Sobolev embeddings with weights in complete riemannian manifolds

TL;DR

This paper establishes weighted Sobolev embedding theorems and Gaffney's inequality for vector bundles on complete Riemannian manifolds, improving classical results under weaker geometric assumptions.

Contribution

It introduces weighted Sobolev embeddings and Gaffney's inequality in complete Riemannian manifolds, extending and strengthening previous theorems under weaker geometric conditions.

Findings

Weighted Sobolev embeddings for vector bundles proved

General Gaffney's inequality with weights established

Improved classical Sobolev embeddings under weak bounded geometry

Abstract

We prove Sobolev embedding Theorems with weights for vector bundles in a complete riemannian manifold. We also get general Gaffney's inequality with weights. As a consequence, under a "weak bounded geometry" hypothesis, we improve classical Sobolev embedding Theorems for vector bundles in a complete riemannian manifold. We also improve known results on Gaffney's inequality in a complete riemannian manifold.

Equations

$${\Gamma }_{ij}^k=\frac{1}{2}{g}^{il}(\frac{\partial g_{kl}}{\partial x^j}+\frac{\partial g_{lj}}{\partial x^k}-\frac{\partial g_{jk}}{\partial x^l}).$$

$$\left|\nabla \left|{\nabla }^m\omega \right|\right|\le \left|{\nabla }^{m+1}\omega \right|.$$

$${\Vert {\nabla }^m\omega \Vert }_{L^{s_0}(M,w^{\prime })}\le A({\Vert {\nabla }^{m+1}\omega \Vert }_{L^{r_0}(M,w)}+{\Vert {\nabla }^m\omega \Vert }_{L^{r_0}(M,w)}).$$

$$\forall v\in W^{1,r}(B_1),{\Vert v\Vert }_{L^t(B_1)}\le C{\Vert v\Vert }_{W^{1,r}(B_1)}$$

$${\Vert \partial u\Vert }_{L^r(B_R)}=R^{-1+n/r}{\Vert \partial v\Vert }_{L^r(B_1)}.$$

$${\Vert u\Vert }_{L^t(B_R)}=R^{n/t}{\Vert v\Vert }_{L^t(B_1)}\le C{R}^{n/t}{\Vert v\Vert }_{W^{1,r}(B_1)}.$$

$$\left|\nabla \omega (y)\right|\le \left|\partial v(z)\right|+C\eta \left|v(z)\right|.$$

$$\left|{\nabla }_M(\chi \omega )\right|\le \left|\partial (\chi \omega )\right|+c\frac{ϵ}{R}\left|\chi \omega \right|.$$

$$\sum _{x\in \mathcal{D}(ϵ),j\le 1,}R(x{)}^{\gamma }{\Vert {\nabla }^{j}u\Vert }_{L_G^r(B(x))}^r\le C{\Vert u\Vert }_{W_G^{1,r}(M,w)}^r.$$

$$\sum _{B\in {\mathcal{C}}_{ϵ},j=0,1}R(x{)}^{\gamma }{\Vert {\nabla }^{j}u\Vert }_{L_G^r(B)}^r\simeq {\Vert u\Vert }_{W_G^{1,r}(M,w)}^r$$

$$R(x{)}^{2+\gamma /r}{\Vert u\Vert }_{L_G^s(B(x,R))}\le CR(x{)}^{\gamma /r}{\Vert u\Vert }_{W_G^{1,r}(B(x,R))}.$$

$$\sum _{B\in {\mathcal{C}}_{ϵ}}R(x{)}^{\nu }{\Vert u\Vert }_{L_G^s(B)}^s\le C\sum _{B\in {\mathcal{C}}_{ϵ}}R(x{)}^{s\gamma /r}{\Vert u\Vert }_{W_G^{1,r}(B(x,R))}^s.$$

Full text

1. 1 Introduction.

2. 2 Admissible balls.

3. 3 Vector bundle.

4. 4 Sobolev spaces for sections of $G$ with weight.

5. 5 Local estimates.

6. 5.1 Sobolev comparison estimates for functions.

7. 5.2 Sobolev comparison estimates for sections of $G.$

8. 5.3 Local Gaffney type inequality in $L^{r}.$

9. 6 Vitali covering.

10. 7 Global results.

11. 7.1 Global estimates for sections of $G.$

12. 7.2 Global Gaffney type inequality in $L^{r}.$

13. 8 Applications.

14. 8.1 Weak bounded geometry

15. 8.2 Examples of manifolds of bounded geometry.

16. 8.3 Compact riemannian manifold with smooth boundary.

17. 8.4 Hyperbolic manifolds.

1 Introduction.

Let $(M,g)$ be a complete riemann manifold. The Sobolev inequalities in $M$ for functions play a major role in the study of differential operators and nonlinear functional analysis. They are valid in ${\mathbb{R}}^{n}$ or if $M$ is compact. More generally we have:

Theorem 1.1**.**

Let $(M,g)$ be a complete riemannian manifold of dimension $n$ with Ricci curvature bounded from below. The Sobolev embeddings for functions are valid for $M$ if and only if there is a uniform lower bound for the volume of balls which is independent of their center, namely if and only if $\inf_{x\in M}\mathrm{V}\mathrm{o}\mathrm{l}(B(x,1))>0.$

The necessity is a well known fact, see p.18 in [Hebey, 1996] and was generalised by Carron [Carron, 1994].

The sufficiency of Theorem 1.1 was done by Varopoulos [Varopoulos, 1989], see also Theorem 3.18, p. 37 in [Hebey, 1996], based on the work of Coulhon and Saloff-Coste [Coulhon and Saloff-Coste, 1993].

In this work we study the Sobolev embeddings for the case of vector bundles over $M.$

So let $G:=(H,\pi,M)$ be a complex ${\mathcal{C}}^{m}$ vector bundle over $M$ of rank $N$ with fiber $H$ with a smooth scalar product $(\ ,\ ).$ We make the hypothesis that we have a metric connection $\nabla^{G}$ on $G,$ i.e. with $d$ the exterior derivation on $M,$ we have $d(u,v)=(\nabla^{G}u,v)+(u,\nabla^{G}v)$ for any smooth sections $u,\ v$ of $G.$ We shall call a vector bundle with these properties an adapted vector bundle.

We shall introduce weights, given by the geometry, on a complete riemannian manifold $(M,g),$ in order to have Sobolev embeddings on vector bundles, always valid with these weights, without any curvature conditions nor volume control.

In order to state the results, we need some definitions.

Definition 1.2**.**

Let $(M,g)$ be a riemannian manifold and $\displaystyle x\in M.$ We shall say that the geodesic ball $\displaystyle B(x,R)$ is $(0,\epsilon)$-admissible if there is a chart $\displaystyle(B(x,R),\varphi)$ such that:

() $\displaystyle(1-\epsilon)\delta_{ij}\leq g_{ij}\leq(1+\epsilon)\delta_{ij}$ in $\displaystyle B(x,R)$ as bilinear form.*

We shall say that the geodesic ball $\displaystyle B(x,R)$ is $(1,\epsilon)$-admissible if moreover

(**) $\displaystyle\ \sum_{\left|{\beta}\right|=1}{R\sup\ _{i,j=1,...,n,\ y\in B_{x}(R)}\left|{\partial^{\beta}g_{ij}(y)}\right|}\leq\epsilon.$

We shall denote ${\mathcal{A}}(0,\epsilon)$ the set of $(0,\epsilon)$-admissible balls on $M$ and ${\mathcal{A}}(1,\epsilon)$ the set of $(1,\epsilon)$-admissible balls on $M.$

The $(0,\epsilon)$-admissible balls will be adapted to functions and the $(1,\epsilon)$-admissible balls will be adapted to sections of the vector bundle $G.$

Definition 1.3**.**

Let $\displaystyle x\in M,$ we set $R^{\prime}(x)=\sup\ \{R>0::B(x,R)\in{\mathcal{A}}(\epsilon)\}.$ We shall say that $\displaystyle R_{\epsilon}(x):=\min\ (1,R^{\prime}(x)/2)$ is the $\epsilon$-admissible radius at $\displaystyle x.$

Here ${\mathcal{A}}(\epsilon)$ will be either ${\mathcal{A}}(0,\epsilon)$ or ${\mathcal{A}}(1,\epsilon)$ depending on the context, functions or sections of $G.$ The notation $R_{\epsilon}(x)$ will means either $R_{0,\epsilon}(x)$ or $R_{1,\epsilon}(x)$ depending on the choice of ${\mathcal{A}}(\epsilon).$

Our general result with weights is:

Theorem 1.4**.**

Let $(M,g)$ be a complete riemannian manifold of dimension $n.$ Let $G:=(H,\pi,M)$ be a complex smooth adapted vector bundle over $M.$ Let $r\geq 1,\ m\geq k\geq 0$ and $\frac{1}{s}=\frac{1}{r}-\frac{(m-k)}{n}>0.$ Let $\displaystyle w(x):=R_{\epsilon}(x)^{\gamma}$ and $\displaystyle w^{\prime}:=R_{\epsilon}(x)^{\nu}$ with $\displaystyle\nu:=s(2+\gamma/r).$ Then $\displaystyle W_{G}^{m,r}(M,w)$ is embedded in $\displaystyle W_{G}^{k,s}(M,w^{\prime})$ and:

$\displaystyle\forall u\in W_{G}^{m,r}(M,w),\ {\left\|{u}\right\|}_{W_{G}^{k,s}(M,w^{\prime})}\leq C{\left\|{u}\right\|}_{W_{G}^{m,r}(M,w)}.$**

Because the admissible radius $R_{\epsilon}(x)$ is always smaller than $1,$ to get rid of the weights we just need that: $\exists\delta>0::\forall x\in M,\ R(x)\geq\delta.$

This is precisely what is given by a Theorem of Hebey-Herzlich  [Hebey and Herzlich, 1997, Corollary, p. 7], which has the easy Corollary:

Corollary 1.5**.**

Let $(M,g)$ be a complete riemannian manifold. If the injectivity radius verifies $r_{inj}(x)\geq i>0$ and the Ricci curvature verifies $Rc_{(M,g)}(x)\geq\lambda g_{x}$ for some $\lambda\in{\mathbb{R}}$ and all $x\in M,$ then there exists a positive constant $\delta>0,$ depending only on $n,\epsilon,\lambda,i$ such that for any $x\in M,\ R_{0,\epsilon}(x)\geq\delta.$

If moreover we have $\left|{Rc_{(M,g)}(x)}\right|\leq C$ for all $x\in M,$ then there exists a positive constant $\delta>0,$ depending only on $n,\epsilon,i$ and $C,$ such that for any $x\in M,\ R_{1,\epsilon}(x)\geq\delta.$

So the following results will be consequences of the Theorem 1.4 and of the Theorem of Hebey-Herzlich.

To state them precisely we shall need to weaken the definition of bounded geometry.

Definition 1.6**.**

A riemannian manifold $M$ has $k$-order weak bounded geometry if:

$\bullet$* the injectivity radius $r(x)$ at $x\in M$ is bounded below by some constant $\delta>0$ for any $\displaystyle x\in M$*

$\bullet$* for $0\leq j\leq k,$ the covariant derivatives $\nabla^{j}R_{c}$ of the Ricci curvature tensor are bounded in $L^{\infty}(M)$ norm.*

Recall the classical Theorem of Cantor [Cantor, 1974]:

Theorem 1.7**.**

Let $(M,g)$ be a complete riemannian manifold. Let $G:=(H,\pi,M)$ be a complex smooth adapted vector bundle over $M.$ Suppose $(M,g)$ verifies:

C1: the injectivity radius of $M$ is bounded away from zero.

C2: There is a $\delta$ such that for each $x\in M$ and $V,W\in T_{x}M,$ the sectional curvature $\left|{K_{x}(V,W)}\right|<\delta.$

Let $\displaystyle 0\leq k<m$ and $\displaystyle 1/s=1/r-(m-k)/n.$ Let $\displaystyle u\in W^{m,r}_{G}(M).$ Then we have $u\in W^{k,s}_{G}(M)$ with the control:

$\displaystyle{\left\|{u}\right\|}^{s}_{W^{k,s}_{G}(M)}\leq C{\left\|{u}\right\|}^{r}_{W^{m,r}_{G}(M)}.$**

Here we prove:

Theorem 1.8**.**

Let $(M,g)$ be a complete riemannian manifold. Let $G:=(H,\pi,M)$ be a complex smooth adapted vector bundle over $M.$ Suppose $(M,g)$ has a [math]-order weak bounded geometry. Let $\displaystyle 0\leq k<m$ and $\displaystyle 1/s=1/r-(m-k)/n.$ Let $\displaystyle u\in W^{m,r}_{G}(M).$ Then we have $u\in W^{k,s}_{G}(M)$ with the control:

$\displaystyle{\left\|{u}\right\|}^{s}_{W^{k,s}_{G}(M)}\leq C{\left\|{u}\right\|}^{r}_{W^{m,r}_{G}(M)}.$**

In the case of functions instead of sections of $G,$ the conditions are weaker: if the injectivity radius of $M$ is bounded away from zero and if the Ricci curvature verifies $Rc_{(M,g)}(x)\geq\lambda g_{x}$ for some $\lambda\in{\mathbb{R}}$ and all $x\in M,$ we get

$\displaystyle{\left\|{u}\right\|}^{s}_{W^{k,s}(M)}\leq C{\left\|{u}\right\|}^{r}_{W^{m,r}(M)}.$**

This improves the Theorem by Cantor because he used the hypothesis that all the sectional curvatures are bounded and here we need only that the Ricci curvature be bounded.

