Regularity of Weighted Tensorized Fractional Brownian Fields and associated function spaces

TL;DR

This paper studies a new class of self-similar Gaussian fields called Weighted Tensorized Fractional Brownian Fields, analyzing their regularity and introducing new function spaces that generalize classical Besov spaces.

Contribution

It introduces Weighted Tensorized Besov Spaces and characterizes their properties, extending the understanding of regularity in fractional Brownian fields beyond tensor-product structures.

Findings

Characterization of regularity properties of WTFBS

Introduction of Weighted Tensorized Besov Spaces

Analysis using Littlewood-Paley and hyperbolic wavelet techniques

Abstract

We investigate a new class of self-similar fractional Brownian fields, called Weighted Tensorized Fractional Brownian Fields (WTFBS). These fields, introduced in the companion paper \cite{ELLV}, generalize the well-known fractional Brownian sheet (FBs) by relaxing its tensor-product structure, resulting in new self-similar Gaussian fields with stationary rectangular increments that differ from the FBs. We analyze the local regularity properties of these fields and introduce a new concept of regularity through the definition of Weighted Tensorized Besov Spaces. These spaces combine aspects of mixed dominating smoothness spaces and hyperbolic Besov spaces, which are similar in structure to classical Besov spaces. We provide a detailed characterization of these spaces using Littlewood-Paley theory and hyperbolic wavelet analysis.