Sobolev embeddings with weights in complete Riemannian manifolds

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.

Introduction

In the following $M≔(M,g)$ will be a ${\mathcal{C}}^{\infty }$ smooth connected complete Riemannian manifold without boundary unless otherwise stated. We shall just say “Riemannian manifold” to mean it.

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 $M$ be a 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 ${inf}_{x\in M}\mathrm{V}\mathrm{o}\mathrm{l}\left(B(x,1)\right)>0$.

The necessity is a well known fact, see p.18 in [13] and was generalized by Carron [5].

The sufficiency of Theorem 1.1 was done by Varopoulos [22], see also Theorem 3.18, p. 37 in [13], based on the work of Coulhon and Saloff-Coste [7].

Let $M$ be a Riemannian manifold and let $G\to M$ be a smooth complex vector bundle over $M$ with a metric connection ${\nabla }^G$ on $G$, i.e. for any vector field $X$ on $M$ we have $X(u,v)=({\nabla }_{X}u,v)+(u,{\nabla }_{X}v)$ 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 the Riemannian manifold $M$, 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 $M$ be a Riemannian manifold and $x\in M$. We shall say that the geodesic ball $B(x,R)$ is $(0,ϵ)$-admissible if there is a chart $(B(x,R),\phi )$ such that:

• $(1-ϵ){\delta }_{ij}\le g_{ij}\le (1+ϵ){\delta }_{ij}$ in $B(x,R)$ as bilinear form.

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

• ${\sum }_{\left|\beta \right|=1}{R}^{}sup{}_{i,j=1,\dots ,n,y\in B_x\left(R\right)}\left|{\partial }^{\beta }{g}_{ij}\left(y\right)\right|\le ϵ$.

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

The $(0,ϵ)$-admissible balls will be adapted to functions and the $(1,ϵ)$-admissible balls will be adapted to sections of the vector bundle $G$. We shall use the shortcut $s.t$. to mean such that in formula.

Definition 1.3

Let $x\in M$, we set $R^{\prime }\left(x\right)≔sup\{R>0s.t.B(x,R)\in \mathcal{A}\left(ϵ\right)\}$. We shall say that $R_{ϵ}\left(x\right)≔min(1,R^{\prime }\left(x\right)∕2)$ is the $ϵ$-admissible radius at $x$.

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

Let $w$ be a weight on $M$, i.e. a positive measurable function on $M$. If $k\in \mathbb{N}$ and $r\ge 1$ are given, we denote by ${\mathcal{C}}_G^{k,r}(M,w)$ the space of smooth sections $\omega$ of $G$ in ${\mathcal{C}}^{\infty }\left(M\right)$ such that $\left|{\nabla }^j\omega \right|\in L^r(M,w)$ for $j=0,\dots ,k$ with the pointwise modulus associated to the pointwise scalar product and where $\nabla$ is the connection associated to $G$. Let $W_G^{k,r}(M,w)$ be the completion of ${\mathcal{C}}_G^{k,r}(M,w)$. See the precise Definition 4.1.

Our general result with weights is:

Theorem 1.4

Let $M$ be a Riemannian manifold of dimension $n$. Let $G$ be an adapted vector bundle over $M$. Let $r\ge 1,m\ge k\ge 0$ and $\frac{1}{s}=\frac{1}{r}-\frac{(m-k)}{n}>0$. Let $w\left(x\right)≔R_{ϵ}{\left(x\right)}^{\gamma }$ and $w^{\prime }≔R_{ϵ}{\left(x\right)}^{\nu }$ with $\nu ≔s(2+\gamma ∕r)$. Then $W_G^{m,r}(M,w)$ is embedded in $W_G^{k,s}(M,w^{\prime })$ and:

$$\forall u\in W_G^{m,r}(M,w),{\Vert u\Vert }_{W_G^{k,s}(M,w^{\prime })}\le C{\Vert u\Vert }_{W_G^{m,r}(M,w)}.$$

Because the admissible radius $R_{ϵ}\left(x\right)$ is always smaller than $1$, to get rid of the weights we just need that there exists a $\delta >0$ such that for any $x\in M$, we have $R\left(x\right)\ge \delta$.

This is precisely what is given by a Theorem of Hebey–Herzlich  [14, Corollary, p. 7], which has the easy Corollary:

Theorem 1.5

Let $M$ be a Riemannian manifold. If the injectivity radius verifies $r_{inj}\left(x\right)\ge i>0$ and the Ricci curvature verifies $R{c}_{(M,g)}\left(x\right)\ge \lambda g_x$ for some $\lambda \in \mathbb{R}$ and all $x\in M$, then there exists a positive constant $\delta >0$, depending only on $n,ϵ,\lambda ,i$ such that for any $x\in M,R_{0,ϵ}\left(x\right)\ge \delta$.

