma }^1(\Omega )}^p=c_3|f{|}_{{\mathcal{L}}_{1,\gamma }^1(\Omega )}^p,\end{array}$$

$$\Vert f{\Vert }_{L^p(B_{\gamma })}\le |B_{\gamma }{|}^{1/p}\left(\frac{\Vert f{\Vert }_{L^1(B_{\gamma })}}{|B_{\gamma }|}+c_3^{1/p}|f{|}_{{\mathcal{L}}_{1,\gamma }^1(\Omega )}\right).$$

Full text

Keywords: Campanato spaces, BMO spaces, extension operators, boundary singularities, power-type cusps.

2010 Mathematics Subject Classification: 46E30, 46E35, 42B35

1 Introduction

Given an open set $\Omega$ in $\mathbb{R}^{N}$, $1\leq p<\infty$ and $\lambda>0$ the Campanato space ${{\mathcal{L}}^{\lambda}_{p}(\Omega)}$ is the space of functions $f\in L^{p}_{loc}(\Omega)$ such that the seminorn defined by

[TABLE]

is finite. Here the supremum is taken over all balls $B$ with center in $\Omega$ and radius not exceeding the diameter of $\Omega$ and $|B\cap\Omega|$ denotes the Lebesgue measure of $B\cap\Omega$. The importance of Campanato spaces is well-known in the literature, in particular because they include other classes of function spaces, such as Morrey spaces if $0<\lambda<1$, $B\!M\!O$ spaces if $\lambda=1$ and spaces of Hölder continuous functions if $\lambda>1$. This equivalence is established by the celebrated Campanato’s embedding Theorem under suitable assumptions on the open set $\Omega$, see Theorem 1 for a statement.

It is important to note that the space $B\!M\!O(\Omega)$ of functions with bounded mean oscillation can be formally defined in the same way by setting $p=\lambda=1$ in (1). However, in the classical definition one takes the supremum over all balls contained in $\Omega$. This subtle distinction is obviously not relevant if $\Omega=\mathbb{R}^{N}$ but also if $\Omega$ is a bounded open set with Lipschitz boundary or the half space: in these cases one can identify the two spaces by setting $B\!M\!O(\Omega)={{\mathcal{L}}^{1}_{1}(\Omega)}$, see Lemma 1. However, in general the two spaces are different, see Example 1.

It is impossible to give an account of all possible applications of Campanato spaces but we would like to highlight the fact that the integral characterisation of Hölder continuous functions by means of the seminorm (1) is of fundamental importance in the study of apriori estimates for elliptic and parabolic equations, in particular in connection with the variational approach developed by Ennio De Giorgi333In fact, it is acknowledged in [campa63, campa64] that the study of these function spaces was suggested to Sergio Campanato by Ennio De Giorgi. This historical fact doesn’t seem much known outside Italy, probably because papers [campa63, campa64] are written in Italian. for the solution of the nineteenth Hilbert’s problem. We refer to [beveza] for a recent article on apriori estimates in Campanato spaces including further references.

In this paper, we discuss the problem of extending a function $f\in{\mathcal{L}^{\lambda}_{p}(\Omega)}$ to the whole of $\mathbb{R}^{N}$ by a function $Tf$ belonging to ${\mathcal{L}}^{\lambda}_{p}(\mathbb{R}^{N})$. The focus is on open sets $\Omega$ with boundaries of class $C^{0,\gamma}$, $\gamma\in]0,1]$. This means that $\Omega$ is locally described at the boundary as the subgraph of a Hölder continuous function with exponent $\gamma$. In particular, if $\gamma<1$ the boundary may exhibits singular points such as power-type cusps, see Figure 1.

The case $\gamma=1$ corresponds to the class of domains with Lipschitz continuous boundaries which, from the point of view of extension theory, represents the regular case and is already discussed in the literature, see [jones, stri] for the case $\lambda=p=1$. We refer to [gallia] for recent results on the extension for $B\!M\!O$ spaces and their variants on uniform domains, as well as to [bib2] for a remarkable decomposition theorem for the space $B\!M\!O$ on arbitrary domains.

We note from the very beginning that the case $\lambda<1$ is the easy one, at least in principle. Indeed, for $\lambda<1$ Campanato spaces coincide with Morrey spaces and it is simple to prove that functions in the Morrey spaces can be extended by zero to the whole of $\mathbb{R}^{N}$, see Lemma LABEL:extensionbyzero. Also the case $\lambda>1$ is not difficult if we identify Campanato spaces with spaces of Hölder continuous functions: these functions can be extended by means of the Björk’s Theorem (see e.g. [bib4, Thm. 1.8.3]). On the other hand, it is well-known that the extension problem for $B\!M\!O$ functions is highly non-trivial. Note that in general $B\!M\!O$ functions cannot be extended by zero. The classical example is provided by the function $f(x)=\log x$, with $x>0$, considered as a function defined on $(0,\infty)$. The extension by zero of $f$ defined by setting $f(x)=0$ for all $x<0$ does not belong to $B\!M\!O(\mathbb{R})$, see Example LABEL:log_0.

