Characterization of functions with zero traces via the distance function and Lorentz spaces

Abstract

Consider a regular domain $\Omega \subset {\mathbb{R}}^N$ and let $d(x)=\mathrm{dist}(x,\partial \Omega )$. Denote $L_a^{1,\infty }(\Omega )$ the space of functions from $L^{1,\infty }(\Omega )$ having absolutely continuous quasinorms. This set is essentially smaller than $L^{1,\infty }(\Omega )$ but, at the same time, essentially larger than a union of all $L^{1,q}(\Omega )$, $q\in [1,\infty )$. A classical result of late 1980's states that for $p\in (1,\infty )$ and $m\in \mathbb{N}$, $u$ belongs to the Sobolev space $W_0^{m,p}(\Omega )$ if and only if $u/d^m\in L^p(\Omega )$ and $\left|{\nabla }^{m}u\right|\in L^p(\Omega )$. During the consequent decades, several authors have spent considerable effort in order to relax the characterizing condition. Recently, it was proved that $u\in W_0^{m,p}(\Omega )$ if and only if $u/d^m\in L^1(\Omega )$ and $\left|{\nabla }^{m}u\right|\in L^p(\Omega )$. In this paper we show that for $N\ge 1$ and $p\in (1,\infty )$ we have $u\in W_0^{1,p}(\Omega )$ if and only if $u/d\in L_a^{1,\infty }(\Omega )$ and $\left|\nabla u\right|\in L^p(\Omega )$. Moreover, we present a counterexample which demonstrates that after relaxing the condition $u/d\in L_a^{1,\infty }(\Omega )$ to $u/d\in L^{1,\infty }(\Omega )$ the equivalence no longer holds.

25 pages, 4 figures

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