Particle systems, Dipoles and Besov spaces of distributions

TL;DR

This paper develops a framework for defining and analyzing Besov spaces of distributions with negative smoothness on abstract measure spaces, using wavelet and dipole bases, and explores their duality with Hölder spaces.

Contribution

It introduces a novel approach to Besov spaces on abstract measure spaces using Schauder bases of Haar wavelets and dipoles, establishing duality results.

Findings

Defined distributions on measure spaces with partitions

Introduced Besov spaces with negative smoothness in this setting

Established duality between Besov and Hölder spaces

Abstract

We define distributions on an abstract measure space endowed with a sequence of partitions, and introduce analogues of Besov spaces with negative smoothness in this setting. In particular, we describe these spaces of distributions using unconditional Schauder bases consisting either of Haar wavelets or of pairs of Dirac masses (dipoles). This framework allows us to obtain duality results between Besov spaces of negative smoothness and H\"older spaces of functions with respect to an appropriately defined pseudo-metric.