We also prove a global Gaffney’s type inequality:

Theorem 1.9**.**

Let $(M,g)$ be a complete riemannian manifold. Let $d$ be the exterior derivation on $M$ and $d^{*}$ its formal adjoint. Let $\omega$ be a $p$-differential form in $M.$ If $(M,g)$ has a [math]-order weak bounded geometry, then we have, with $r\geq 1$:

$\displaystyle{\left\|{\omega}\right\|}_{W_{p}^{1,r}(M)}\leq C({\left\|{d(\omega)}\right\|}_{L_{p+1}^{r}(M)}+{\left\|{d^{*}(\omega)}\right\|}_{L_{p-1}^{r}(M)}+{\left\|{\omega}\right\|}_{L_{p}^{r}(M)}).$**

And, as an easy corollary:

Corollary 1.10**.**

Let $(M,g)$ be a complete riemannian manifold with a [math]-order weak bounded geometry. Let $r\geq 1$and let $\omega$ be a $p$-differential form in $M.$ We have:

$\displaystyle{\left\|{\omega}\right\|}_{L_{p}^{s}(M)}\leq C({\left\|{d(\omega)}\right\|}_{L_{p+1}^{r}(M)}+{\left\|{d^{*}(\omega)}\right\|}_{L_{p-1}^{r}(M)}+{\left\|{\omega}\right\|}_{L_{p}^{r}(M)})$**

with $\frac{1}{s}=\frac{1}{r}-\frac{1}{n}>0.$

N. Lohoué [Lohoué, 1985] proved the same result under the stronger hypothesis that $(M,g)$ has a $2$-order bounded geometry plus some other hypotheses on the laplacian and the range of $r.$ Here already [math]-order weak bounded geometry is enough.

This work is presented as follow:

$\bullet$ In the next Section we study the main properties of the $\epsilon$-admissible balls.

$\bullet$ In Section 3 we define the vector bundle $G$ we are interested in and precise the metric connexion $\nabla^{G}$ on it we shall use.

$\bullet$ In Section 4 we define the Sobolev spaces of smooth sections of $G,$ with weights. We prove in this Section a generalisation of a nice result of T. Aubin [Aubin, 1982] which says that, in order to prove Sobolev embeddings for sections of $G$ with weights, we have just to prove them at the first level. This is crucial for our estimates.

$\bullet$ In Section 5 we prove the local estimates, i.e. for sections of $G$ in the $\epsilon$-admissible balls.

$\bullet$ In SubSection 5.3 we also prove a local Gaffney type inequality, using a result by C. Scott [Scott, 1995].

$\bullet$Then in order to get global results we group the $\epsilon$-admissible balls via a Vitali type covering in Section 6.

$\bullet$ In Section 7 we prove the global estimates for functions, sections of $G$ and Gaffney type in $L^{r}.$

$\bullet$ In Section 8 we improve the classical Sobolev embeddings to the case of riemannian manifolds with weak bounded geometry, by use of a Theorem of Hebey-Herzlich [Hebey and Herzlich, 1997]. This implies the validity of Sobolev embeddings for vector bundles in compact riemannian manifold without boundary.

$\bullet$ In SubSection 8.3 we deduce from the compact case without boundary the validity of Sobolev embeddings for vector bundles in compact riemannian manifold with smooth boundary. We use here the method of the "double" manifold.

$\bullet$ Finally in SubSection 8.4 we introduce the lifted doubling property and we study the case of hyperbolic manifolds.

2 Admissible balls.

Recall the definition of the admissible radius:

Definition 2.1**.**

Let $\displaystyle x\in M,$ we set $R^{\prime}(x)=\sup\ \{R>0::B(x,R)\in{\mathcal{A}}(\epsilon)\}.$ We shall say that $\displaystyle R_{\epsilon}(x):=\min\ (1,R^{\prime}(x)/2)$ is the $\epsilon$-admissible radius at $\displaystyle x.$

Then clearly if $B(x,R)\in{\mathcal{A}}(\epsilon),$ i.e. is $\epsilon$-admissible, then so is $B(x,S)$ if $S\leq R.$

Remark 2.2**.**

Let $\displaystyle x,y\in M.$ Suppose that $R^{\prime}(x)>d_{g}(x,y),$ where $d_{g}(x,y)$ is the riemannian distance between $x$ and $y.$ Consider the ball $B(y,\rho)$ of center $y$ and radius $\rho:=R^{\prime}(x)-d_{g}(x,y).$ This ball is contained in $B(x,R^{\prime}(x))$ hence, by definition of $R^{\prime}(x),$ we have that all the points in $B(y,\rho)$ verify the conditions 1) and 2) so, by definition of $R^{\prime}(y),$ we have that $R^{\prime}(y)\geq R^{\prime}(x)-d_{g}(x,y).$ If $\displaystyle R^{\prime}(x)\leq d_{g}(x,y)$ this is also true because $\displaystyle R^{\prime}(y)>0.$ Exchanging $x$ and $y$ we get that $\left|{R^{\prime}(y)-R^{\prime}(x)}\right|\leq d_{g}(x,y).$

Hence $R^{\prime}(x)$ is $1$-lipschitzian so it is continuous. So the $\epsilon$-admissible radius $R_{\epsilon}(x)$ is also continuous.

Remark 2.3**.**

Because on our admissible ball $B(x,R_{\epsilon}(x))$ there is a diffeomorphism from $\displaystyle B(x,R)$ to $\varphi(B(x,R))\subset{\mathbb{R}}^{n},$ i.e. on an open set in the tangent space $T_{x}M,$ we get that the injectivity radius $r_{inj}(x)$ always verifies $\displaystyle r_{inj}(x)\geq R_{\epsilon}(x).$

Lemma 2.4**.**

(Slow variation of the admissible radius) Let $(M,g)$ be a riemannian manifold then with $R(x)=R_{\epsilon}(x)=$ the $\epsilon$-admissible radius at $x\in M.$ We get:

$\forall y\in B(x,R(x))$* we have $R(x)/2\leq R(y)\leq 2R(x).$*

Proof.

Let $x,y\in M$ and $d(x,y)$ the riemannian distance on $(M,g).$ Let $y\in B(x,R(x))$ then $d(x,y)\leq R(x)$ and suppose first that $R(x)\geq R(y).$ Then, because $R(x)=R^{\prime}(x)/2,$ we get $y\in B(x,R^{\prime}(x)/2)$ hence we have $B(y,R^{\prime}(x)/2)\subset B(x,R^{\prime}(x)).$ But by the definition of $R^{\prime}(x),$ the ball $\displaystyle B(x,R^{\prime}(x))$ is admissible and this implies that the ball $\displaystyle B(y,R^{\prime}(x)/2)$ is also admissible for exactly the same constants and the same chart; this implies that $R^{\prime}(y)\geq R^{\prime}(x)/2$ hence $R(y)\geq R(x)/2,$ so $R(x)\geq R(y)\geq R(x)/2.$

If $R(x)\leq R(y)$ then $d(x,y)\leq R(x)\Rightarrow d(x,y)\leq R(y)\Rightarrow x\in B(y,R^{\prime}(y)/2)\Rightarrow B(x,R^{\prime}(y)/2)\subset B(y,R^{\prime}(y)).$ Hence the same way as above we get $R(y)\geq R(x)\geq R(y)/2\Rightarrow R(y)\leq 2R(x).$ So in any case we proved that

$\forall y\in B(x,R(x))$ we have $R(x)/2\leq R(y)\leq 2R(x).$ $\hfill\blacksquare$

3 Vector bundle.

Let $(M,g)$ be a complete riemannian manifold and let $G:=(H,\pi,M)$ be an adapted complex ${\mathcal{C}}^{m}$ vector bundle over $M$ of rank $N$ with fiber $H.$ Recall that this means that $G$ has a smooth scalar product $(\ ,\ )$ and a metric connection $\nabla^{G}:{\mathcal{C}}^{\infty}(M,G)\rightarrow{\mathcal{C}}^{\infty}(M,G\otimes T^{*}M),$ i.e. verifying $\displaystyle d(u,v)=(\nabla^{G}u,v)+(u,\nabla^{G}v),$ where $d$ is the exterior derivative on $M.$ See  [Taylor, 2000, Section 13].

Lemma 3.1**.**

The $\epsilon$-admissible balls $B(x,R_{\epsilon}(x))$ trivialise the bundle $G.$

Proof.

Because if $B(x,R)$ is an $\epsilon$-admissible ball, we have by Remark 2.3 that $R\leq r_{inj}(x).$ Then, one can choose a local frame field for $G$ on $\displaystyle B(x,R)$ by radial parallel translation, as done in [Taylor, 2000, Section 13, p.86-87], see also [Mazzucato and Nistor, 2006, p. 4, eq. (1.3)]. This means that the $\epsilon$-admissible ball also trivialises the bundle $G.$ $\hfill\blacksquare$

If $\partial_{j}:=\partial/\partial x_{j}$ in a coordinate system on, say $B(x_{0},R),$ and with a local frame $\{e_{\alpha}\}_{\alpha=1,...,N},$ we have, for a smooth sections of $G,$ $u=u^{\alpha}e_{\alpha}$ with the Einstein summation convention. We set:

$\displaystyle\ \nabla_{\partial_{j}}u=(\partial_{j}u^{\alpha}+u^{\beta}\Gamma^{G,\alpha}_{\beta j})e_{\alpha},$

the Christoffel coefficients $\displaystyle\Gamma^{G,\alpha}_{\beta j}$ being defined by $\displaystyle\nabla_{\partial_{j}}e_{\beta}=\Gamma^{G,\alpha}_{\beta j}e_{\alpha}.$

We shall make the following Control by the Metric Tensor hypothesis, for $B(x_{0},R)\in{\mathcal{A}}(1,\epsilon)$:

(CMT) $\forall x\in B(x_{0},R),\ \left|{\Gamma^{G,\alpha}_{\beta j}(x)}\right|\leq C(n,G,\epsilon)\sum_{\left|{\beta}\right|=1}{\sup\ _{i,j=1,...,n,}\left|{\partial^{\beta}g_{ij}(x)}\right|},$

the constant $C$ depending only on $n,\epsilon$ and $G.$

This hypothesis is natural in the sense that it is true for tensor bundles over $M.$

Lemma 3.2**.**

Let $F$ be a tensor bundle over $M.$ Then the hypothesis (CMT) is true.

Proof.

Let $\displaystyle\Gamma^{k}_{lj}$ be the Christoffel coefficients of the Levi-Civita connexion on the tangent bundle $TM.$ We have

[TABLE]

Now on $B(x_{0},R)\in{\mathcal{A}}(\epsilon),$ we have $(1-\epsilon)\delta_{ij}\leq g_{ij}\leq(1+\epsilon)\delta_{ij}$ as bilinear forms. Hence

$\displaystyle\forall x\in B(x_{0},R),\ \ \left|{\Gamma^{k}_{ij}(x)}\right|\leq\frac{3}{2}(1+\epsilon)\sum_{\left|{\beta}\right|=1}{\sup\ _{i,j=1,...,n,}\left|{\partial^{\beta}g_{ij}(x)}\right|}$

in a coordinates chart on $B(x_{0},R).$

Let $\omega:=\omega_{k}dx^{k}$ be a $1$-form in this chart. Then $\nabla_{\partial_{j}}\omega=(\partial_{j}\omega_{k}-\Gamma^{l}_{kj}a_{l})dx^{k}.$ In particular we have $\nabla_{\partial_{j}}dx^{m}=(-\Gamma^{m}_{kj})dx^{k}.$ Hence (CMT) is true. The same if $X$ is a vector field, $X:=X^{k}\partial_{k}$ we have $\nabla_{\partial_{j}}X=(\partial_{j}X^{k}+\Gamma^{k}_{lj}X^{l})\partial_{k}.$ So $\nabla_{\partial_{j}}\partial_{l}=(\Gamma^{k}_{lj})\partial_{k}$ and (CMT) is still true.

Now on $(p,q)$ tensors, using the fact that $\nabla$ is a derivation, we have that

$\displaystyle\nabla_{\partial_{j}}(dx^{k_{1}}\otimes\cdot\cdot\cdot\otimes dx^{k_{q}}\otimes\partial_{l_{1}}\otimes\cdot\cdot\cdot\otimes\partial_{l_{p}})=$

$\displaystyle=(\nabla_{\partial_{j}}dx^{k_{1}})\otimes\cdot\cdot\cdot\otimes dx^{k_{q}}\otimes\partial_{l_{1}}\otimes\cdot\cdot\cdot\otimes\partial_{l_{p}}+\cdot\cdot\cdot$

$\displaystyle\cdot\cdot\cdot+dx^{k_{1}}\otimes\cdot\cdot\cdot\otimes dx^{k_{q}}\otimes\partial_{l_{1}}\otimes\cdot\cdot\cdot\otimes(\nabla_{\partial_{j}}\partial_{l_{p}})$

is a linear combination of $\displaystyle\Gamma^{k}_{lj}$ with constant coefficients, hence the (CMT) is also true, with a constant depending only on the dimension $n$ of $M$ and $p,q,\epsilon.$

A special mention for the bundle $\Lambda^{p}(M)$ of $p$-forms on $M,$ which is a sub bundle of the $(0,p)$ tensors bundle. It also has the (CMT) property.

The proof is complete. $\hfill\blacksquare$

4 Sobolev spaces for sections of $G$ with weight.

We have seen that $\nabla^{G}:\ {\mathcal{C}}^{\infty}(M,G)\rightarrow{\mathcal{C}}^{\infty}(M,G\otimes T^{*}M).$ On the tensor product of two Hilbert spaces we put the canonical scalar product $(u\otimes\omega,v\otimes\mu):=(u,v)(\omega,\mu),$ with $u\otimes\omega\in G\otimes T^{*}M,$ and completed by linearity to all elements of the tensor product. On $T^{*}M$ we have the Levi-Civita connexion $\nabla^{M},$ which is of course a metric one, and on $G$ we have the metric connexion $\nabla^{G}$ so we define a connexion on the tensor product $G\otimes T^{*}M$:

$\displaystyle\nabla^{G\otimes T^{*}M}(u\otimes\omega)=(\nabla^{G}u)\otimes\omega+u\otimes(\nabla^{T^{*}M}\omega)$

by asking that this connexion be a derivation. We get easily that

$\displaystyle\nabla^{G\otimes T^{*}M}\ :\ {\mathcal{C}}^{\infty}(M,G\otimes T^{*}M)\rightarrow{\mathcal{C}}^{\infty}(M,G\otimes(T^{*}M)^{\otimes 2})$

is still a metric connexion, i.e.

