Sobolev embeddings with weights in complete Riemannian manifolds
Introduction
In the following will be a smooth connected complete Riemannian manifold without boundary unless otherwise stated. We shall just say “Riemannian manifold” to mean it.
The Sobolev inequalities in for functions play a major role in the study of differential operators and nonlinear functional analysis. They are valid in or if is compact. More generally we have:
Theorem 1.1 Let be a Riemannian manifold of dimension with Ricci curvature bounded from below. The Sobolev embeddings for functions are valid for 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 .
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 be a Riemannian manifold and let be a smooth complex vector bundle over with a metric connection on , i.e. for any vector field on we have for any smooth sections of . 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 , 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 be a Riemannian manifold and . We shall say that the geodesic ball is -admissible if there is a chart such that: in as bilinear form. We shall say that the geodesic ball is -admissible if moreover .
We shall denote with the set of -admissible balls on and the set of -admissible balls on .
The -admissible balls will be adapted to functions and the -admissible balls will be adapted to sections of the vector bundle . We shall use the shortcut . to mean such that in formula.
Definition 1.3 Let , we set . We shall say that is the -admissible radius at .
Here will be either or depending on the context, functions or sections of . The notation will mean either or depending on the choice of .
Let be a weight on , i.e. a positive measurable function on . If and are given, we denote by the space of smooth sections of in such that for with the pointwise modulus associated to the pointwise scalar product and where is the connection associated to . Let be the completion of . See the precise Definition 4.1.
Our general result with weights is:
Theorem 1.4 Let be a Riemannian manifold of dimension . Let be an adapted vector bundle over . Let and . Let and with . Then is embedded in and:
Because the admissible radius is always smaller than , to get rid of the weights we just need that there exists a such that for any , we have .
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 be a Riemannian manifold. If the injectivity radius verifies and the Ricci curvature verifies for some and all , then there exists a positive constant , depending only on such that for any . If moreover we have for all , then there exists a positive constant , depending only on and , such that for any .
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 has -order weak bounded geometry if: the injectivity radius at is bounded below by some constant for any for , the covariant derivatives of the Ricci curvature tensor are bounded in norm.
Recall the classical Theorem of Cantor [4]:
Theorem 1.7 Let be a Riemannian manifold. Let be a complex smooth adapted vector bundle over . Suppose moreover that verifies: C1: The injectivity radius of is bounded away from zero. C2: There is a such that for each and , the sectional curvature satisfies . Let and . Let . Then we have with the control:
Here we prove:
Theorem 1.8 Let be a Riemannian manifold. Let be a complex smooth adapted vector bundle over . Suppose has a -order weak bounded geometry. Let and . Let . Then we have with the control: In the case of functions instead of sections of , the conditions are weaker: If the injectivity radius of is bounded away from zero and if the Ricci curvature verifies for some and all , we get
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 be a Riemannian manifold. Let be the exterior derivation on and its formal adjoint. Let be a -differential form in . If has a -order weak bounded geometry, then we have, with :
And, as an easy corollary:
Corollary 1.10 Let be a Riemannian manifold with a -order weak bounded geometry. Let and let be a -differential form in . We have: with .
N. Lohoué [17] proved the same result under the stronger hypothesis that has a -order bounded geometry plus some other hypotheses on the Laplacian and the range of . Here already -order weak bounded geometry is enough.
This work is presented as follows:
In the next Section we study the main properties of the -admissible balls.
In Section 3 we define the vector bundle we are interested in and precise the metric connection on it we shall use.
In Section 4 we define the Sobolev spaces of smooth sections of , 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 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 in the -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 -admissible balls via a Vitali type covering in Section 6.
In Section 7 we prove the global estimates for functions, sections of and Gaffney type in .
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.
Section snippets
Admissible balls
Recall the definition of the admissible radius:
Definition 2.1 Let be a Riemannian manifold. Let , we set . We shall say that is the -admissible radius at .
Then clearly if , i.e. is -admissible, then so is if .
Remark 2.2 Let . Suppose that , where is the Riemannian distance between and . Consider the ball of center and radius . This ball is contained in hence, by definition of
Vector bundle
Let be a Riemannian manifold and let be an adapted complex vector bundle over . Recall that this means that has a smooth scalar product and a metric connection , i.e. verifying , where is the exterior derivative on . See [21, Section 13].
Lemma 3.1 The -admissible balls trivialize the bundle .
Proof Because if is an -admissible ball, we have by Remark 2.3 that . Then, one can choose a local frame field for on by
Sobolev spaces for sections of with weight
We have seen that . On the tensor product of two Hilbert spaces we put the canonical scalar product , with , and completed by linearity to all elements of the tensor product. On we have the Levi-Civita connection , which is of course a metric one, and on we have the metric connection so we define a connection on the tensor product : by asking that this connection be a derivation. We get easily that
Sobolev comparison estimates for functions
Lemma 5.1 We have the Sobolev comparison estimates, where is a -admissible ball in and is the admissible chart relative to , and, with the Euclidean ball in centered at and of radius , The constants depending only on and not on .
Proof We have to compare the norms of , with the corresponding ones for in . First we have because
Vitali covering
Lemma 6.1 Let be a collection of balls in a metric space, with . There exists a disjoint subcollection of with the following properties:
This is a well known lemma, see for instance [11], section 1.5.1.
Fix and let where is the admissible radius at . We built a Vitali covering with the collection . The previous lemma gives a disjoint subcollection such that every ball
Global estimates for sections of
Lemma 7.1 We have, for any section of , with and , with and , that:
Proof We consider the function . We set to get an adapted weight. Let , then . We have, because is a covering of ,
Because, by Lemma 2.4, we have that for any , then we get
Applications
We shall give some examples where we have classical estimates using that for any , via [14, Corollary, p. 7] (see also Theorem 1.3 in the book by Hebey [13]).
We have:
Corollary 8.1 Let be a Riemannian manifold. If the injectivity radius verifies and the Ricci curvature verifies for some and all , then there exists a positive constant , depending only on such that for any . Hence for all . If we have for
References (23)
- et al.
Calderon-Zygmund inequality and Sobolev spaces on noncompact riemannian manifolds
Adv. Math.
(2015) Solutions of elliptic equation in a complete riemannian manifold
J. Geom. Anal.
(2018)Sobolev solutions of parabolic equation in a complete riemannian manifold
(2019)Sobolev inequalities for riemannian bundles
Bull. Amer. Math. Soc.
(1974)Inégalités Isopérimétriques sur les Variétés Riemanniennes
(1994)- et al.
Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete riemannian manifolds
J. Differential Geom.
(1982) - et al.
Isopérimétrie pour les groupes et les variétés
Rev. Math. Iberoam.
(1993) Differential forms in manifolds with boundary
Ann. Math.
(1952)