The extension problem for classical $B\!M\!O$ spaces (where balls $B$ in (1) are contained in $\Omega)$ was studied in the fundamental paper [jones] by Peter Jones who characterised the domains which allow the extension of $B\!M\!O$ functions. These are known in the literature as uniform domains and include bounded domains with Lipschitz boundaries but not domains of class $C^{0,\gamma}$ if $\gamma<1$. Up to our knowledge, not much is available in the literature in this case.

As discussed in [lamves], the pioneering analysis carried out in [barozzi] and [dapra] suggests that in the case of domains of class $C^{0,\gamma}$ the Euclidean metric should be replaced by a metric depending on $\gamma$. This metric, denoted by $\delta_{\gamma}$, is anisotropic and its definition depends on the direction of the possible cusps of the boundary. For example, if the boundary of $\Omega$ is represented by the cusp of equation $x_{N}=|\bar{x}|^{\gamma}$ where the elements of ${\mathbb{R}}^{N}$ for $N\geq 2$ are written in the form $x=(\overline{x},x_{N})$, with $\overline{x}=(x_{1},\dots,x_{N-1})\in{\mathbb{R}}^{N-1}$ and $x_{N}\in{\mathbb{R}}$, then the natural metric to be used is

[TABLE]

for all $x,y\in{\mathbb{R}}^{N}$. It is important to note that the Lebesgue measure of the corresponding balls $B_{\gamma}$ of radius $r$ behaves like $r^{N_{\gamma}}$ where

[TABLE]

Moreover, if $\Omega$ is of class $C^{0,\gamma}$ also the Lebesgue measure of $B_{\gamma}\cap\Omega$ behaves like $r^{N_{\gamma}}$: this property plays a crucial role in the analysis of Campanato spaces on domains and is called Property (A) in the literature.

By replacing the balls $B$ in (1) by the anisotropic balls $B_{\gamma}$ we obtain the corresponding Campanato spaces ${{\mathcal{L}}^{\lambda}_{p,\gamma}(\Omega)}$ depending on $\gamma$. These spaces are better suited for the validity of embedding and extension theorems in the spirit of classical results. Indeed, by using the general result of [dapra] one can prove Campanato’s embedding Theorem in the non-critical case $\lambda\neq 1$, see [lamves]. On the other hand, in the critical case $\lambda=1$ it is possible to use the generalised John-Niremberg inequality from [aalto] to prove that the spaces ${{\mathcal{L}}^{1}_{p,\gamma}(\Omega)}$ are independent of $p$, a well-known fact in the Lipschitz case $\gamma=1$. See Theorem 1 for a detailed proof.

The importance of using the metric $\delta_{\gamma}$ is also evident in the problem of extension. The simplest extension operator is provided by reflection. For example, given a cusp of the form $\Omega=\{(\bar{x},x_{N})\in\mathbb{R}^{N}:\ x_{N}<|\bar{x}|^{\gamma}\}$ one would extend a function $f(\bar{x},x_{N})$ defined in $\Omega$ by setting

[TABLE]

It is clear that the natural introduction of the term $|\bar{x}|^{\gamma}$ in formula (4) deteriorates the degree of smoothness of $f$: for example, a Lipschitz function $f$ would be transformed into a Hölder continuous function of exponent $\gamma$. (This deterioration phenomenon is well-known and is studied in [bu] for Sobolev spaces of arbitrary order.) This example gives a strong hint about the fact that incorporating $|\bar{x}|^{\gamma}$ in the metric as in (2) allows to keep the same smoothness exponent in the extension of functions.

In this paper, we consider this problem for Campanato spaces of arbitrary order $\lambda$ (including the theoretically difficult case $\lambda=1$), not only for the sake of uniform treatment but also to emphasise the preservation of the exponent $\gamma$ in the extension by reflection.

First, we consider the case of elementary unbounded domains of class $C^{0,\gamma}$. These domains are defined as subgraphs of Hölder continuous functions with exponent $\gamma$, defined on $\mathbb{R}^{N-1}$. As done in [stri] for $B\!M\!O$ functions in the Lipschitz case $\gamma=1$, in Theorem LABEL:estensioneholder we prove that the even extension (that is, the reflection with respect to the boundary) preserves the Campanato spaces ${{\mathcal{L}}^{\lambda}_{p,\gamma}}$ (note that the odd extension wouldn’t work).