$\displaystyle\ d(u\otimes\omega,v\otimes\mu)=(\nabla^{G\otimes T^{*}M}(u\otimes\omega),\ v\otimes\mu)+(u\otimes\omega,\ \nabla^{G\otimes T^{*}M}(v\otimes\mu)).$

We define by iteration $\nabla^{j}u:=\nabla(\nabla^{j-1}u)$ on the section $u$ of $G$ and the associated pointwise scalar product $(\nabla^{j}u(x),\nabla^{j}v(x))$ which is defined on $G\otimes(T^{*}M)^{\otimes j},$ with again the metric connection

$\displaystyle d(\nabla^{j}u,\nabla^{j}v)(x)=(\nabla^{j+1}u,\nabla^{j}v)(x)+(\nabla^{j}u,\nabla^{j+1}v)(x).$

Let $w$ be a weight on $M,$ i.e. a positive measurable function on $M.$ If $k\in{\mathbb{N}}$ and $r\geq 1$ are given, we denote by ${\mathcal{C}}^{k,r}_{G}(M,w)$ the space of smooth sections of $G\ \omega\in{\mathcal{C}}^{\infty}(M)$ such that $\ \left|{\nabla^{j}\omega}\right|\in L^{r}(M,w)$ for $j=0,...,k$ with the pointwise modulus associated to the pointwise scalar product. Hence

$\displaystyle{\mathcal{C}}^{k,r}_{G}(M,w):=\{\omega\in{\mathcal{C}}_{G}^{\infty}(M),\ \forall j=0,...,k,\ \int_{M}{\left|{\nabla^{j}\omega}\right|^{r}(x)w(x)dv(x)}<\infty\},$

with $dv$ the volume measure on $(M,g).$

Now we have, see M. Cantor  [Cantor, 1974, Definition 1 & 2, p. 240] for the case without weight:

Definition 4.1**.**

The Sobolev space $W_{G}^{k,r}(M,w)$ is the completion of ${\mathcal{C}}^{k,r}_{G}(M,w)$ with respect to the norm:

$\displaystyle{\left\|{\omega}\right\|}_{W_{G}^{k,r}(M,w)}=\sum_{j=0}^{k}{{\left({\int_{M}{\left|{\nabla^{j}\omega(x)}\right|^{r}w(x)dv(x)}}\right)}^{1/r}}.$**

The usual case is when $w\equiv 1.$ Then we write simply $W_{G}^{k,r}(M).$

We shall apply these well known facts to generalise a nice result of T. Aubin.

Let $w(x),\ w^{\prime}(x)$ be weights on the complete riemannian manifold $(M,g).$ We have:

Proposition 4.2**.**

Let $(M,g)$ be a complete riemannian manifold. If $W_{G}^{1,r_{0}}(M,w)$ is embedded in $L_{G}^{s_{0}}(M,w^{\prime}),$ with $\frac{1}{s_{0}}=\frac{1}{r_{0}}-\frac{1}{n}\ (1\leq r_{0}<n),$ then $W_{G}^{k,r}(M,w)$ is embedded in $W_{G}^{l,s_{l}}(M,w^{\prime}),$ with $\frac{1}{s_{l}}=\frac{1}{r}-\frac{(k-l)}{n}>0.$

Proof.

We shall copy the proof of Proposition 2.11, p. 36 in [Aubin, 1982], replacing $L^{s}(M)$ by $L_{G}^{s}(M,w^{\prime})$ and $W^{k,r}(M)$ by $W_{G}^{k,r}(M,w)$ and simplifying a little bit the argument.

Let $m$ be an integer and let $\omega\in{\mathcal{C}}_{G}^{m+1}.$ We have pointwise:

[TABLE]

To see this, by $\left|{\nabla^{m}\omega}\right|^{2}(x)=(\nabla^{m}\omega,\nabla^{m}\omega)(x),$ we have

$\displaystyle\nabla\left|{\nabla^{m}\psi}\right|^{2}=\nabla(\nabla^{m}\omega,\nabla^{m}\omega)(x)=2(\nabla^{m+1}\omega,\nabla^{m}\omega),$

the last equality because $\nabla$ is a metric connection.

By the Cauchy-Schwartz inequality, we get

$\displaystyle\ \left|{\nabla\left|{\nabla^{m}\omega}\right|^{2}}\right|(x)\leq 2\left|{\nabla^{m+1}\omega}\right|\left|{\nabla^{m}\omega}\right|(x).$

Setting $F(x):=\left|{\nabla^{m}\psi}\right|(x),$ because $\nabla$ is a derivation, we also have $\nabla F^{2}(x)=2F(x)\nabla F(x),$ hence:

$\displaystyle\left|{\nabla\left|{\nabla^{r}\psi}\right|^{2}}\right|=2\left|{F}\right|\left|{\nabla\left|{F}\right|}\right|\leq 2\left|{\nabla^{m+1}\omega}\right|\left|{\nabla^{m}\omega}\right|,$

so $\left|{\nabla\left|{\nabla^{m}\omega}\right|}\right|\leq\left|{\nabla^{m+1}\omega}\right|.$

Since $W_{G}^{1,r_{0}}(M,w)$ is embedded in $L_{G}^{s_{0}}(M,w^{\prime}),$ there exists a constant $A,$ such that for all $\varphi\in W_{G}^{1,r_{0}}(M,w)$: (for now on, we do not indicate the subscript to ease the notation.)

$\displaystyle\ {\left\|{\varphi}\right\|}_{L^{s_{0}}(M,w^{\prime})}\leq A({\left\|{\nabla\varphi}\right\|}_{L^{r_{0}}(M,w)}+{\left\|{\varphi}\right\|}_{L^{r_{0}}(M,w)}).$

Let us apply this inequality with $\varphi=\left|{\nabla^{m}\omega}\right|,$ assuming $\varphi,$ which is a function now, belongs to $W^{1,r_{0}}(M,w)$:

$\displaystyle{\left\|{\nabla^{m}\omega}\right\|}_{L^{s_{0}}(M,w^{\prime})}\leq A({\left\|{\nabla\left|{\nabla^{m}\omega}\right|}\right\|}_{L^{r_{0}}(M,w)}+{\left\|{\nabla^{m}\omega}\right\|}_{L^{r_{0}}(M,w)})$

hence, using (4.2) and integrating,

[TABLE]

Now let $\omega\in W_{G}^{k,r}(M,w)\cap{\mathcal{C}}_{G}^{\infty}(M).$ Applying inequalities (4.2) and (4.3) with $r=r_{0}$ and $m=k-1,\ k-2,...,$ we find, for $\nabla^{j}\omega,$

$\displaystyle\ {\left\|{\nabla^{k-1}\omega}\right\|}_{L^{s_{k-1}}(M,w^{\prime})}\leq A({\left\|{\nabla^{k}\omega}\right\|}_{L^{r}(M,w)}+{\left\|{\nabla^{k-1}\omega}\right\|}_{L^{r}(M,w)}),$

$\displaystyle{\left\|{\nabla^{k-2}\omega}\right\|}_{L^{s_{k-1}}(M,w^{\prime})}\leq A({\left\|{\nabla^{k-1}\omega}\right\|}_{L^{r}(M,w)}+{\left\|{\nabla^{k-2}\omega}\right\|}_{L^{r}(M,w)}),$

$\displaystyle{\left\|{\omega}\right\|}_{L^{s_{k-1}}(M,w^{\prime})}\leq A({\left\|{\nabla\omega}\right\|}_{L^{r}(M,w)}+{\left\|{\psi}\right\|}_{L^{r}(M,w)}).$

Thus

$\displaystyle\ {\left\|{\omega}\right\|}_{W^{k-1,s_{k-1}}(M,w^{\prime})}\leq 2A{\left\|{\omega}\right\|}_{W^{k,r}(M,w)}.$

Therefore a Cauchy sequence in $W_{G}^{k,r}(M,w)$ of sections of $G$ is a Cauchy sequence in $W_{G}^{k-1,s_{k-1}}(M,w^{\prime}),$ and the preceding inequality holds for all $\psi\in W_{G}^{k,r}(M,w)$ and we get:

$\displaystyle W_{G}^{k,r}(M,w)\subset W_{G}^{k-1,s_{k-1}}(M,w^{\prime}).$

Now with $w=w^{\prime}$ we prove similarly the following embeddings:

$\displaystyle W_{G}^{k-1,s_{k-1}}(M,w^{\prime})\subset W_{G}^{k-2,s_{k-2}}(M,w^{\prime})\subset\cdot\cdot\cdot\subset W_{G}^{l,s_{l}}(M,w^{\prime}).$

Hence $W_{G}^{k,r}(M,w)\subset W_{G}^{l,s_{l}}(M,w^{\prime}).$ $\hfill\blacksquare$

Proposition 4.2 says that, in order to prove Sobolev embeddings with weights, we have just to prove that $W_{G}^{1,r}(M,w)$ is embedded in $L_{G}^{s}(M,w^{\prime}),$ with $\frac{1}{s}=\frac{1}{r}-\frac{1}{n}\ (1\leq r<n).$

This is very important here because we shall have just to deal with first order Sobolev spaces. Hence we have to work only with $\nabla^{G}u$ which, by our assumption (CMT), implies at most the first order derivatives of the metric tensor.

The aim now is to prove that $W_{G}^{1,r_{0}}(M,w)$ is embedded in $L_{G}^{s_{0}}(M,w^{\prime}),$ with $\frac{1}{s_{0}}=\frac{1}{r_{0}}-\frac{1}{n}\ (1\leq r_{0}<n)$ then we shall be able to apply Proposition 4.2.

5 Local estimates.

5.1 Sobolev comparison estimates for functions.

Lemma 5.1**.**

We have the Sobolev comparison estimates where $B(x,R)$ is a $(0,\epsilon)$-admissible ball in $M$ and $\varphi\ :\ B(x,R)\rightarrow{\mathbb{R}}^{n}$ is the admissible chart relative to $B(x,R),$

$\displaystyle\forall u\in W^{1,r}(B(x,R)),\ {\left\|{u}\right\|}_{W^{1,r}(B(x,R))}\leq C{\left\|{u\circ\varphi^{-1}}\right\|}_{W^{1,r}(\varphi(B(x,R)))},$**

and, with $B_{e}(0,t)$ the euclidean ball in ${\mathbb{R}}^{n}$ centered at [math] and of radius $t,$

$\displaystyle{\left\|{v}\right\|}_{W^{1,r}(B_{e}(0,(1-\epsilon)R))}\leq C^{\prime}{\left\|{u}\right\|}_{W^{1,r}(B(x,R))}.$**

The constants $C,C^{\prime}$ depending only on $\epsilon,n$ and not on $B\in{\mathcal{A}}(\epsilon).$

Proof.

We have to compare the norms of $u,\ \nabla u,$ with the corresponding ones for $v:=u\circ\varphi^{-1}$ in ${\mathbb{R}}^{n}.$

First we have because $(1-\epsilon)\delta_{ij}\leq g_{ij}\leq(1+\epsilon)\delta_{ij}$ in $B(x,R)$:

$\displaystyle B_{e}(0,(1-\epsilon)R)\subset\varphi(B(x,R))\subset B_{e}(0,(1+\epsilon)R).$

For functions, in a coordinates chart, we have $(\nabla u)_{j}:=\partial_{j}u$ hence with $\forall y\in B(x,R),\ z:=\varphi(y),$ we get:

$\displaystyle\forall y\in B(x,R),\ \left|{u(y)}\right|=\left|{v(z)}\right|,\ \ \left|{\nabla u(y)}\right|\leq\left|{\partial v(z)}\right|.$

Integrating this we get:

$\displaystyle{\left\|{\nabla u}\right\|}_{L^{r}(B(x,R))}\leq{\left\|{\partial v}\right\|}_{L^{r}(B_{e}(0,(1+\epsilon)R))}.$

So, using that

$\displaystyle{\left\|{u}\right\|}_{W^{1,r}(B(x,R))}={\left\|{\nabla u}\right\|}_{L^{r}(B(x,R))}+{\left\|{u}\right\|}_{L^{r}(B(x,R))},$

we get

$\displaystyle{\left\|{u}\right\|}_{W^{1,r}(B(x,R))}\leq C({\left\|{\partial v}\right\|}_{L^{r}(B_{e}(0,(1+\epsilon)R))}+{\left\|{v}\right\|}_{L^{r}(B_{e}(0,(1+\epsilon)R))}),$

hence

$\displaystyle{\left\|{u}\right\|}_{W^{1,r}(B(x,R))}\leq C{\left\|{v}\right\|}_{W^{1,r}(\varphi(B(x,R))}.$

Of course all these estimates can be reversed so we also have

$\displaystyle{\left\|{v}\right\|}_{W^{1,r}(B_{e}(0,(1-\epsilon)R))}\leq C{\left\|{u}\right\|}_{W^{1,r}(B(x,R))}.$

This ends the proof of the lemma. $\hfill\blacksquare$

We have to study the behavior of the Sobolev embeddings w.r.t. the radius. Set $B_{R}:=B_{e}(0,R)$ an euclidean ball in ${\mathbb{R}}^{n}.$ For this purpose we have by  [Amar, 2018a, Lemma 7.7] in the special case $m=1.$

Lemma 5.2**.**

We have, with $t::\frac{1}{t}=\frac{1}{r}-\frac{1}{n},$

$\displaystyle\forall R,\ 0<R\leq 1,\ \forall u\in W^{1,r}(B_{R}),\ {\left\|{u}\right\|}_{L^{t}(B_{R})}\leq CR^{-1}\ {\left\|{u}\right\|}_{W^{1,r}(B_{R})}$**

the constant $C$ depending only on $\displaystyle n,\ r.$

Proof.

