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Traces of functions in Besov spaces in Gibbs environment

Quentin Rible, Stéphane Seuret

Abstract

This paper investigates the traces of functions belonging to the inhomogeneous Besov spaces B $ξ$ p,q , where $ξ$ is a product of capacities defined as powers of Gibbs measures. We first establish that the traces of functions in B $ξ$ 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 $ξ$ $\infty$,q . This bound is then refined for a prevalent set of functions in B $ξ$ $\infty$,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.

Traces of functions in Besov spaces in Gibbs environment

Abstract

This paper investigates the traces of functions belonging to the inhomogeneous Besov spaces B p,q , where is a product of capacities defined as powers of Gibbs measures. We first establish that the traces of functions in B 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 ,q . This bound is then refined for a prevalent set of functions in B ,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.

Paper Structure

This paper contains 19 sections, 36 theorems, 177 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Capacities, wavelets and multifractal formalism
  3. Capacities and multifractal formalism
  4. Gibbs measure and related capacity
  5. Product capacity on [0,1] hat D
  6. Multi-resolution wavelet analysis
  7. Traces and inhomogeneous Besov spaces
  8. Wavelet characterisation of traces
  9. Proof of Theorem \ref{['theo:incl_trace']}
  10. Upper bound for the spectrum of traces of Btilde xi inf,q functions
  11. A saturating function in B mu inf,q
  12. Construction of the saturating function Gq
  13. Singularity spectrum of the traces G(q-a) of the saturating function
  14. Prevalent singularity spectrum of traces
  15. Prevalence property of an auxiliary set
  16. ...and 4 more sections

Key Result

Theorem 1.9

If $\xi\in \mathcal{C}([0,1]^D)$ is doubling and satisfies the SMF, then there exists a prevalent set of functions $f\in\widetilde{B}^{\xi}_{\infty,q}([0,1]^D)$ such that $\sigma_f=\tau_\xi^*=\sigma_\xi$.

Figures (3)

  • Figure 1.1: Upper-bound for $\sigma_{f_a}(h)$ for all $a\in[0,1]^{d'}$ in grey, for $\nu_r$-almost all $a\in[0,1]^{d'}$ in red.
  • Figure 1.2: Spectrum $\sigma_{f_a}(h)$ for almost all $f$ and $\nu_r$-almost all $a\in[0,1]^{d'}$.
  • Figure 2.1: Spectrum $\sigma_{\nu}$ for a Gibbs measure $\nu$.

Theorems & Definitions (83)

  • Definition 1.1
  • Remark 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Remark 1.7
  • Definition 1.8
  • Theorem 1.9: Rible:2025:Prevalence_inhomogeneous_Besov
  • Proposition 1.10
  • ...and 73 more