Traces of functions in Besov spaces in Gibbs environment

TL;DR

This paper studies the behavior of functions in Besov spaces with Gibbs measure-based capacities, focusing on their traces along hyperplanes and revealing sensitivity of regularity to hyperplane choice.

Contribution

It establishes the Besov space classification of traces of functions in Gibbs environment and computes their singularity spectrum, highlighting the influence of hyperplane selection.

Findings

Traces of Besov functions along hyperplanes belong to specific Besov spaces.

An upper bound for the singularity spectrum of traces is derived.

Regularity of traces is highly sensitive to the hyperplane chosen.

Abstract

This paper investigates the traces of functions belonging to the inhomogeneous Besov spaces B $\xi$ p,q , where $\xi$ is a product of capacities defined as powers of Gibbs measures. We first establish that the traces of functions in B $\xi$ p,q along affine hyperplanes belong to another inhomogeneous Besov space. Furthermore, we derive an upper bound for the singularity spectrum of the traces of all functions in B $\xi$ $\infty$,q . This bound is then refined for a prevalent set of functions in B $\xi$ $\infty$,q , for which we explicitly compute the singularity spectrum of their traces. Notably, our analysis reveals that the regularity properties of these affine traces are highly sensitive to the choice of the hyperplane along which the trace is taken.