Sobolev embeddings on domains involving two types of symmetries

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.