Inhomogeneous Sobolev and Besov Spaces: Embeddings and prevalent smoothness

2025/12/15 by Quentin Rible, Rible, Quentin
Mathematics · #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2512.13160

openalex created_date 2025/12/15 · openalex publication_date 2025/12/15 · openalex updated_date 2026/07/28

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 μ, referred to as an environment. In the case where μ is an almost doubling set function, we relate these new spaces with inhomogeneous Besov spaces recently introduced by Barral-Seuret in 2023. When μ 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.

Citations

• Multifractal analysis based on the weak scaling exponent and applications to MEG recordings in neuroscience
• A review of univariate and multivariate multifractal analysis illustrated by the analysis of marathon runners physiological data
• The least doubling constant of a metric measure space
• Multifractal behavior of polynomial Fourier series
• Local behavior of traces of Besov functions: Prevalent results

Related

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