Weighted estimates for time-fractional parabolic equations with VMO coefficients

TL;DR

This paper establishes weighted estimates and solvability results for time-fractional parabolic equations with VMO coefficients, extending to half-space domains using advanced harmonic analysis tools.

Contribution

It introduces new weighted estimates for time-fractional parabolic equations with VMO coefficients and addresses solvability in half-space domains.

Findings

Weighted estimates for equations with VMO coefficients

Solvability results in whole space and half-space

Application of harmonic analysis techniques

Abstract

This paper is devoted to the weighted estimates and the solvability of time-fractional parabolic equations. The leading coefficients \(a^{ij}(t,x)\) are assumed to have small mean oscillations in \((t,x)\) locally, in both non-divergence and divergence forms, in the whole space. By employing appropriate odd and even extensions along with suitable boundary value conditions, we derive the corresponding results for the half-space. The proofs rely on the application of the Fefferman-Stein theorem and the Hardy-Littlewood maximal function theorem in the context of weighted mixed spaces.