Next, we consider the general case where different parts of the boundary have to be rotated in different directions in order to be represented as graphs of Hölder continuous functions with exponent $\gamma$, see Figure 1. In this case the underlying metric has to be adapted to the different directions and this leads to the analysis of Campanato spaces $\mathcal{L}^{\lambda,\mathcal{R}}_{p,\gamma}(\Omega)$ obtained as finite sums of Campanato spaces. Here ${\mathcal{R}}$ is the finite collection of rotations used to represent the boundary of $\Omega$ as mentioned above.

Note that for $\gamma=1$ the space $\mathcal{L}^{\lambda,\mathcal{R}}_{p,\gamma}(\Omega)$ is independent of ${\mathcal{R}}$ and coincides with the usual Campanato space since all metrics involved are strongly equivalent to the Euclidean metric, see Remark LABEL:patcheq. On the other hand, since for $\gamma\neq 1$ the metrics arising from different directions are not strongly equivalent, we believe that it would be difficult (and probably not even particularly useful in applications) to give an intrinsic definition of the space $\mathcal{L}^{\lambda,\mathcal{R}}_{p,\gamma}(\Omega)$ independent of ${\mathcal{R}}$.

We conclude the paper by formulating an extension problem for a natural $B\!M\!O_{\gamma}$ space associated with the metric $\delta_{\gamma}$ as in (LABEL:bmogamma), which we believe is open.

We refer to the classical monograph [bib4] for a clear introduction to the theory of Campanato spaces and to [samko] for a recent survey. We also quote [bib12] as a standard reference for $B\!M\!O$ spaces and [sawano] for an extensive study of Morrey spaces. Finally, we quote papers [fanlam], [lamves], [lamvio] for related recent results concerning the extension problem for Sobolev-Morrey spaces.

This paper is organized as follows. Section 2 is devoted to preliminary and auxiliary results which have also their own interest. In particular, we discuss Campanato’s embedding Theorem, the relation between $B\!M\!O$ spaces and Campanato spaces and the multiplication by $C^{\infty}$-functions in Campanato spaces. In Section 3 we discuss the extension problem, we prove Theorem LABEL:estensioneholder concerning the extension of Campanato spaces on elementary domains; moreover, we define the Campanato spaces $\mathcal{L}^{\lambda,\mathcal{R}}_{p,\gamma}(\Omega)$ and we prove the corresponding extension theorem; we also formulate an open problem.

2 Preliminary results and embeddings

2.1 Notation

As mentioned in the introduction, we denote the elements of ${\mathbb{R}}^{N}$ for $N\geq 2$ by $x=(\overline{x},x_{N})$, with $\overline{x}=(x_{1},\dots,x_{N-1})\in{\mathbb{R}}^{N-1}$ and $x_{N}\in{\mathbb{R}}$. For any $\gamma\in]0,1]$, we consider the metric $\delta_{\gamma}$ in ${\mathbb{R}}^{N}$ defined by (2). The corresponding open balls with radius $r$, centred at $x$ are defined by

[TABLE]

It is useful to observe that the Lebesgue measure of $B_{\gamma}(x,y)$ equals $2\omega_{N-1}r^{N_{\gamma}}$ where $\omega_{N-1}$ is the measure of the unit ball in ${\mathbb{R}}^{N-1}$ and $N_{\gamma}$ is defined in (3). The diameter of a set $A$ in ${\mathbb{R}}^{N}$ with respect to the metric $\delta_{\gamma}$ is denoted by $\delta_{\gamma}(A)$. Namely, $\delta_{\gamma}(A)=\sup\{\delta_{\gamma}(x,y):\ x,y\in A\}$. Finally, by $\delta_{\gamma}(A,B)$ we denote the distance between two sets $A,B$ in $\mathbb{R}^{N}$ defined by $\sup\{\delta_{\gamma}(x,y):\ x\in A,\ y\in B\}$.

