Sobolev embeddings with weights in complete Riemannian manifolds (Q2182007)

Let \(M:=(M,g)\) be a \(C^\infty\) smooth connected complete Riemannian manifold without boundary. It is well known that Sobolev inequalities in \(M\) for functions (which play an important role in the study of differential operators and nonlinear functional analysis) are valid in \(\mathbb R^n\) or if \(M\) is compact. In the present paper, the author proves Sobolev embedding theorems with weights for vector bundles in a complete Riemannian manifold. He also proves a general Gaffney-type inequality with weights. He improvesthe classical Sobolev embeddings to the case of Riemannian manifolds with weak bounded geometry. Then he deduces the validity of Sobolev embeddings for vector bundles in compact Riemannian manifolds with smooth boundary. Finally, he introduces the lifted doubling property and studies the case of hyperbolic manifolds.