Traces of weighted Sobolev spaces with Muckenhoupt weight. The case $p=1$

Abstract

A complete description of traces on ${\mathbb{R}}^n$ of functions from the weighted Sobolev space $W_1^l({\mathbb{R}}^{n+1},\gamma )$, $l\in \mathbb{N}$, with weight $\gamma \in A_1^{\mathrm{loc}}({\mathbb{R}}^{n+1})$ is obtained. In the case $l=1$ the proof of the trace theorems is based on a~special nonlinear algorithm for constructing a~system of tilings of the space~${\mathbb{R}}^n$. As the trace of the space $W_1^1({\mathbb{R}}^{n+1},\gamma )$ we have the new function space $Z(\{{\gamma }_{k,m}\})$.

arXiv commentAccepted in Nonlinear Analysis: Theory, Methods and Applications (final version)