Poincar\'e-type inequalities in Musielak spaces

TL;DR

This paper extends Poincaré-type inequalities to Musielak spaces, broadening their applicability beyond classical Sobolev and Orlicz spaces by establishing new conditions on Musielak functions.

Contribution

It introduces conditions on Musielak functions that ensure Poincaré inequalities hold, expanding the theoretical framework of functional inequalities in Musielak spaces.

Findings

Established Poincaré inequalities in Musielak spaces.

Identified conditions on Musielak functions for inequalities to hold.

Connected null trace functions with Musielak space structures.

Abstract

In this paper we investigate Poincar\'e-type integral inequalities in the functional Musielak structure. We extend the ones already well known in Sobolev, Orlicz and variable exponent Sobolev spaces. We introduce conditions on the Musielak functions under which they hold. The identification with null trace functions space is given.