Start with $R=1,$ then we have, by the Sobolev embeddings which are valid on balls in ${\mathbb{R}}^{n},$ with $t::\frac{1}{t}=\frac{1}{r}-\frac{1}{n},$

[TABLE]

where $\displaystyle C$ depends only on $n$ and $\displaystyle r.$ For $u\in W^{1,r}(B_{R})$ we set

$\displaystyle\forall x\in B_{1},\ y:=Rx\in B_{R},\ v(x):=u(y).$

Then we have

$\displaystyle\partial v(x)=\partial u(y){\times}\frac{\partial y}{\partial x}=R\partial u(y);$

So we get, because the jacobian for this change of variables is $R^{-n},$

$\displaystyle{\left\|{\partial v}\right\|}_{L^{r}(B_{1})}^{r}=\int_{B_{1}}{\left|{\partial v(x)}\right|^{r}dm(x)}=\int_{B_{R}}{\left|{\partial u(y)}\right|^{r}\frac{R^{r}}{R^{n}}dm(x)}=R^{r-n}{\left\|{\partial u}\right\|}_{L^{r}(B_{R})}^{r}.$

So

[TABLE]

And of course ${\left\|{u}\right\|}_{L^{r}(B_{R})}=R^{n/r}{\left\|{v}\right\|}_{L^{r}(B_{1})}.$

So with 5.4 we get

[TABLE]

But

$\displaystyle{\left\|{u}\right\|}_{W^{1,r}(B_{R})}={\left\|{u}\right\|}_{L^{r}(B_{R})}+{\left\|{\partial u}\right\|}_{L^{r}(B_{R})},$

and

$\displaystyle{\left\|{v}\right\|}_{W^{1,r}(B_{1})}={\left\|{v}\right\|}_{L^{r}(B_{1})}+{\left\|{\partial v}\right\|}_{L^{r}(B_{1})},$

so

$\displaystyle{\left\|{v}\right\|}_{W^{1,r}(B_{1})}:=R^{-n/r}{\left\|{u}\right\|}_{L^{r}(B_{R})}+R^{1-n/r}{\left\|{\partial u}\right\|}_{L^{r}(B_{R})}.$

Because we have $\displaystyle R\leq 1,$ we get

$\displaystyle{\left\|{v}\right\|}_{W^{1,r}(B_{1})}\leq R^{-n/r}({\left\|{u}\right\|}_{L^{r}(B_{R})}+{\left\|{\partial u}\right\|}_{L^{r}(B_{R})})=R^{-n/r}{\left\|{u}\right\|}_{W^{1,r}(B_{R})}.$

Putting it in (5.6) we get

$\displaystyle{\left\|{u}\right\|}_{L^{t}(B_{R})}\leq CR^{n/t}{\left\|{v}\right\|}_{W^{1,r}(B_{1})}\leq CR^{-n(\frac{1}{r}-\frac{1}{t})}{\left\|{u}\right\|}_{W^{1,r}(B_{R})}.$

But, because $t::\frac{1}{t}=\frac{1}{r}-\frac{1}{n},$ we get $(\frac{1}{r}-\frac{1}{t})=\frac{1}{n}$ and

$\displaystyle{\left\|{u}\right\|}_{L^{t}(B_{R})}\leq CR^{-1}{\left\|{u}\right\|}_{W^{1,r}(B_{R})}.$

The constant $C$ depends only on $n,r.$ The proof is complete. $\hfill\blacksquare$

Lemma 5.3**.**

Let $x\in M$ and $B(x,R)$ be a $(0,\epsilon)$-admissible ball in the complete riemannian manifold $(M,g)$; we have, with $1/s=1/r-1/n,$

$\displaystyle\forall u\in W^{1,r}(B(x,R)),\ {\left\|{u}\right\|}_{L^{s}(B(x,R))}\leq CR^{-2}{\left\|{u}\right\|}_{W^{1,r}(B(x,R))},$**

the constant $\displaystyle C$ depending only on $n,\ r$ and $\displaystyle\epsilon.$

Proof.

By Lemma 5.2, we get in ${\mathbb{R}}^{n}$:

$\displaystyle{\left\|{u}\right\|}_{L^{s}(B_{R})}\leq CR^{-1}\ {\left\|{u}\right\|}_{W^{1,r}(B_{R})}$

so we can apply the comparison Lemma 5.1:

$\displaystyle{\left\|{u}\right\|}_{L^{s}(B(x,R))}\leq C{\left\|{u\circ\varphi^{-1}}\right\|}_{L^{s}(B_{R}))},$

which gives:

$\displaystyle{\left\|{u}\right\|}_{L^{s}(B(x,R))}\leq CR^{-1}\ {\left\|{u}\right\|}_{W^{1,r}(B_{R})}.$

Again with the reverse inequalities in the comparison Lemma 5.1:

$\displaystyle{\left\|{u}\right\|}_{W^{1,r}(B_{R})}\leq CR^{-1}{\left\|{u}\right\|}_{W^{1,r}(B(x,R))}.$

So we get

$\displaystyle{\left\|{u}\right\|}_{L^{s}(B(x,R))}\leq CR^{-2}\ {\left\|{u}\right\|}_{W^{1,r}(B(x,R)}.$

The constant $C$ being independent of $x\in M$ and of $R.$ The proof is complete. $\hfill\blacksquare$

5.2 Sobolev comparison estimates for sections of $G.$

Lemma 5.4**.**

We have the Sobolev comparison estimates where $B(x,R)$ is a $(1,\epsilon)$-admissible ball in $M$ and $\varphi\ :\ B(x,R)\rightarrow{\mathbb{R}}^{n}$ is the admissible chart relative to $B(x,R).$ Set $v:=\varphi^{*}\omega,$ then:

$\displaystyle\forall\omega\in W_{G}^{1,r}(B(x,R)),\ {\left\|{\omega}\right\|}_{W_{G}^{1,r}(B(x,R))}\leq(1+C\epsilon)R^{-1}{\left\|{v}\right\|}_{W^{1,r}(\varphi(B(x,R)))},$**

and, with $B_{e}(0,t)$ the euclidean ball in ${\mathbb{R}}^{n}$ centered at [math] and of radius $t,$

$\displaystyle{\left\|{v}\right\|}_{W^{1,r}(B_{e}(0,(1-\epsilon)R))}\leq(1+C\epsilon)R^{-1}{\left\|{\omega}\right\|}_{W^{1,r}(B(x,R))}.$**

We also have:

$\displaystyle\forall\omega\in L_{G}^{r}(B(x,R)),\ {\left\|{\omega}\right\|}_{L_{G}^{r}(B(x,R))}\leq(1+C\epsilon){\left\|{v}\right\|}_{L^{r}(\varphi(B(x,R)))},$**

and

$\displaystyle{\left\|{v}\right\|}_{L^{r}(B_{e}(0,(1-\epsilon)R))}\leq(1+C\epsilon){\left\|{\omega}\right\|}_{L_{G}^{r}(B(x,R))}.$**

Proof.

We have to compare the norms of $\omega,\ \nabla\omega,$ with the corresponding ones for $v:=\varphi^{*}\omega$ in ${\mathbb{R}}^{n}.$

By Lemma 3.1 the $\epsilon$-admissible ball $B(x,R)$ trivialises the bundle $G$ hence the image of a section of $G$ in ${\mathbb{R}}^{n}$ is just vectors of functions. Precisely $v:=\varphi^{*}\omega\in\varphi(B(x,R)){\times}{\mathbb{R}}^{N}.$

We have because $(1-\epsilon)\delta_{ij}\leq g_{ij}\leq(1+\epsilon)\delta_{ij}$ in $B(x,R)$:

$\displaystyle B_{e}(0,(1-\epsilon)R)\subset\varphi(B(x,R))\subset B_{e}(0,(1+\epsilon)R).$

Let $\omega$ be a section of $G$ in $M.$ By our assumption (CMT) we have that $\nabla\omega$ depends on the first order derivatives of the metric tensor $g.$

Because of (3.1) we get, with the fact that $B(x,R)$ is $(1,\epsilon)$-admissible, with $\eta:=\frac{\epsilon}{R},$

$\displaystyle\sum_{\left|{\beta}\right|=1}{R\sup\ _{i,j=1,...,n,\ y\in B_{x}(R)}\left|{\partial^{\beta}g_{ij}(y)}\right|}\leq\epsilon\Rightarrow\left|{\Gamma^{k}_{ij}}\right|\leq C\eta.$

Hence

$\displaystyle\forall y\in B(x,R),\ \left|{\omega(y)}\right|=\left|{v(z)}\right|,\ \ \left|{\nabla\omega(y)}\right|\leq\left|{\partial\omega}\right|+\left|{\Phi}\right|,$

where $\Phi$ depends on the coefficients of $\omega$ and on the first order derivatives of the metric tensor $g.$

So

[TABLE]

Integrating this we get

$\displaystyle{\left\|{\nabla\omega(y)}\right\|}_{L^{r}(B(x,R))}\leq{\left\|{\partial v}\right\|}_{L^{r}(B_{e}(0,(1+\epsilon)R))}+C\eta{\left\|{v}\right\|}_{L^{r}(B_{e}(0,(1+\epsilon)R))}.$

We also have the reverse estimates

$\displaystyle{\left\|{\partial v}\right\|}_{L^{r}(B_{e}(0,(1-\epsilon)R))}\leq{\left\|{\nabla\omega}\right\|}_{L^{r}(B(x,R))}+C\eta{\left\|{\omega}\right\|}_{L^{r}(B(x,R))}.$

So, using that

$\displaystyle{\left\|{\omega}\right\|}_{W_{G}^{1,r}(B(x,R))}={\left\|{\nabla\omega}\right\|}_{L_{G}^{r}(B(x,R))}+{\left\|{\omega}\right\|}_{L_{G}^{r}(B(x,R))},$

we get

$\displaystyle{\left\|{\omega}\right\|}_{W_{G}^{1,r}(B(x,R))}\leq{\left\|{\partial v}\right\|}_{L^{r}(B_{e}(0,(1+\epsilon)R))}+C\eta{\left\|{v}\right\|}_{L^{r}(B_{e}(0,(1+\epsilon)R))}\leq$

$\displaystyle\leq(1+\eta C){\left\|{v}\right\|}_{W^{1,r}(B_{e}(0,(1+\epsilon)R))}.$

Now for $R\leq 1,$ we get:

$\displaystyle 1+C\eta=1+C\frac{\epsilon}{R}=R^{-1}(R+C\epsilon)\leq R^{-1}(1+C\epsilon).$

The second part follows the same lines but is easier because there is no derivations.

The proof of the lemma is complete. $\hfill\blacksquare$

Lemma 5.5**.**

Let $x\in M$ and $B(x,R)$ be a $(1,\epsilon)$-admissible ball in the complete riemannian manifold $(M,g)$; we have, with $1/s=1/r-1/n,$

$\displaystyle\forall\omega\in W_{G}^{1,r}(B(x,R)),\ {\left\|{\omega}\right\|}_{L_{G}^{s}(B(x,R))}\leq CR^{-2}{\left\|{\omega}\right\|}_{W_{G}^{1,r}(B(x,R))}.$**

the constant $\displaystyle C$ depending only on $n,\ r$ and $\displaystyle\epsilon.$

Proof.

Because the image of a section of $G$ in ${\mathbb{R}}^{n}$ is just vectors of functions by Lemma 3.1, Lemma 5.2 is also true for $\varphi^{*}\omega$:

$\displaystyle{\left\|{\varphi^{*}\omega}\right\|}_{L^{s}(B_{R})}\leq CR^{-1}\ {\left\|{\varphi^{*}\omega}\right\|}_{W^{1,r}(B_{R})}$

so we can apply the second part in the comparison Lemma 5.4:

$\displaystyle{\left\|{\omega}\right\|}_{L_{G}^{s}(B(x,R))}\leq C{\left\|{\varphi^{*}\omega}\right\|}_{L^{s}(B_{R}))},$

which gives:

$\displaystyle{\left\|{\omega}\right\|}_{L_{G}^{s}(B(x,R))}\leq CR^{-1}\ {\left\|{\varphi^{*}\omega}\right\|}_{W^{1,r}(B_{R})}.$

Again with the reverse inequalities in the comparison Lemma 5.4:

$\displaystyle{\left\|{\varphi^{*}\omega}\right\|}_{W^{1,r}(B_{R})}\leq CR^{-1}{\left\|{\omega}\right\|}_{W_{G}^{1,r}(B(x,R))}.$

So we get

$\displaystyle{\left\|{\omega}\right\|}_{L_{G}^{s}(B(x,R))}\leq CR^{-2}\ {\left\|{\omega}\right\|}_{W_{G}^{1,r}(B(x,R)}.$

The constant $C$ is independent of $x\in M$ and of $R.$ The proof is complete. $\hfill\blacksquare$

5.3 Local Gaffney type inequality in $L^{r}.$

We shall restrict here to the case of the bundle $\Lambda^{p}(M)$ of $p$-forms on $M.$

Of course the operator $d$ on $p$-forms is local and so is $\displaystyle d^{*}$ as a first order differential operator on $M.$

Let $B:=B(x_{0},R)$ be a $(1,\epsilon)$-admissible ball in the complete riemannian manifold $(M,g)$ and $(B,\varphi)$ be a coordinates chart. Let $\omega$ be a $p$-form in $M.$ Let $\chi$ be a smooth cut-off function, $\chi\in{\mathcal{C}}_{0}^{1}(B),\ 0\leq\chi\leq 1,\ \chi\equiv 1$ in $B^{1}:=B(x_{0},R/2).$ We consider the $p$-form $\chi\omega.$

Read in the chart $(B,\varphi)$ with the local coordinates $x,$ we get $\omega=a_{J}dx^{J}$ with $J=(j_{1},...,j_{p})$ is a multi-index of length $p$ and the functions $a_{J}$ are in $W^{1,r}(B).$ We get then that $d(\chi\omega)=\chi da_{J}\wedge dx^{J}+a_{J}d\chi\wedge dx^{J},$ hence, with $da_{J}=\frac{\partial a_{J}}{\partial x^{j}}dx^{j},$ we deduce $d\omega=\frac{\partial a_{J}}{\partial x^{j}}dx^{j}\wedge dx^{J}$ and $d(\chi\omega)=\chi\frac{\partial a_{J}}{\partial x^{j}}dx^{j}\wedge dx^{J}+a_{J}d\chi\wedge dx^{J}.$

We shall take the following notation from the book by C. Voisin [Voisin, 2002].

