Weighted Sobolev-type embedding theorems for functions with symmetries

Abstract:

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 $\mathcal {M}$ and a compact group of isometries $G$. They showed that the limit Sobolev exponent increases if there are no points in $\mathcal {M}$ with discrete orbits under the action of $G$.

In the paper, the situation where $\mathcal {M}$ contains points with discrete orbits is considered. It is shown that the limit Sobolev exponent for $W^1_p({\mathcal {M}})$ increases in the case of embeddings into weighted spaces $L_q({\mathcal {M}},w)$ instead of the usual $L_q$ 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 $W^1_p({\mathcal {M}})$ into weighted Hölder and Orlicz spaces are treated.