On weighted critical imbeddings of Sobolev spaces

Abstract

Our concern in this paper lies with two aspects of weighted exponential spaces connected with their role of target spaces for critical imbeddings of Sobolev spaces. We characterize weights which do not change an exponential space up to equivalence of norms. Specifically, we first prove that $${L_{\exp t^{\alpha}}(\chi_B)=L_{\exp t^{\alpha}}(\rho)}$$ if and only if $${\rho^q \in L_q}$$ with some q > 1. Second, we consider the Sobolev space $${W^{1}_{N}(\varOmega),}$$ where Ω is a bounded domain in $${\mathbb{R}^{N}}$$ with a sufficiently smooth boundary, and its imbedding into a weighted exponential Orlicz space $${L_{\exp t^{p'}}(\varOmega,\rho)}$$ , where ρ is a radial and non-increasing weight function. We show that there exists no effective weighted improvement of the standard target $${L_{\exp t^{N'}}(\varOmega)=L_{\exp t^{N'}}(\varOmega,\chi_{\varOmega})}$$ in the sense that if $${W^{1}_{N}(\varOmega)}$$ is imbedded into $${L_{\exp t^{p'}}(\varOmega,\rho),}$$ then $${L_{\exp t^{p'}}(\varOmega,\rho)}$$ and $${L_{\exp t^{N'}}(\varOmega)}$$ coincide up to equivalence of the norms; that is, we show that there exists no effective improvement of the standard target space. The same holds for critical cases of higher-order Sobolev spaces and even Besov and Lizorkin–Triebel spaces.