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

So the following results will be consequences of 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:

$•$ the injectivity radius $r\left(x\right)$ at $x\in M$ is bounded below by some constant $\delta >0$ for any $x\in M$

$•$ for $0\le j\le k$, the covariant derivatives ${\nabla }^j{R}_c$ of the Ricci curvature tensor are bounded in $L^{\infty }\left(M\right)$ norm.

Recall the classical Theorem of Cantor [4]:

Theorem 1.7

Let $(M,g)$ be a Riemannian manifold. Let $G\to M$ be a complex smooth adapted vector bundle over $M$. Suppose moreover that $M$ 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 satisfies $\left|K_x(V,W)\right|<\delta$.

Let $0\le k<m$ and $1∕s=1∕r-(m-k)∕n$. Let $u\in W_G^{m,r}\left(M\right)$. Then we have $u\in W_G^{k,s}\left(M\right)$ with the control:

$${\Vert u\Vert }_{W_G^{k,s}\left(M\right)}^s\le C{\Vert u\Vert }_{W_G^{m,r}\left(M\right)}^r.$$

Here we prove:

Theorem 1.8

Let $M$ be a Riemannian manifold. Let $G\to M$ be a complex smooth adapted vector bundle over $M$. Suppose $M$ has a $0$-order weak bounded geometry. Let $0\le k<m$ and $1∕s=1∕r-(m-k)∕n$. Let $u\in W_G^{m,r}\left(M\right)$. Then we have $u\in W_G^{k,s}\left(M\right)$ with the control:

$${\Vert u\Vert }_{W_G^{k,s}\left(M\right)}^s\le C{\Vert u\Vert }_{W_G^{m,r}\left(M\right)}^r.$$

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 $R{c}_{(M,g)}\left(x\right)\ge \lambda g_x$ for some $\lambda \in \mathbb{R}$ and all $x\in M$, we get

$${\Vert u\Vert }_{W^{k,s}\left(M\right)}^s\le C{\Vert u\Vert }_{W^{m,r}\left(M\right)}^r.$$

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 $M$ be a Riemannian manifold. Let $d$ be the exterior derivation on $M$ and $d^{\ast }$ its formal adjoint. Let $\omega$ be a $p$-differential form in ${\mathcal{C}}_0^{\infty }\left({\Lambda }^p\left(M\right)\right)$. If $M$ has a $0$-order weak bounded geometry, then we have, with $r\ge 1$:

$${\Vert \omega \Vert }_{W_p^{1,r}\left(M\right)}\le C({\Vert d\left(\omega \right)\Vert }_{L_{p+1}^r\left(M\right)}+{\Vert d^{\ast }\left(\omega \right)\Vert }_{L_{p-1}^r\left(M\right)}+{\Vert \omega \Vert }_{L_p^r\left(M\right)}).$$

And, as an easy corollary:

Corollary 1.10

Let $M$ be a Riemannian manifold with a $0$-order weak bounded geometry. Let $r\ge 1$ and let $\omega$ be a $p$-differential form in ${\mathcal{C}}_0^{\infty }\left({\Lambda }^p\left(M\right)\right)$. We have:

$${\Vert \omega \Vert }_{L_p^s\left(M\right)}\le C({\Vert d\left(\omega \right)\Vert }_{L_{p+1}^r\left(M\right)}+{\Vert d^{\ast }\left(\omega \right)\Vert }_{L_{p-1}^r\left(M\right)}+{\Vert \omega \Vert }_{L_p^r\left(M\right)})$$

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

N. Lohoué [17] proved the same result under the stronger hypothesis that $M$ has a $2$-order bounded geometry plus some other hypotheses on the Laplacian and the range of $r$. Here already $0$-order weak bounded geometry is enough.

This work is presented as follows:

$•$ In the next Section we study the main properties of the $ϵ$-admissible balls.

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

$•$ In Section 4 we define the Sobolev spaces of smooth sections of $G$, with weights. We prove in this Section a generalization of a nice result of T. Aubin [3] 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.

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

$•$ In Section 5.3 we also prove a local Gaffney type inequality, using a result by C. Scott [19].

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

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

$•$ 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 [14]. This implies the validity of Sobolev embeddings for vector bundles in compact Riemannian manifold without boundary.

$•$ In Section 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.

$•$ Finally in Section 8.4 we introduce the lifted doubling property and we study the case of hyperbolic manifolds.