Inhomogeneous Sobolev and Besov Spaces: Embeddings and prevalent smoothness

TL;DR

This paper introduces a new class of inhomogeneous Sobolev spaces governed by a set function, explores their relation to Besov spaces, and characterizes the multifractal nature of prevalent elements within them.

Contribution

It generalizes Sobolev spaces using a set function environment, links these to recent Besov space developments, and analyzes the multifractal properties of typical elements.

Findings

Relation established between inhomogeneous Sobolev and Besov spaces for almost doubling set functions.

Prevalent elements in these spaces are shown to be multifractal with a determined singularity spectrum.

Completes previous generic results by characterizing multifractality in the context of these new spaces.

Abstract

In this article, we introduce inhomogeneous Sobolev spaces that naturally generalise the standard Sobolev-Slobodeckij spaces. The inhomogeneity of these spaces is governed by a set function $\mu$, referred to as an environment. In the case where $\mu$ is an almost doubling set function, we relate these new spaces with inhomogeneous Besov spaces recently introduced by Barral-Seuret in 2023. When $\mu$ is in addition a capacity, wee also prove that prevalent elements in such spaces are multifractal (with a singularity spectrum that we determine), completing previous Baire generic results already obtained.