Let $\Omega$ be an open set in ${\mathbb{R}}^{N}$. To avoid taking care of minor details, we shall always assume that $\Omega$ is a domain, which means that $\Omega$ is a connected open set. If $f$ is a real-valued function defined in $\Omega$, we denote by $f_{0}$ the extension by zero of $f$. Namely,

[TABLE]

Let $p\in[1,\infty[$ and $\lambda\in]0,\infty[$. If $f$ is such that $f_{0}\in L^{p}_{loc}({\mathbb{R}}^{N})$ we set

[TABLE]

and

[TABLE]

The corresponding Morrey spaces are defined by

[TABLE]

Similarly, the Campanato spaces are defined by

[TABLE]

Note that if $\Omega$ is bounded then $L^{\lambda}_{p,\gamma}(\Omega),{\mathcal{L}}^{\lambda}_{p,\gamma}(\Omega)\subset L^{p}(\Omega)$, $\|\cdot\|_{L^{\lambda}_{p,\gamma}(\Omega)}$ defines a norm in $L^{\lambda}_{p,\gamma}(\Omega)$ while $|\cdot|_{{\mathcal{L}}^{\lambda}_{p,\gamma}(\Omega)}$ is a seminorm in ${\mathcal{L}}^{\lambda}_{p,\gamma}(\Omega)$. In order to have a normed space, it is customary to endow the Campanato space ${\mathcal{L}}^{\lambda}_{p,\gamma}(\Omega)$ with the norm defined by

[TABLE]

for all $f\in{\mathcal{L}}^{\lambda}_{p,\gamma}(\Omega)$.

We observe that ${L^{\lambda}_{p,1}(\Omega)}$, ${{\mathcal{L}}^{\lambda}_{p,1}(\Omega)}$ are the classical Morrey and Campanato spaces and we recall that ${L^{\lambda}_{p,1}(\Omega)}$ contains only the zero function for $\lambda>1$ and it coincides with $L^{\infty}(\Omega)$ for $\lambda=1$.

2.2 Campanato vs $B\!M\!O$ spaces

In the literature the Campanato space $\mathcal{L}^{1}_{1,1}(\mathbb{R}^{N})$ is better known as the space of functions with bounded mean oscillation and is denoted by $B\!M\!O(\mathbb{R}^{N})$. In the case of a domain $\Omega$ strictly contained in $\mathbb{R}^{N}$, the definition of $B\!M\!O(\Omega)$ is not univocal. One classical definition is the following

[TABLE]

where

[TABLE]

and the supremum is taken on all cubes $Q$ with sides parallel to the coordinate planes (or, equivalently, all Euclidean balls contained in $\Omega$), see e.g., [jones]. Other authors identify $B\!M\!O(\Omega)$ with $\mathcal{L}^{1}_{1,1}(\Omega)$, see e.g., [samko]. See also e.g., [aalto] for a general definition in metric spaces. From the definitions it’s clear that $\mathcal{L}^{1}_{1,1}(\Omega)$ is a subspace of $B\!M\!O(\Omega)$ no matter what $\Omega$ is. Nevertheless, there are cases where the two spaces coincide and cases where they don’t.

Lemma 1**.**

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^{N}$. Then,

[TABLE]

Proof.

It’s enough to prove the inclusion $B\!M\!O(\Omega)\subset\mathcal{L}^{1}_{1,1}(\Omega)$. Under our assumptions on $\Omega$, by [jones, Theorem $1$] it follows that every function $f\in B\!M\!O(\Omega)$ can be extended to a function $\tilde{f}\in B\!M\!O(\mathbb{R}^{N})$. On the other hand, since $\Omega$ satisfies condition (8) with $\gamma=1$, it’s easy to see that the restriction $\tilde{f}|_{\Omega}=f$ belongs to $\mathcal{L}^{1}_{1,1}(\Omega)$ (see Lemma LABEL:restrizioneholder below). ∎

Example 1**.**

Let $\Omega=\mathbb{R}\times]-1,1[$ and $f(x_{1},x_{2})=x_{1}$. It’s easy to see that $f\in B\!M\!O(\Omega)$ (see [bolk, Remark $1$]). However, if we consider a family of squares $Q=]-r,r[^{\,2}$ centred at the origin and having edges of lenght $2r$, $r>1$ we obtain

[TABLE]

which diverges as $r$ goes to infinity. Thus $f\notin\mathcal{L}^{1}_{1,1}(\Omega)$.

2.3 Embedding theorems and equivalent norms

Most results of this paper are based on the validity of the so-called condition (A) on the domain $\Omega$. For $\gamma=1$ this terminology was used already in the paper [campa63]. Namely, it is required that there exists a positive constant $c$ independent of $x$ such that

[TABLE]

for all $x\in\overline{\Omega}$ and $r\in]0,\delta_{\gamma}(\Omega)]$.

Remark 1**.**

If the domain $\Omega$ satisfies inequality (8), we can define the spaces $\mathcal{L}^{\lambda}_{p,\gamma}(\Omega)$ (respectively $L^{\lambda}_{p,\gamma}(\Omega)$) by replacing the quantity $|B_{\gamma}(x,r)\cap\Omega|^{\lambda}$ by $r^{\lambda N_{\gamma}}$ in (6) (respectively (5)).