With a local Hodge $*$ operator $\Lambda^{p}\rightarrow\Lambda^{n-p}$ (locally $M$ is always orientable), i.e. in a coordinates chart $U$ with $\omega=a_{J}dx^{J}\in\Lambda^{p},$ it is defined as:

$\displaystyle*\omega:=(-1)^{\sigma(\ J)}a_{J^{c}}dx^{J^{c}}\in\Lambda^{n-p}$

with $J^{c}$ is the complement of $J$ in $(1,2,...,n)$ and $\sigma(J)$ is [math] or $1.$ We have: $\int_{U}{\omega\wedge\omega^{*}}=\int_{U}{\sum_{J,\ \left|{J}\right|=p}{}\left|{a_{J}}\right|^{2}}dv.$

Using the link between the $*$ Hodge operator and the adjoint $d^{*}$ of $d,$ (see  [Voisin, 2002, Section 5.1.2, p. 118]), we get: $d^{*}=(-1)^{p}*^{-1}d*$ on $\Lambda^{p}.$

Hence here we have:

$\displaystyle d^{*}(\chi\omega)=(-1)^{p(n-p)-n+p-1}(*d(*(\chi\omega)))=L_{K}(\frac{\partial a_{J}}{\partial x^{j}})dx^{K}+F_{K}(a_{J},\frac{\partial\chi}{\partial x^{j}})dx^{K},$

where $L_{K}$ is linear in the $\frac{\partial a_{J}}{\partial x^{j}}$ and $F_{K}$ is linear in the $a_{J}$ and in the $\frac{\partial\chi}{\partial x^{j}},$ and $\left|{K}\right|=p-1.$ Moreover $L_{K},\ F_{K}$ have compact support in $B.$

Now for the covariant derivative $\nabla_{M}$ on $M,$ we have, in our chart $(B,\varphi),$ using (5.7)

[TABLE]

On the other hand, in ${\mathbb{R}}^{n}$ we have $\left|{\nabla_{{\mathbb{R}}^{n}}(\chi\omega)}\right|=\left|{\partial(\chi\omega)}\right|$ just because $\nabla_{{\mathbb{R}}^{n}}(\chi\omega)=\partial(\chi\omega).$

Now we are in position to apply the following Proposition 4.3 in [Scott, 1995]:

Proposition 5.6**.**

(Gaffney type inequality for $L^{r}$)

$\displaystyle\ {\left\|{\nabla_{{\mathbb{R}}^{n}}\omega}\right\|}_{r}^{r}\leq C{\left\|{d\omega}\right\|}_{L^{r}({\mathbb{R}}^{n})}^{r}+{\left\|{d^{*}\omega}\right\|}_{L^{r}({\mathbb{R}}^{n})}^{r}$**

for $\omega\in{\mathcal{C}}^{\infty}_{0}(\Lambda^{p}({\mathbb{R}}^{n}))$ and $C=C(n,r).$

From it, we get in ${\mathbb{R}}^{n},$

$\displaystyle{\left\|{\partial(\chi\omega)}\right\|}_{L^{r}(\varphi(B))}\leq C({\left\|{d(\chi\omega)}\right\|}_{L^{r}(\varphi(B))}+{\left\|{d^{*}(\chi\omega)}\right\|}_{L^{r}(\varphi(B))}).$

So, by (5.8), we get, because $\chi\equiv 1$ in $B^{1}$:

$\displaystyle\ {\left\|{\nabla_{M}\omega}\right\|}_{L^{r}(B^{1})}\leq C({\left\|{d(\chi\omega)}\right\|}_{L^{r}(B)}+{\left\|{d^{*}(\chi\omega)}\right\|}_{L^{r}(B)})+c\epsilon R^{-1}{\left\|{\chi\omega}\right\|}_{L^{r}(B)},$

because, by condition (*) in the definition of the $\epsilon$-admissible ball $B,$ we have that the Lebesgue measure in $\varphi(B)$ and the volume measure on $B$ are equivalent.

Now, because ${\left\|{d\chi}\right\|}_{\infty}\leq cR^{-1}$ in $B$:

$\displaystyle{\left\|{d(\chi\omega)}\right\|}_{L^{r}(B)}\leq{\left\|{d\chi}\right\|}_{\infty}{\left\|{\omega}\right\|}_{L^{r}(B)}+{\left\|{d\omega}\right\|}_{L^{r}(B)}\leq cR^{-1}{\left\|{\omega}\right\|}_{L^{r}(B)}+{\left\|{d\omega}\right\|}_{L^{r}(B)},$

and the same

$\displaystyle{\left\|{d^{*}(\chi\omega)}\right\|}_{L^{r}(B)}\leq cR^{-1}{\left\|{\omega}\right\|}_{L^{r}(B)}+{\left\|{d^{*}\omega}\right\|}_{L^{r}(B)}.$

Hence, with new constants $c,C$ depending only on $n,\ r,\ p,\ \epsilon,$

$\displaystyle{\left\|{\nabla_{M}\omega}\right\|}_{L^{r}(B^{1})}\leq C({\left\|{d(\omega)}\right\|}_{L^{r}(B)}+{\left\|{d^{*}(\omega)}\right\|}_{L^{r}(B)})+cR^{-1}{\left\|{\omega}\right\|}_{L^{r}(B)}.$

So we proved the corollary of Scott’s Proposition 5.6:

Corollary 5.7**.**

Let $B:=B(x_{0},R)$ be a $(1,\epsilon)$-admissible ball in the complete riemannian manifold $(M,g)$ and set $B^{1}:=B(x_{0},R/2).$ Let $\omega$ be a $p$-form in $M.$ We have the local $L^{r}$ Gaffney’s inequality:

$\displaystyle{\left\|{\nabla_{M}\omega}\right\|}_{L^{r}(B^{1})}\leq C({\left\|{d(\omega)}\right\|}_{L^{r}(B)}+{\left\|{d^{*}(\omega)}\right\|}_{L^{r}(B)})+cR^{-1}{\left\|{\omega}\right\|}_{L^{r}(B)},$**

the constants $c,C$ depending only on $n,\ r,\ p,\ \epsilon.$

6 Vitali covering.

Lemma 6.1**.**

Let ${\mathcal{F}}$ be a collection of balls $\{B(x,r(x))\}$ in a metric space, with $\forall B(x,r(x))\in{\mathcal{F}},\ 0<r(x)\leq R.$ There exists a disjoint subcollection ${\mathcal{G}}$ of ${\mathcal{F}}$ with the following properties:

every ball $B$ in ${\mathcal{F}}$ intersects a ball $C$ in ${\mathcal{G}}$ and $B\subset 5C.$

This is a well known lemma, see for instance  [Evans and Gariepy, 1992], section 1.5.1.

Fix $\epsilon>0$ and let \forall x\in M,\ r(x):=R_{\epsilon}(x)/10,\where $R_{\epsilon}(x)$ is the admissible radius at $\displaystyle x,$ we built a Vitali covering with the collection ${\mathcal{F}}:=\{B(x,r(x))\}_{x\in M}.$ The previous lemma gives a disjoint subcollection ${\mathcal{G}}$ such that every ball $B$ in ${\mathcal{F}}$ intersects a ball $C$ in ${\mathcal{G}}$ and we have $\displaystyle B\subset 5C.$ We set ${\mathcal{D}}(\epsilon):=\{x\in M::B(x,r(x))\in{\mathcal{G}}\}$ and ${\mathcal{C}}_{\epsilon}:=\{B(x,5r(x)),\ x\in{\mathcal{D}}(\epsilon)\}$: we shall call ${\mathcal{C}}_{\epsilon}$ a $\epsilon$-admissible covering of $(M,g).$ We notice that $B(x,5r(x))=B(x,R_{\epsilon}(x)/2).$

Then we have [Amar, 2018b, Proposition 6.12],

Proposition 6.2**.**

Let $(M,g)$ be a riemannian manifold, then the overlap of a $\epsilon$-admissible covering ${\mathcal{C}}(\epsilon)$ is less than $T=\frac{(1+\epsilon)^{n/2}}{(1-\epsilon)^{n/2}}(100)^{n},$ i.e.

$\forall x\in M,\ x\in B(y,5r(y)),$* where $B(y,r(y))\in{\mathcal{G}}(\epsilon),$ for at most $T$ such balls.*

Moreover we have

$\displaystyle\forall f\in L^{1}(M),\ \sum_{j\in{\mathbb{N}}}{\int_{B(x_{j},r(x_{j}))}{\left|{f(x)}\right|dv_{g}(x)}}\leq T{\left\|{f}\right\|}_{L^{1}(M)}.$**

And its Corollary [Amar, 2018b, Corollary 6.13],

Corollary 6.3**.**

Let $(M,g)$ be a complete riemannian manifold. Consider the covering by the balls $\{B(x,R_{\epsilon}(x)),\ x\in{\mathcal{D}}(\epsilon)\}.$ Then the overlap of the associated covering verifies:

$\displaystyle T_{1}\leq\frac{(1+\epsilon)^{n/2}}{(1-\epsilon)^{n/2}}(100)^{n}{\times}2^{n}.$**

7 Global results.

7.1 Global estimates for sections of $G.$

Lemma 7.1**.**

We have, for any section $f$ of $G$ with $w(x):=R(x)^{\mu}$ and $B(x):=B(x,R(x)),$ with $x\in{\mathcal{D}}(\epsilon)$ and $R(x):=R_{\epsilon}(x),$ that:

$\displaystyle\forall\tau\geq 1,\ {\left\|{f}\right\|}_{L^{\tau}_{G}(M,\ w)}^{\tau}\simeq\sum_{x\in{\mathcal{D}}(\epsilon)}{R(x)^{\mu}{\left\|{f}\right\|}_{L^{\tau}_{G}(B(x))}^{\tau}}.$**

Proof.

We consider the function $\left|{f}\right|.$ We set $w(x):=R(x)^{\mu}$ to get an adapted weight. Let $x\in{\mathcal{D}}(\epsilon),$ then $B(x):=B(x,R(x))\in{\mathcal{C}}_{\epsilon}.$

We have, because ${\mathcal{C}}_{\epsilon}$ is a covering of $M,$

$\displaystyle{\left\|{f}\right\|}_{L^{\tau}(M,w)}^{\tau}:=\int_{M}{\left|{f(x)}\right|^{\tau}w(x)dv(x)}\leq\sum_{x\in{\mathcal{D}}(\epsilon)}{\int_{B(x)}{\left|{f(y)}\right|^{\tau}R(y)^{\mu}}dv(y)}.$

Because we have, by Lemma 2.4, $\forall y\in B,\ R(y)\leq 2R(x),$ we get

$\displaystyle\sum_{x\in{\mathcal{D}}(\epsilon)}{\int_{B(x)}{\left|{f(y)}\right|^{\tau}R(y)^{\mu}}dv(y)}\leq$

$\displaystyle\leq\sum_{x\in{\mathcal{D}}(\epsilon)}{2^{\mu}R(x)^{\mu}\int_{B(x)}{\left|{f(y)}\right|^{\tau}}dv(y)}\leq 2^{\mu}\sum_{x\in{\mathcal{D}}(\epsilon)}{R(x)^{\mu}{\left\|{f}\right\|}_{L^{\tau}(B(x))}^{\tau}}.$

Hence we get

$\displaystyle{\left\|{f}\right\|}_{L^{\tau}(M,w)}^{\tau}\leq 2^{\mu}\sum_{x\in{\mathcal{D}}(\epsilon)}{R(x)^{\mu}{\left\|{f}\right\|}_{L^{\tau}(B)}^{\tau}}.$

To get the reverse inequality we still use Lemma 2.4, to have $\forall y\in B,\ R(x)\leq 2R(y)$ so we get:

$\displaystyle\sum_{x\in{\mathcal{D}}(\epsilon)}{R(x)^{\mu}\int_{B(x)}{\left|{f(y)}\right|^{\tau}}dv(y)}\leq 2^{\mu}\sum_{x\in{\mathcal{D}}(\epsilon)}{\int_{B(x)}{R(y)^{\mu}\left|{f(y)}\right|^{\tau}}dv(y)}.$

Now we use the fact that the overlap of ${\mathcal{C}}_{\epsilon}$ is bounded by $T,$

$\displaystyle\sum_{x\in{\mathcal{D}}(\epsilon)}{\int_{B(x)}{R(y)^{\mu}\left|{f(y)}\right|^{\tau}}dv(y)}\leq 2^{\mu}T\int_{M}{R(y)^{\mu}\left|{f(y)}\right|^{\tau}}dv(y)=2^{\mu}T{\left\|{f}\right\|}_{L^{\tau}(M,w)}^{\tau}.$

So we get

$\displaystyle\sum_{x\in{\mathcal{D}}(\epsilon)}{R(x)^{\mu}{\left\|{f}\right\|}_{L^{\tau}(B)}}^{\tau}\leq 2^{\mu}T{\left\|{f}\right\|}_{L^{\tau}(M,w)}^{\tau}.$

The proof is complete. $\hfill\blacksquare$

Lemma 7.2**.**

Let $u\in W^{1,r}_{G}(M,w)$ where $w(x)=R(x)^{\gamma}.$ We have:

$\displaystyle\sum_{B\in{\mathcal{C}}_{\epsilon},\ j=0,1}{R(x)^{\gamma}{\left\|{\nabla^{j}u}\right\|}^{r}_{L^{r}_{G}(B)}}\simeq{\left\|{u}\right\|}^{r}_{W^{1,r}_{G}(M,w)}$**

and

$\displaystyle\sum_{B\in{\mathcal{C}}_{\epsilon}}{R(x)^{\gamma}{\left\|{u}\right\|}^{r}_{W^{1,r}_{G}(B)}}\simeq{\left\|{u}\right\|}^{r}_{W^{1,r}_{G}(M,w)}.$**

Proof.

