Parabolic weighted Sobolev-Poincar\'e type inequalities

TL;DR

This paper establishes weighted Sobolev-Poincaré inequalities tailored for parabolic PDEs, utilizing weights from the parabolic Muckenhoupt class to extend classical inequalities to more general weighted function spaces.

Contribution

It introduces new weighted inequalities for parabolic PDEs with weights in the parabolic A_p class, broadening the scope of Sobolev-Poincaré inequalities in this context.

Findings

Derived weighted inequalities for parabolic PDEs

Extended classical inequalities to parabolic Muckenhoupt weights

Applicable to function spaces with space-time dependent weights

Abstract

We derive weighted Sobolev-Poincar\'e type inequalities in function spaces concerned with parabolic partial differential equations. We consider general weights depending on both space and time variables belonging to a Muckenhoupt class, so-called the parabolic $A_p$-class, where only the parabolic cubes are involved in the definition.