Anisotropic Sobolev spaces with weights

TL;DR

This paper develops a framework for Sobolev spaces with weights tailored to singular elliptic operators in a half-space, enhancing understanding of their functional analytic properties and potential applications.

Contribution

It introduces anisotropic weighted Sobolev spaces adapted to specific singular elliptic operators in the half-space setting.

Findings

Characterization of weighted Sobolev spaces for singular elliptic operators

Analysis of regularity properties in weighted spaces

Potential applications to boundary value problems with singular coefficients

Abstract

We study Sobolev spaces with weights in the half-space $\mathbb{R}^{N+1}_+=\{(x,y): x \in \mathbb{R}^N, y>0\}$, adapted to the singular elliptic operators \begin{equation*} \mathcal L =y^{\alpha_1}\Delta_{x} +y^{\alpha_2}\left(D_{yy}+\frac{c}{y}D_y -\frac{b}{y^2}\right). \end{equation*}