We start with $u\in W^{1,r}_{G}(M,w)$ where $w(x)=R(x)^{\gamma}.$ This means:

$\displaystyle\forall j\in{\mathbb{N}},\ j\leq 1,\ \int_{M}{\left|{\nabla^{j}u(y)}\right|^{r}R(y)^{\gamma}dv(y)}\leq C{\left\|{u}\right\|}^{r}_{W^{1,r}_{p}(M,w)}.$

Take a Vitali $\epsilon$-covering ${\mathcal{C}}_{\epsilon}$ of $M.$ Let $B:=B(x,R(x))\in{\mathcal{C}}_{\epsilon}$; because $u\in W^{1,r}_{G}(M,w),$ we have $u\in W_{G}^{1,r}(B,w).$

By Lemma 7.1 we have, with $f:=\left|{\nabla^{j}u}\right|$ and $\tau=r,$

$\displaystyle\int_{M}{\left|{\nabla^{j}u(y)}\right|^{r}R(y)^{\gamma}dv(y)}\simeq\sum_{x\in{\mathcal{D}}(\epsilon)}{R(x)^{\gamma}{\left\|{\nabla^{j}u}\right\|}_{L^{r}_{G}(B(x))}^{r}}.$

Hence, adding on $j,\ j\leq 1,$

[TABLE]

Using that ${\left\|{u}\right\|}_{W^{1,r}(B(x,R))}={\left\|{\nabla u}\right\|}_{L^{r}(B(x,R))}+{\left\|{u}\right\|}_{L^{r}(B(x,R))},$ we deduce from (7.9):

$\displaystyle\sum_{B\in{\mathcal{C}}_{\epsilon}}{R(x)^{\gamma}{\left\|{u}\right\|}^{r}_{W^{1,r}_{G}(B)}}\leq C{\left\|{u}\right\|}^{r}_{W^{1,r}_{G}(M,w)}.$

For the converse, because:

$\displaystyle{\left\|{u}\right\|}^{r}_{W^{1,r}_{G}(M,w)}\leq\sum_{j=0}^{1}{\int_{M}{\left|{\nabla^{j}u(y)}\right|^{r}R(y)^{\gamma}dv(y)}}$

we get again with Lemma 7.1,

$\displaystyle\int_{M}{\left|{\nabla^{j}u(y)}\right|^{r}R(y)^{\gamma}dv(y)}\simeq\sum_{x\in{\mathcal{D}}(\epsilon)}{R(x)^{\gamma}{\left\|{\nabla^{j}u}\right\|}_{L^{r}_{G}(B(x))}^{r}}.$

Hence

$\displaystyle{\left\|{u}\right\|}^{r}_{W^{1,r}_{G}(M,w)}\leq\sum_{x\in{\mathcal{D}}(\epsilon),\ j=0,1}{R(x)^{\gamma}{\left\|{\nabla^{j}u}\right\|}_{L^{r}_{G}(B(x))}^{r}}\lesssim\sum_{x\in{\mathcal{D}}(\epsilon)}{R(x)^{\gamma}{\left\|{u}\right\|}_{W^{1,r}_{G}(B(x))}^{r}}.$

Now from

[TABLE]

the constants underlying $\simeq$ being independent of $B,$ and with:

$\displaystyle\sum_{\ j=0,1}{{\left\|{\nabla^{j}u}\right\|}^{r}_{L^{r}_{G}(B)}}={\left\|{u}\right\|}^{r}_{W^{1,r}_{G}(B)},$

we get

$\displaystyle\sum_{\ j=0,1}{R(x)^{\gamma}{\left\|{\nabla^{j}u}\right\|}^{r}_{L^{r}_{G}(B)}}\simeq R(x)^{\gamma}{\left\|{u}\right\|}^{r}_{W^{1,r}_{G}(B)}.$

So, porting in (7.10), we get

$\displaystyle\sum_{B\in{\mathcal{C}}_{\epsilon},}{R(x)^{\gamma}{\left\|{u}\right\|}^{r}_{W^{1,r}_{G}(B)}}\simeq{\left\|{u}\right\|}^{r}_{W^{1,r}_{G}(M,w)}.$

The proof is complete. $\hfill\blacksquare$

Lemma 7.3**.**

We have:

$\displaystyle r\geq s\Rightarrow\sum_{j\in{\mathbb{N}}}{a_{j}^{r}}\leq{\left({\sum_{j\in{\mathbb{N}}}{a_{j}^{s}}}\right)}^{r/s};\ \ r\leq s\Rightarrow\sum_{j\in{\mathbb{N}}}{a_{j}^{r}}\geq{\left({\sum_{j\in{\mathbb{N}}}{a_{j}^{s}}}\right)}^{r/s}.$**

Proof.

The comparison of the norms of $\ell^{r}({\mathbb{N}})$ and $\displaystyle\ell^{s}({\mathbb{N}})$ gives the result. $\hfill\blacksquare$

Lemma 7.4**.**

Let $B=B(x,R)$ be an $\epsilon$-admissible ball in $M.$ We have, for $\omega\in L^{s}_{G}(B)$ with $s\geq r$:

$\displaystyle{\left\|{\omega}\right\|}_{L_{G}^{r}(B))}\leq c(n,\epsilon)R^{\frac{n}{r}-\frac{n}{s}}{\left\|{\omega}\right\|}_{L_{G}^{s}(B)},$**

with $c$ depending only on $n,\ \epsilon$ and $G.$

Proof.

First suppose that $B$ is in ${\mathbb{R}}^{n}$ with the Lebesgue measure. Let $\omega\in L^{s}(B)).$ Because $\frac{dv}{\left|{B}\right|}$ is a probability measure on $B,$ where $\left|{B}\right|$ is the volume of the ball $B$ and $s\geq r,$ we get

$\displaystyle{\left({\int_{B}{\left|{\omega(y)}\right|^{r}\frac{dv(y)}{\left|{B}\right|}}}\right)}^{1/r}\leq{\left({\int_{B}{\left|{\omega(y)}\right|^{s}\frac{dv(y)}{\left|{B}\right|}}}\right)}^{1/s},$

hence

$\displaystyle{\left\|{\omega}\right\|}_{L^{r}(B)}\leq\left|{B}\right|^{\frac{1}{r}-\frac{1}{s}}{\left\|{\omega}\right\|}_{L^{s}(B)}.$

Now back to the manifold, for $B_{x}:=B(x,R)$ a $\epsilon$-admissible ball we get

$\displaystyle\forall y\in B_{x},\ (1-\epsilon)^{n}\leq\left|{\mathrm{d}\mathrm{e}\mathrm{t}g(y)}\right|\leq(1+\epsilon)^{n},$

hence we have, comparing the Lebesgue measure in ${\mathbb{R}}^{n}$ with the volume measure in $M,$

$\displaystyle\forall x\in M,\ (1-\epsilon)^{n/2}\nu_{n}R^{n}\leq\mathrm{V}\mathrm{o}\mathrm{l}(B(x,\ R_{\epsilon}(x)))\leq(1+\epsilon)^{n/2}\nu_{n}R^{n},$

where $\nu_{n}$ is the volume of the unit ball in ${\mathbb{R}}^{n}.$ So, because the ball $B$ trivialises the bundle $G$ by Lemma 3.1, on the manifold $M,$ we have

$\displaystyle{\left\|{\omega}\right\|}_{L_{G}^{r}(B))}\leq c(n,\epsilon,G)R^{\frac{n}{r}-\frac{n}{s}}{\left\|{\omega}\right\|}_{L_{G}^{s}(B)}$

with $c$ depending only on $n,\ \epsilon$ and $G.$ $\hfill\blacksquare$

Now we fix $x\in{\mathcal{D}}(\epsilon)$ hence the ball $B:=B(x,R_{\epsilon}(x))$ is fixed. We have, with $1/s=1/r-1/n,$ that $W^{1,r}(B)\subset L^{s}(B),$ by Lemma 5.3 for functions or by Lemma 5.5 for sections of $G$ with:

$\displaystyle\forall u\in W_{G}^{1,r}(B(x,R)),\ {\left\|{u}\right\|}_{L_{G}^{s}(B(x,R))}\leq CR(x)^{-2}{\left\|{u}\right\|}_{W_{G}^{1,r}(B(x,R))}.$

So

[TABLE]

Now the aim is to get the global form of the previous result.

Let $w^{\prime}(x):=R(x)^{\nu}$ with $\nu:=s(2+\gamma/r).$ Using (7.11), raising to the power $s$ and adding, we get:

[TABLE]

Because $s\geq r,$ the Lemma 7.3 gives:

$\displaystyle\sum_{B\in{\mathcal{C}}_{\epsilon}}{R(x)^{s\gamma/r}{\left\|{u}\right\|}^{s}_{W^{1,r}(B(x,R))}}\leq{\left({\sum_{B\in{\mathcal{C}}_{\epsilon}}{R(x)^{\gamma}{\left\|{u}\right\|}^{r}_{W^{1,r}(B(x,R))}}}\right)}^{s/r}.$

Now we use Lemma 7.2 with $w(x):=R(x)^{\gamma},$

$\displaystyle\sum_{B\in{\mathcal{C}}_{\epsilon}}{R(x)^{\gamma}{\left\|{u}\right\|}^{r}_{W^{1,r}_{G}(B)}}\simeq{\left\|{u}\right\|}^{r}_{W^{1,r}_{G}(M,w)}$

to get, with Lemma 7.1, with $\mu=\nu:=2s+s\gamma/r,$

$\displaystyle{\left\|{u}\right\|}^{s}_{L^{s}_{G}(M,w^{\prime})}\simeq\sum_{B\in{\mathcal{C}}_{\epsilon}}{R(x)^{\nu}{\left\|{u}\right\|}^{s}_{L_{G}^{s}(B(x,R))}}$

hence, by (7.12),

$\displaystyle{\left\|{u}\right\|}^{s}_{L^{s}_{G}(M,w^{\prime})}\leq C{\left({\sum_{B\in{\mathcal{C}}_{\epsilon}}{R(x)^{\gamma}{\left\|{u}\right\|}^{r}_{W_{G}^{1,r}(B(x,R))}}}\right)}^{s/r}\simeq C{\left\|{u}\right\|}^{s}_{W^{1,r}_{G}(M,w)}.$

So we proved the following Sobolev embedding Theorem with weights:

Theorem 7.5**.**

Let $1/s=1/r-1/n>0.$ Let $u\in W^{1,r}_{G}(M,w)$ with $w(x):=R(x)^{\gamma}.$ Then we have $u\in L^{s}_{G}(M,w^{\prime}),$ with $\nu:=s(2+\gamma/r)$ and $w^{\prime}:=R(x)^{\nu},$ with the control:

$\displaystyle{\left\|{u}\right\|}_{L^{s}_{G}(M,w^{\prime})}\leq C{\left\|{u}\right\|}_{W^{1,r}_{G}(M,w)}.$**

Theorem 7.5 with Proposition 4.2 give:

Theorem 7.6**.**

Let $(M,g)$ be a complete riemannian manifold. Let $G:=(H,\pi,M)$ be a complex smooth adapted vector bundle over $M.$ Let $w(x):=R(x)^{\gamma}$ and $w^{\prime}:=R(x)^{\nu}$ with $\nu:=s(2+\gamma/r).$ Then $W_{G}^{m,r}(M,w)$ is embedded in $W_{G}^{k,s}(M,w^{\prime}),$ with $\frac{1}{s}=\frac{1}{r}-\frac{(m-k)}{n}>0$ and:

$\displaystyle\forall u\in W_{G}^{m,r}(M,w),\ {\left\|{u}\right\|}_{W_{G}^{k,s}(M,w^{\prime})}\leq C{\left\|{u}\right\|}_{W_{G}^{m,r}(M,w)}.$**

Remark 7.7**.**

The weights, as function of the $\epsilon$-admissible radius $R_{\epsilon}(x),$ do not depend on the fact that we work with functions or sections of $G$ but the radius itself depends on that fact. The $(1,\epsilon)$ admissible radius for the sections of $G$ is smaller than the $(0,\epsilon)$ one for functions.

