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

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