In particular, under assumption (8) Campanato’s embedding Theorem holds. Here $C^{0,\alpha}(\overline{\Omega},\delta_{\gamma})$ denotes the space of Hölder continuous functions with respect to the metric $\delta_{\gamma}$, with exponent $\alpha\in]0,1]$, endowed with the norm

[TABLE]

Theorem 1**.**

(Campanato-Da Prato-John-Nirenberg) Let $\Omega$ be a bounded domain in $\mathbb{R}^{N}$ satisfying condition (8). The following statements hold:

• $(i)$

If $0<\lambda<1$ then $\mathcal{L}^{\lambda}_{p,\gamma}(\Omega)\simeq L^{\lambda}_{p,\gamma}(\Omega)$;

• $(ii)$

If $\lambda=1$ then $\mathcal{L}^{\lambda}_{p,\gamma}(\Omega)\simeq\mathcal{L}^{\lambda}_{1,\gamma}(\Omega)$ for all $p\in[1,\infty[$;

• $(ii)$

If $\lambda>1$ then $\mathcal{L}^{\lambda}_{p,\gamma}(\Omega)\simeq C^{0,\alpha}(\overline{\Omega},\delta_{\gamma})$ where $\alpha=\frac{N_{\gamma}(\lambda-1)}{p}.$

It is understood that if $\gamma\alpha>1$ the space $C^{0,\alpha}(\overline{\Omega},\delta_{\gamma})$ consists of constant functions only. Here the symbol $\simeq$ is used to indicate that two spaces coincide (possibly identifying functions which coincide almost everywhere) and the corresponding norms are equivalent. The proofs of statements $(i)$ and $(iii)$ in the previous theorem can be found in [dapra]. The proof of statement $(ii)$ can be performed by using the celebrated John-Niremberg inequality and (in a form or another) is known in the literature for the Euclidean metric, see e.g., [samko, Thm. 4.3] and [bolk, Thm. 14]. Since we do not have a specific reference for a proof of statement $(ii)$ above for $\gamma\neq 1$, we find it convenient to include it here.

Proof.

We prove statement $(ii)$ in Theorem 1. Let $f\in\mathcal{L}^{1}_{p,\gamma}(\Omega)$. By Hölder’s inequality, we deduce that

[TABLE]

which proves that $\mathcal{L}^{1}_{p,\gamma}(\Omega)\subset\mathcal{L}^{1}_{1,\gamma}(\Omega)$. The corresponding estimate for the norms which proves that this embedding is continuous can be easily deduced by the previous inequality and Hölder’s inequality.

We now prove the reverse inclusion. Let $f\in\mathcal{L}^{1}_{1,\gamma}(\Omega)$. Then, by the general result in [aalto, Theorem $5.2$] applied to the metric space $(\Omega,{\delta_{\gamma}}_{|\Omega})$, it follows that

[TABLE]

where $a=|f|_{\mathcal{L}^{1}_{1,\gamma}(\Omega)}$, $B_{\gamma}$ is any ball in the metric space $(\Omega,{\delta_{\gamma}}_{|\Omega})$ and $c_{1},c_{2}$ are positive constants independent of $f$, $t$ and $B_{\gamma}$. (Note that since $\Omega$ satisfies (8) then the measure ${\delta_{\gamma}}_{|\Omega}$ is doubling as required in [aalto].) Thus,

[TABLE]

where we have set $c_{3}=c_{1}c^{-p}_{2}\Gamma(p+1)$. In order to estimate the $L^{p}$-norm of the function $f$, we simply use triangle inequality to deduce by (10) that

[TABLE]

By (10) and inequality (11) applied to a ball with radius sufficiently large to ensure444recall that the balls $B_{\gamma}$ used in this part of the proof are restrictions to $\Omega$ of balls in ${\mathbb{R}}^{N}$ endowed with the metric $\delta_{\gamma}$ that $B_{\gamma}=\Omega$, we deduce that $\mathcal{L}^{1}_{1,\gamma}(\Omega)\subset\mathcal{L}^{1}_{p,\gamma}(\Omega)$, the embedding being continuous. ∎

The previous theorems points out also the fact that the spaces $\mathcal{L}^{\lambda}_{p,\gamma}(\Omega)$ heavily depends on $\gamma$. The following example clarifies this by showing that in general $\mathcal{L}^{\lambda}_{p,\gamma}(\Omega)\neq\mathcal{L}^{\lambda}_{p,1}(\Omega)$ if $\gamma\neq 1$.