In  [Amar, 2018b, Theorem 6.23, p. 21], we proved, with the bundle $\Lambda^{p}(M)$ of $p$-forms, the following:

Theorem 7.8**.**

Let $M$ be a complete non compact riemannian manifold of class ${\mathcal{C}}^{2}$ without boundary. Let $\alpha>0,\ r\geq 2$ and $k$ the smallest integer such that, with $\frac{1}{r_{k}}=\frac{1}{2}-\frac{2k}{n},$ we have $r_{k}\geq r.$ For any $\omega\in L^{r}([0,T+\alpha],L^{r}_{p}(M,w_{1}))\cap L^{r}([0,T+\alpha],L^{2}_{p}(M))$ there is a $u\in L^{r}([0,T],W^{2,r}_{p}(M,w_{2}))$ such that $\partial_{t}u+\Delta u=\omega$ and:

$\displaystyle{\left\|{\partial_{t}u}\right\|}_{L^{r}([0,T],L^{r}_{p}(M,w_{2}))}+{\left\|{u}\right\|}_{L^{r}([0,T],W^{2,r}_{p}(M,w_{2}))}\leq$**

$\displaystyle\leq c_{1}{\left\|{\omega}\right\|}_{L^{r}([0,T+\alpha],L^{r}_{p}(M,w_{1}))}+c_{2}{\sqrt{T+\alpha}}{\left\|{\omega}\right\|}_{L^{r}([0,T+\alpha],L^{2}_{p}(M))}$**

where the weights functions are: $w_{1}(x)=R(x)^{(\frac{n}{2}-\frac{n}{r}+2)}$ and$\ w_{2}(x)=R(x)^{(3+8k)}$ if we work only with functions and $w_{2}(x)=R(x)^{(3+12k)}$ for any $p$-forms, $p\geq 1.$

As a corollary we get, if we are interested in estimates $L^{r}-L^{s},$

Corollary 7.9**.**

Let $M$ be a complete non compact riemannian manifold of class ${\mathcal{C}}^{2}$ without boundary. Let $\alpha>0,\ r\geq 2$ and $k$ the smallest integer such that, with $\frac{1}{r_{k}}=\frac{1}{2}-\frac{2k}{n},$ we have $r_{k}\geq r.$ For any $\omega\in L^{r}([0,T+\alpha],L^{r}_{p}(M,w_{1}))\cap L^{r}([0,T+\alpha],L^{2}_{p}(M))$ there is a $u\in L^{r}([0,T],L^{s}_{p}(M,w_{2}))$ such that $\partial_{t}u+\Delta u=\omega$ and:

$\displaystyle{\left\|{\partial_{t}u}\right\|}_{L^{r}([0,T],L^{r}_{p}(M,w_{2}))}+{\left\|{u}\right\|}_{L^{r}([0,T],L^{s}_{p}(M,w_{2}))}\leq$**

$\displaystyle\leq c_{1}{\left\|{\omega}\right\|}_{L^{r}([0,T+\alpha],L^{r}_{p}(M,w_{1}))}+c_{2}{\sqrt{T+\alpha}}{\left\|{\omega}\right\|}_{L^{r}([0,T+\alpha],L^{2}_{p}(M))}$**

with $\frac{1}{s}=\frac{1}{r}-\frac{2}{n}>0$ and where the weights functions are: $w_{1}(x)=R(x)^{(\frac{n}{2}-\frac{n}{r}+2)}$ and $w_{2}(x)=R(x)^{s(2+(3+8k)/r)}$ if we work only with functions and $w_{2}(x)=R(x)^{s(2+(3+12k)/r)}$ for any $p$-forms, $p\geq 1.$

Proof.

By Theorem 7.6 we have $W_{G}^{m,r}(M,w)\subset W_{G}^{k,s}(M,w^{\prime}),$ with $\frac{1}{s}=\frac{1}{r}-\frac{(m-k)}{n}>0.$ So we choose $m=2,\ k=0$ and $w=R(x)^{(3+12k)}$ which gives $w^{\prime}(x)=R(x)^{\nu}$ with $\nu=s(2+(3+12k)/r).$ We get then, again with $G=\Lambda^{p}(M)$:

$\displaystyle\forall u\in W_{p}^{2,r}(M,w),\ {\left\|{u}\right\|}_{L_{p}^{s}(M,w^{\prime})}\leq C{\left\|{u}\right\|}_{W_{p}^{2,r}(M,w)}.$

If we work with functions, we have $\nu=s(2+(3+8k)/r).$

Putting this in Theorem 7.8, this finishes the proof of the corollary. $\hfill\blacksquare$

7.2 Global Gaffney type inequality in $L^{r}.$

Let $B:=B(x,R)$ be a $(1,\epsilon)$-admissible ball in the complete riemannian manifold $(M,g)$ and set $B^{1}:=B(x,R/2).$ Let $\omega$ be a $p$-form in $M.$ We have the local $L^{r}$ Gaffney’s inequality by Corollary 5.7:

$\displaystyle{\left\|{\nabla_{M}\omega}\right\|}_{L^{r}(B^{1})}\leq C({\left\|{d(\omega)}\right\|}_{L^{r}(B)}+{\left\|{d^{*}(\omega)}\right\|}_{L^{r}(B)})+cR^{-1}{\left\|{\omega}\right\|}_{L^{r}(B)},$

the constants $c,C$ depending only on $n,\ r,\ p,\ \epsilon.$ Because $R\leq 1,$ we get, with another constant $c$:

$\displaystyle{\left\|{\omega}\right\|}_{W^{1,r}(B^{1})}\leq{\left\|{\nabla_{M}\omega}\right\|}_{L^{r}(B^{1})}+{\left\|{\omega}\right\|}_{L^{r}(B^{1})}\leq$

$\displaystyle\leq C({\left\|{d(\omega)}\right\|}_{L^{r}(B)}+{\left\|{d^{*}(\omega)}\right\|}_{L^{r}(B)})+cR^{-1}{\left\|{\omega}\right\|}_{L^{r}(B)},$

To globalise this, we proceed as above by summing over the balls $B$ and $B^{1}.$ Raising to the power $r$ and adding, we get

$\displaystyle\sum_{x\in{\mathcal{D}}(\epsilon)}{{\left\|{\omega}\right\|}^{r}_{W^{1,r}(B^{1})}}\leq C\sum_{x\in{\mathcal{D}}(\epsilon)}{({\left\|{d(\omega)}\right\|}^{r}_{L^{r}(B)}+{\left\|{d^{*}(\omega)}\right\|}^{r}_{L^{r}(B)}+cR^{-r}{\left\|{\omega}\right\|}^{r}_{L^{r}(B)})}$

As above we have

$\displaystyle{\left\|{\omega}\right\|}^{r}_{W^{1,r}(M)}\leq\sum_{x\in{\mathcal{D}}(\epsilon)}{{\left\|{\omega}\right\|}^{r}_{W^{1,r}(B^{1})}}$

and, because the overlap of the covering $\{B(x,R_{\epsilon}(x)),\ x\in{\mathcal{D}}(\epsilon)\}$ is bounded by $T_{1}$ by Corollary 6.3, we get

$\displaystyle\sum_{x\in{\mathcal{D}}(\epsilon)}{({\left\|{d(\omega)}\right\|}^{r}_{L^{r}(B)}+{\left\|{d^{*}(\omega)}\right\|}^{r}_{L^{r}(B)})}\leq T_{1}({\left\|{d(\omega)}\right\|}^{r}_{L^{r}(M)}+{\left\|{d^{*}(\omega)}\right\|}^{r}_{L^{r}(M)})$

and

$\displaystyle\sum_{x\in{\mathcal{D}}(\epsilon)}{R_{\epsilon}^{-r}(x){\left\|{\omega}\right\|}^{r}_{L^{r}(B(x,R_{\epsilon}(x)))})}\leq T_{1}{\left\|{\omega}\right\|}^{r}_{L^{r}(M,w)},$

where $w(x)$ is the weight $w(x):=R^{-r}_{\epsilon}(x).$ So we proved the global Gaffney’s type inequality with weight:

Theorem 7.10**.**

Let $(M,g)$ be a complete riemannian manifold. Let $r\geq 1$ and $w(x):=R(x)^{-r}.$ Let $\omega$ be a $p$-form in $M.$ We have:

$\displaystyle{\left\|{\omega}\right\|}_{W_{p}^{1,r}(M)}\leq C({\left\|{d(\omega)}\right\|}_{L_{p+1}^{r}(M)}+{\left\|{d^{*}(\omega)}\right\|}_{L_{p-1}^{r}(M)}+{\left\|{\omega}\right\|}_{L_{p}^{r}(M,w)}).$**

So using Theorem 7.6 with $\gamma=0,\ m=1,\ k=0,\ \frac{1}{s}=\frac{1}{r}-\frac{1}{n}>0,\ w^{\prime}(x)=R_{\epsilon}(x)^{2s},$ plus Theorem 7.10, we get

Corollary 7.11**.**

Let $(M,g)$ be a complete riemannian manifold. Let $r\geq 1$ and $w(x):=R(x)^{-r}.$ Let $\omega$ be a $p$-form in $M.$ We have:

$\displaystyle{\left\|{\omega}\right\|}_{L_{p}^{s}(M,w^{\prime})}\leq C({\left\|{d(\omega)}\right\|}_{L_{p+1}^{r}(M)}+{\left\|{d^{*}(\omega)}\right\|}_{L_{p-1}^{r}(M)}+{\left\|{\omega}\right\|}_{L_{p}^{r}(M,w)})$**

with $\frac{1}{s}=\frac{1}{r}-\frac{1}{n}>0,\ w^{\prime}(x)=R_{\epsilon}(x)^{2s}.$

This corollary is a kind of N. Lohoué’s result [Lohoué, 1985] with weights and without any geometric conditions on the riemannian manifold $(M,g).$

8 Applications.

We shall give some examples where we have classical estimates using that $\forall x\in M,\ R_{\epsilon}(x)\geq\delta,$ via [Hebey and Herzlich, 1997, Corollary, p. 7] (see also Theorem 1.3 in the book by Hebey [Hebey, 1996]).

We have:

Corollary 8.1**.**

Let $(M,g)$ be a complete riemannian manifold. If the injectivity radius verifies $r_{inj}(x)\geq i>0$ and the Ricci curvature verifies $Rc_{(M,g)}(x)\geq\lambda g_{x}$ for some $\lambda\in{\mathbb{R}}$ and all $x\in M,$ then there exists a positive constant $\delta>0,$ depending only on $n,\epsilon,\lambda,\alpha,i$ such that for any $x\in M,\ r_{H}(1+\epsilon,0,\alpha)(x)\geq\delta.$

Hence $R_{0,\epsilon}(x)\geq\delta$ all $x\in M.$

If we have $\left|{Rc_{(M,g)}(x)}\right|\leq C$ for all $x\in M,$ then there exists a positive constant $\delta>0,$ depending only on $n,\epsilon,C,\alpha,i$ and $c,$ such that for any $x\in M,\ r_{H}(1+\epsilon,1,\alpha)(x)\geq\delta.$

Hence $R_{1,\epsilon}(x)\geq\delta$ all $x\in M.$

Proof.

The Theorem of Hebey and Herzlich gives that, under these hypotheses, for any $\alpha\in(0,1)$ that $\forall x\in M,\ r_{H}(1+\epsilon,0,\alpha)(x)\geq\delta.$

Now recall that $r_{H}(1+\epsilon,0,\alpha)(x)$ is the sup of $S$ such that, in an harmonic coordinates patch,

1. $(1-\epsilon)\delta_{ij}\leq g_{ij}\leq(1+\epsilon)\delta_{ij}$ in $B(x,S)$ as bilinear forms,

2. $S^{\alpha}(x)\sup\ _{i,j=1,...,n,\ y\neq z\in B(x,S)}\left|{\frac{g_{ij}(y)-g_{ij}(z)}{d_{g}(y,z)^{\alpha}}}\right|\leq\epsilon.$

So when we take the sup for $R_{0,\epsilon}(x)$ on any smooth coordinates patch we get $r_{H}(1+\epsilon,0,\alpha)(x)\leq R_{0,\epsilon}(x).$

The same way we get $r_{H}(1+\epsilon,1,\alpha)(x)\leq R_{1,\epsilon}(x).$

The proof is complete. $\hfill\blacksquare$

As a corollary we retrieve Corollary 3.19, p. 38, in [Hebey, 1996]:

Corollary 8.2**.**

Let $(M,g)$ be a complete riemannian manifold. If we have the injectivity radius bounded below and the Ricci curvature verifying $Rc(x)\geq\lambda g_{x}$ for some $\lambda\in{\mathbb{R}}$ and all $x\in M$ then the Sobolev embeddings for functions are valid in $(M,g).$

Remark 8.3**.**

Because the proof of the Theorem of Hebey and Herzlich does not use the Theorem of Varopoulos, we get here a different proof of Corollary 8.2.

We get also a Sobolev embedding for sections of $G.$

Corollary 8.4**.**

Let $(M,g)$ be a complete riemannian manifold. Let $G:=(H,\pi,M)$ be a complex ${\mathcal{C}}^{m}$ adapted vector bundle over $M.$ If $M$ has [math]-order weak bounded geometry, then the "classical" Sobolev embeddings for sections of $G$ are valid in $(M,g).$

As already said in the introduction, this improves a very well known Theorem by M. Cantor [Cantor, 1974]. Also our proof is completely different.

We also have a global Gaffney’s type inequality:

Corollary 8.5**.**

Let $(M,g)$ be a complete riemannian manifold with a [math]-order weak bounded geometry, then the global Gaffney’s type inequality in $L^{r}$ is valid:

$\displaystyle{\left\|{\omega}\right\|}_{W_{p}^{1,r}(M)}\leq C({\left\|{d(\omega)}\right\|}_{L_{p+1}^{r}(M)}+{\left\|{d^{*}(\omega)}\right\|}_{L_{p-1}^{r}(M)}+{\left\|{\omega}\right\|}_{L_{p}^{r}(M)}).$**

Also Corollary 7.11 take the form:

Corollary 8.6**.**

Let $(M,g)$ be a complete riemannian manifold with a [math]-order weak bounded geometry. Let $r\geq 1$and let $\omega$ be a $p$-form in $M.$ We have:

$\displaystyle{\left\|{\omega}\right\|}_{L_{p}^{s}(M)}\leq C({\left\|{d(\omega)}\right\|}_{L_{p+1}^{r}(M)}+{\left\|{d^{*}(\omega)}\right\|}_{L_{p-1}^{r}(M)}+{\left\|{\omega}\right\|}_{L_{p}^{r}(M)})$**

with $\frac{1}{s}=\frac{1}{r}-\frac{1}{n}>0.$

N. Lohoué [Lohoué, 1985] proved the same result under the stronger hypothesis that $(M,g)$ has a $2$-order bounded geometry plus some other hypotheses on the laplacian and the range of $r.$

8.1 Weak bounded geometry

The Corollary 8.1 gives, in particular, that if $(M,g)$ has a [math]-order weak bounded geometry, then we get that $\forall x\in M,\ R_{(1,\epsilon)}(x)\geq\delta.$ Hence we get the improved classical Sobolev embedding Theorems:

Theorem 8.7**.**

Let $(M,g)$ be a complete riemannian manifold. Let $G:=(H,\pi,M)$ be a complex smooth adapted vector bundle over $M.$ Suppose $(M,g)$ has a [math]-order weak bounded geometry. Let $0\leq k<m$ and $1/s=1/r-(m-k)/n.$ Let $u\in W^{m,r}_{G}(M).$ Then we have $u\in W^{k,s}_{G}(M)$ with the control:

$\displaystyle{\left\|{u}\right\|}^{s}_{W^{k,s}_{G}(M)}\leq C{\left\|{u}\right\|}^{r}_{W^{m,r}_{G}(M)}.$**

We shall give some examples of such a situation.

