Sobolev-Lorentz capacity and its regularity in the Euclidean setting

Abstract

This paper studies the Sobolev-Lorentz capacity and its regularity in the Euclidean setting for $n \ge 1$ integer. We extend here our previous results on the Sobolev-Lorentz capacity obtained for $n \ge 2.$ Moreover, for $n \ge 2$ integer we obtain a few new results concerning the $n,1$ relative and global capacities. We obtain sharp estimates for the $n,1$ relative capacity of the concentric condensers $(\overline{B}(0,r), B(0,1))$ for all $r$ in $[0,1).$ As a consequence we obtain the exact value of the $n,1$ capacity of a point relative to all its bounded open neighborhoods from ${\mathbf{R}}^n$ when $n \ge 2.$ We also show that this aforementioned constant is the value of the $n,1$ global capacity of any point from ${\mathbf{R}}^n,$ where $n \ge 2$ is integer. This allows us to give a new proof of the embedding $H_{0}^{1,(n,1)}(\Omega) \hookrightarrow C(\overline{\Omega}) \cap L^{\infty}(\Omega),$ where $\Omega \subset {\mathbf{R}}^n$ is open and $n \ge 2$ is an integer. In the penultimate section of our paper we prove a new weak convergence result for bounded sequences in the non-reflexive spaces $H^{1,(p,1)}(\Omega)$ and $H_{0}^{1,(p,1)}(\Omega).$ The weak convergence result concerning the spaces $H^{1,(p,1)}(\Omega)$ is valid whenever $1<p<\infty,$ while the weak convergence result concerning the spaces $H_{0}^{1,(p,1)}(\Omega)$ is valid whenever $1 \le n<p<\infty$ or $1<n=p<\infty.$ As a consequence of the weak convergence result concerning the spaces $H_{0}^{1,(p,1)}(\Omega),$ in the last section of our paper we show that the relative and the global $(p,1)$ and $p,1$ capacities are Choquet whenever $1 \le n<p<\infty$ or $1<n=p<\infty.$