Harnack's Inequality and A Priori Estimates for Fractional Powers of Non-symmetric Differential Operators

TL;DR

This paper develops a new extension theorem in Banach spaces for non-symmetric operators and applies it to derive Harnack inequalities and regularity results for fractional powers of various second-order differential operators, including non-self-adjoint and rough coefficient cases.

Contribution

It introduces a general extension theorem for non-symmetric operators and applies it to establish Harnack estimates and regularity for fractional powers of diverse differential operators.

Findings

Established Harnack inequalities for fractional non-symmetric operators

Derived a priori regularity estimates for solutions

Extended reflection techniques to broader classes of operators

Abstract

We obtain a new general extension theorem in Banach spaces for operators which are not required to be symmetric, and apply it to obtain Harnack estimates and a priori regularity for solutions of fractional powers of several second order differential operators. These include weighted elliptic and subellitptic operators in divergence form (nonnecessarily self-adjoint), and nondivergence form operators with rough coefficients. We utilize the reflection extension technique introduced by Caffarelli and Silvestre.