Particle systems, Dipoles and Besov spaces of distributions

TL;DR

This work extends Besov spaces with negative smoothness to highly irregular measure spaces by introducing a good-grid framework and using unbalanced Haar wavelets. It establishes a duality between ${\mathcal{B}^{-s}_{1,1}}$ and ${\mathcal{B}^s_{\infty,\infty}}$, and shows that a Hölder-like structure emerges under a natural pseudo-metric. A key advance is the unconditional dipole-based basis for ${\mathcal{B}^{-s}_{1,1}}$, enabling Dirac-dipole atomic decompositions and connecting to particle-system spaces ${\mathcal{PS}^s}$ and ${\mathcal{DC}^s}$. These results provide sharp, elementary tools for analysis on irregular spaces and have potential applications to ergodic theory via transfer operators on distribution 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 Hölder spaces of functions with respect to an appropriately defined pseudo-metric.

Theorems & Definitions (39)

• Theorem A
• Theorem B
• Definition 3.2.4: Girardi and Sweldens Sweldens1997
• Definition 4.1.6
• Proposition 4.1.9
• Definition 4.1.10
• Proposition 4.1.11: Theorem A-\ref{['thma1']}
• proof
• Definition 4.2.12: Test functions
• Proposition 4.2.13