On the Interior $W^{1,p}$ estimate of extension problem of fractional elliptic equation

TL;DR

This paper investigates the W^{1,p} regularity of the extension problem associated with fractional elliptic equations, providing L^p estimates for solutions and advancing understanding of fractional operator regularity.

Contribution

It offers new insights into the interior W^{1,p} estimates for the extension problem of fractional elliptic equations, a key step in analyzing fractional operators.

Findings

Established interior W^{1,p} estimates for the extension problem

Derived L^p estimates for solutions of fractional elliptic equations

Enhanced understanding of regularity properties of fractional operators

Abstract

In order to study Fractional operator, Caffarelli introduced the concept of extension problem. Hence, for any fractional elliptic operator, we get a degenerate elliptic equation. By studying the W^{1,p} regularity of extension problem, we can get several L^p estimate of solution of original fractional elliptic equation.