It is well known that Sobolev embeddings can be improved in the presence of symmetries. In this article, we considere the situation in which given a domain $\Omega=\Omega_1 \times \Omega_2$ in $\mathbb{R}^N$ with a cylindrical symmetry, and acting a group $G$ in $\Omega_1$, for this situation it is shown that the critical Sobolev exponent increases in the case of embeddings into weighted spaces $L^{q}_{h}(\Omega)$. In this paper, we will enunciate several results related to compact embeddings of a Sobolev space with radially symmetric functions into some weighted space $L^{q}$, with $q$ higher than the usual critical exponent.
Comment: 17 pages