8.2 Examples of manifolds of bounded geometry.

These examples are precisely taken from  [Eldering, 2012, p. 40], where there is more explanations.

$\bullet$ Euclidean space with the standard metric trivially has bounded geometry.

$\bullet$ A smooth, compact Riemannian manifold $M$ has bounded geometry as well; both the injectivity radius and the curvature including derivatives are continuous functions, so these attain their finite minima and maxima, respectively, on $M.$ If $M\in{\mathcal{C}}^{m+2},$ then it has bounded geometry of order $m.$

$\bullet$ Non compact, smooth Riemannian manifolds that possess a transitive group of isomorphisms (such as the hyperbolic spaces ${\mathbb{H}}^{n}$) have $m$-order bounded geometry since the finite injectivity radius and curvature estimates at any single point translate to a uniform estimate for all points under isomorphisms.

More manifolds of bounded geometry can be constructed with these basic building blocks in the following ways.

$\bullet$ The product of a finite number of manifolds of bounded geometry again has bounded geometry, since the direct sum structure of the metric is inherited by the exponential map and curvature.

$\bullet$ If we take a finite connected sum of manifolds with bounded geometry such that the gluing modifications are smooth and contained in a compact set, then the resulting manifold has bounded geometry again.

8.3 Compact riemannian manifold with smooth boundary.

We have seen that a compact riemannian manifold of class ${\mathcal{C}}^{2}$ without boundary has a [math]-order bounded geometry, so the Sobolev embeddings are valid on it. We shall deduce that they are also valid in the case of a compact riemannian manifold with a ${\mathcal{C}}^{\infty}$ smooth boundary. But first we have, using that a compact riemannian manifold has [math]-order bounded geometry:

Corollary 8.8**.**

Let $(M,g)$ be a compact riemannian manifold without boundary. Let $G:=(H,\pi,M)$ be an adapted complex smooth vector bundle over $M.$

Then the Sobolev embeddings for sections of $G$ are valid in $(M,g).$ Precisely we have: $W_{G}^{m,r}(M)$ is embedded in $W_{G}^{k,s}(M),$ with $\frac{1}{s}=\frac{1}{r}-\frac{(m-k)}{n}>0$ and:

$\displaystyle\forall u\in W_{G}^{m,r}(M),\ {\left\|{u}\right\|}_{W_{G}^{k,s}(M)}\leq C{\left\|{u}\right\|}_{W_{G}^{m,r}(M)}.$**

Now let $M$ be a ${\mathcal{C}}^{\infty}$ smooth connected compact riemannian manifold with a ${\mathcal{C}}^{\infty}$ smooth boundary $\partial M.$ We want to show how the results in case of a compact boundary-less manifold apply to this case.

A classical way to get rid of an "annoying boundary" of a manifold is to use its "double". For instance: Duff [Duff, 1952], Hörmander [Hörmander, 1994, p. 257].

Here we copy the following construction from [Guneysu and Pigola, 2015, Appendix B].

The "Riemannian double" $\Gamma:=\Gamma(M)$ of $M,$ obtained by gluing two copies, $M$ and $M_{2},$ of $M$ along $\partial M,$ is a compact Riemannian manifold without boundary. Moreover, by its very construction, it is always possible to assume that $\displaystyle\Gamma$ contains an isometric copy of the original manifold $M.$ We shall also write $M$ for this isometric copy to ease notation.

We take $u\in W_{G}^{m,r}(M)$ and we want to show that $u\in W_{G}^{k,s}(M),$ with $\frac{1}{s}=\frac{1}{r}-\frac{(m-k)}{n}>0.$

We shall suppose that $G$ extends smoothly to $\Gamma,$ i.e. the connexion is smooth and still is a metric connexion on $\Gamma,$ and the scalar product also is smooth in $\Gamma.$ For instance this is the case if $G=\Lambda^{p}(M),$ the bundle of $p$-forms in $M.$

The Seeley Theorem [Seeley, 1964] in the version of Lions [Lions, 1964], tells us that any function $f\in W^{m,r}(M)$ can be extended to $\Gamma$ as $\tilde{f}\in W^{m,r}(\Gamma).$ By use of a finite covering of $M$ by balls $B(x,R(x))$ with center $x\in M$ and trivializing the bundle $G,$ and an associated partition of unity, this result of Seeley can be made valid to a section $u\in W_{G}^{m,r}(M).$ So we have an extension $\tilde{u}\in W_{G}^{m,r}(\Gamma).$

Using Corollary 8.8 we get that $\tilde{u}\in W_{G}^{k,s}(\Gamma),$ with $\frac{1}{s}=\frac{1}{r}-\frac{(m-k)}{n}>0$ and:

$\displaystyle\forall\tilde{u}\in W_{G}^{m,r}(\Gamma),\ {\left\|{\tilde{u}}\right\|}_{W_{G}^{k,s}(\Gamma)}\leq C{\left\|{\tilde{u}}\right\|}_{W_{G}^{m,r}(\Gamma)}.$

Hence, restricting $\tilde{u}$ to $M$ we get, a fortiori, $u\in W_{G}^{k,s}(M),$ and:

$\displaystyle\forall u\in W_{G}^{m,r}(M),\ {\left\|{u}\right\|}_{W_{G}^{k,s}(M)}\leq C{\left\|{u}\right\|}_{W_{G}^{m,r}(M)}.$

So we proved:

Theorem 8.9**.**

Let $(M,g)$ be a compact riemannian manifold with a smooth boundary. Let $G:=(H,\pi,M)$ be a complex vector bundle over $M,$ which admit a smooth adapted extension to a "double" manifold $\Gamma.$ Then the Sobolev embeddings for sections of $G$ are valid in $(M,g).$ Precisely we have: $W_{G}^{m,r}(M)$ is embedded in $W_{G}^{k,s}(M),$ with $\frac{1}{s}=\frac{1}{r}-\frac{(m-k)}{n}>0$ and:

$\displaystyle\forall u\in W_{G}^{m,r}(M),\ {\left\|{u}\right\|}_{W_{G}^{k,s}(M)}\leq C{\left\|{u}\right\|}_{W_{G}^{m,r}(M)}.$**

8.4 Hyperbolic manifolds.

These are manifolds such that the sectional curvature $K_{M}$ is constantly $-1.$ For them we have first that the Ricci curvature is bounded.

Lemma 8.10**.**

Let $(M,g)$ be a complete Riemannian manifold such that $H\leq K_{M}\leq K$ for constants $H,K\in{\mathbb{R}}.$ Then we have that ${\left\|{Rc}\right\|}_{\infty}\leq\max(\left|{H}\right|,\left|{K}\right|).$

Proof.

Take a tangent vector $v\in T_{x}M,$ with $\left|{v}\right|=1,$ we obtain the Ricci curvature $Rc$ of $v$ at $x$ by extending $v=v_{n}$ to an orthonormal basis $v_{1},...,v_{n}.$ Then we compute the Ricci curvature along $v$:

$\displaystyle Rc_{x}(v)=\frac{1}{n-1}\sum_{j=1}^{n-1}{{\left\langle{R_{x}(v,v_{j})v,v_{j}}\right\rangle}}.$

where $R$ denotes the Riemannian curvature tensor. On the other hand, the sectional curvature $K_{M,x}(v,v_{j})$ for $j<n$ is given by (remember the $v_{j}$ are orthonormal):

$\displaystyle K_{M,x}(v,v_{j})={\left\langle{R_{x}(v,v_{j})v,v_{j}}\right\rangle}$

So we get

$\displaystyle Rc_{x}(v)=\frac{1}{n-1}\sum_{j=1}^{n-1}{{\left\langle{R_{x}(v,v_{j})v,v_{j}}\right\rangle}}=\frac{1}{n-1}\sum_{j=1}^{n-1}{K_{M,x}(v,v_{j})}$

hence $H\leq Rc_{x}(v)\leq K\Rightarrow\left|{Rc_{x}}\right|\leq\max(\left|{H}\right|,\left|{K}\right|).$

To get estimates on the Ricci tensor $Rc_{(M,g)}(x)(u,v),$ we notice that $Rc_{x}(v)=Rc_{(M,g)}(x)(v,v)$ and we get the estimates by polarisation. $\hfill\blacksquare$

To get that the injectivity radius $r_{inj}(x)$ is bounded below we shall use a Theorem by Cheeger, Gromov and Taylor [Cheeger et al., 1982]:

Theorem 8.11**.**

Let $(M,g)$ be a complete Riemannian manifold such that $K_{M}\leq K$ for constants $K\in{\mathbb{R}}.$ Let $0<r<\frac{\pi}{4{\sqrt{K}}}$ if $K>0$ and $r\in(0,\infty)$ if $K\leq 0.$ Then the injectivity radius $r_{inj}(x)$ at $x$ satisfies

$\displaystyle r_{inj}(x)\geq r\frac{\mathrm{V}\mathrm{o}\mathrm{l}(B_{M}(x,r))}{\mathrm{V}\mathrm{o}\mathrm{l}(B_{M}(x,r))+\mathrm{V}\mathrm{o}\mathrm{l}(B_{T_{x}M}(0,2r))},$**

where $B_{T_{x}M}(0,2r))$ denotes the volume of the ball of radius $2r$ in $T_{x}M,$ where both the volume and the distance function are defined using the metric $g^{*}:=\exp_{p}^{*}g$ i.e. the pull-back of the metric $g$ to $T_{x}M$ via the exponential map.

This Theorem leads to the definition:

Definition 8.12**.**

Let $(M,g)$ be a Riemannian manifold. We shall say that it has the lifted doubling property if we have:

$\displaystyle(LDP)\ \ \ \ \ \ \ \ \ \exists\beta,\gamma>0::\forall x\in M,\ \exists r\geq\beta,\ \mathrm{V}\mathrm{o}\mathrm{l}(B_{T_{x}M}(0,2r))\leq\gamma\mathrm{V}\mathrm{o}\mathrm{l}(B_{M}(x,r)),$**

where $B_{T_{x}M}(0,2r))$ denotes the volume of the ball of radius $2r$ in $T_{x}M,$ and both the volume and the distance function are defined on $T_{x}M$ using the metric $g^{*}:=\exp_{p}^{*}g$ i.e. the pull-back of the metric $g$ to $T_{x}M$ via the exponential map.

Hence we get:

Corollary 8.13**.**

Let $(M,g)$ be a complete Riemannian manifold such that $K_{M}\leq K$ for a constant $K\in{\mathbb{R}}.$ For instance an hyperbolic manifold. Suppose moreover that $(M,g),$ has the lifted doubling property. Then:

$\displaystyle\forall x\in M,\ r_{inj}(x)\geq\frac{\beta}{1+\gamma}.$**

Proof.

By the (LDP) we get, for a $r\geq\beta,$

$\displaystyle\mathrm{V}\mathrm{o}\mathrm{l}(B_{T_{x}M}(0,2r))\leq\gamma\mathrm{V}\mathrm{o}\mathrm{l}(B_{M}(x,r)).$

We apply Theorem 8.11 of Cheeger, Gromov and Taylor to get

$\displaystyle r_{inj}(x)\geq r\frac{\mathrm{V}\mathrm{o}\mathrm{l}(B_{M}(x,r))}{\mathrm{V}\mathrm{o}\mathrm{l}(B_{M}(x,r))+\mathrm{V}\mathrm{o}\mathrm{l}(B_{T_{x}M}(0,2r))}.$

So

$\displaystyle\frac{\mathrm{V}\mathrm{o}\mathrm{l}(B_{M}(x,r))}{\mathrm{V}\mathrm{o}\mathrm{l}(B_{M}(x,r))+\mathrm{V}\mathrm{o}\mathrm{l}(B_{T_{x}M}(0,2r))}\geq\frac{1}{1+\gamma}$

hence, because $r\geq\beta,$ we get the result. $\hfill\blacksquare$

So finally we get

Corollary 8.14**.**

Let $(M,g)$ be a complete Riemannian manifold such that $H\leq K_{M}\leq K$ for constants $H,K\in{\mathbb{R}},$ where $K_{M}$ is the sectional curvature of $M.$ Let $G:=(H,\pi,M)$ be an adapted complex smooth vector bundle over $M.$ Suppose moreover that $(M,g)$ has the lifted doubling property. Then $\exists\delta>0,\ \forall x\in M,\ R_{1,\epsilon}(x)\geq\delta.$

This implies that the Sobolev embeddings are valid for sections of $G$ in that case.

Proof.

Because the Lemma 8.10 gives that the Ricci curvature is bounded. The Corollary 8.13 gives $\forall x\in M,\ r_{inj}(x)\geq\frac{\beta}{1+\gamma}.$ So the Corollary 8.1 completes the proof. $\hfill\blacksquare$

Remark 8.15**.**

In the case the hyperbolic manifold $(M,g)$ is simply connected, then by the Hadamard Theorem [do Carmo, 1993, Theorem 3.1, p. 149], we get that the injectivity radius is $\infty,$ so we have also the classical embedding Theorems in this case, even for sections of $G.$