Sobolev embeddings for Fractional Haj{\l}asz-Sobolev spaces in the setting of rearrangement invariant spaces

TL;DR

This paper establishes symmetrization inequalities for fractional Haj{ extl}asz-Sobolev spaces within rearrangement invariant spaces, linking these inequalities to measure bounds, thus advancing the understanding of function space embeddings.

Contribution

It introduces new symmetrization inequalities in fractional Haj{ extl}asz-Sobolev spaces and connects them to measure lower bounds, expanding the theoretical framework of these spaces.

Findings

Symmetrization inequalities are derived for fractional Haj{ extl}asz-Sobolev spaces.

These inequalities are shown to be equivalent to measure lower bounds for a broad class of measures.

The results deepen the understanding of embeddings in rearrangement invariant spaces.

Abstract

We obtain symmetrization inequalities in the context of Fractional Hajlasz-Sobolev spaces in the setting of rearrangement invariant spaces and prove that for a large class of measures our symmetrization inequalities are equivalent to the lower bound of the measure.