Double phase meets Muckenhoupt

TL;DR

This paper extends classical Muckenhoupt weight theory to the double phase model using generalized Orlicz spaces, establishing key boundedness and regularity results for solutions.

Contribution

It introduces a Muckenhoupt-type condition for the double phase model, enabling a comprehensive theory similar to classical weights, including boundedness and regularity results.

Findings

Established boundedness of maximal operator in the new setting

Proved Sobolev-Poincaré estimates for solutions

Demonstrated Hölder continuity of solutions

Abstract

In this paper we generalize the famous result of [FKS] to the double phase model. In particular, we work with minimal assumptions on the modulating coefficient by introducing a Muckenhoupt-type condition on generalized Orlicz spaces. We develop a complete theory equivalent to that of classical Muckenhoupt weights, including the boundedness of the maximal operator and Sobolev-Poincare estimates. We combine this with the De~Giorgi technique to show H\"older continuity of the solutions.