Weighted Sobolev-type Embedding Theorems for Functions with Symmetries

It is well known that Sobolev embeddings can be refined in the presence of symmetries. Hebey and Vaugon (1997) studied this phenomena in the context of an arbitrary Riemannian manifold M and a compact group of isometries G. They showed that the limit Sobolev exponent increases if there are no points in M with discrete orbits under the action of G. In the paper, the situation where M contains points with discrete orbits is considered. It is shown that the limit Sobolev exponent for W 1 p (M) increases in the case of embeddings into weighted spaces Lq(M, w) instead of the usual Lq spaces, where the weight function w(x) is a positive power of the distance from x to the set of points with discrete orbits. Also, embeddings of W 1 p (M) into weighted Hölder and Orlicz spaces are treated. Introduction It is well known that Sobolev embeddings can be refined if we deal with subspaces of functions invariant under some group of symmetries. This phenomenon was used in some particular cases to prove the existence of solutions of various boundary value problems (see, e.g., [1]–[5]; a similar effect for trace embeddings was employed, e.g., in the recent paper [6]). In [7], this problem was studied in the more general context of an arbitrary Riemannian manifold and a compact group of isometries. The critical embedding exponent increases if the manifold contains no points with orbits of dimension zero, due to reduction of the effective manifold dimension. Our goal is to consider the case where the manifold contains points with zero-dimensional orbits. The conventional embedding theorem cannot be refined in this case. For example, if p < n, then, in general, even a radially symmetric function in W 1 p (B R) may fail to be integrable in a power exceeding p∗ = np n−p (here and in the sequel, B n R(X) stands for the ball of radius R in R centered at X, B R = B n R(0)). However, the origin is the only possible singular point for radial functions, and their properties improve after multiplication by a positive power of r = |x|. In [8, Theorem 2.5], it was shown that, for q ≥ p∗ and α > np − n q − 1, the set of radially symmetric functions f ∈ W 1 p (B 1 ) is compactly embedded in Lq(B 1 ; r ). 2000 Mathematics Subject Classification. Primary 46E35; Secondary 58D99.