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Functional calculus and semilinear evolution equations for the Taibleson operator on non-Archimedean local fields

Cédric Arhancet, Christoph Kriegler

TL;DR

The paper proves that the Taibleson operator $D^\alpha$ on vector-valued $L^p$ spaces over a non-Archimedean local field $\mathbb{K}$ admits a bounded $\mathrm{H}^\infty(\Sigma_\theta)$ functional calculus for any $\theta>0$ on $\mathrm{L}^p(\mathbb{K}^n,Y)$ when $1<p<\infty$ and $Y$ is a UMD Banach function space, as well as a Hörmander calculus of order $\tfrac{3}{2}$. The approach relies on $R$-boundedness of families of convolution operators on locally compact Spector-Vilenkin groups, enabling $R$-analyticity and dimension-free estimates on totally disconnected spaces. The results yield maximal $L^q$-regularity and well-posedness results for evolution equations driven by $D^\alpha$, including semilinear problems, with immediate smoothing and robust stability properties. By connecting harmonic analysis on Spector-Vilenkin groups with operator-calculus techniques, the work extends functional-calculus theory to vector-valued settings in ultrametric contexts and provides a framework for linear and nonlinear ultrametric evolution equations with broad applicability.

Abstract

For any non-Archimedean local field $\mathbb{K}$ and any integer $n \geq 1$, we show that the Taibleson operator admits a bounded $\mathrm{H}^\infty(Σ_θ)$ functional calculus on the Bochner space $\mathrm{L}^p(\mathbb{K}^n,Y)$ for any $\mathrm{UMD}$ Banach function space $Y$ and any angle $θ> 0$, where $Σ_θ=\{ z \in \mathbb{C}^*: |\arg z| < θ\}$ and $1 < p < \infty$. Moreover, we prove that it even admits a bounded Hörmander functional calculus of order $\frac{3}{2}$. In our study, we explore harmonic analysis on locally compact Spector-Vilenkin groups establishing the $R$-boundedness of a family of convolution operators. Our results enhance the understanding of functional calculi of operators acting on vector-valued $\mathrm{L}^p$-spaces associated with totally disconnected spaces. As an application, we obtain maximal regularity results and well-posedness for a class of evolution equations driven by the Taibleson operator.

Functional calculus and semilinear evolution equations for the Taibleson operator on non-Archimedean local fields

TL;DR

The paper proves that the Taibleson operator on vector-valued spaces over a non-Archimedean local field admits a bounded functional calculus for any on when and is a UMD Banach function space, as well as a Hörmander calculus of order . The approach relies on -boundedness of families of convolution operators on locally compact Spector-Vilenkin groups, enabling -analyticity and dimension-free estimates on totally disconnected spaces. The results yield maximal -regularity and well-posedness results for evolution equations driven by , including semilinear problems, with immediate smoothing and robust stability properties. By connecting harmonic analysis on Spector-Vilenkin groups with operator-calculus techniques, the work extends functional-calculus theory to vector-valued settings in ultrametric contexts and provides a framework for linear and nonlinear ultrametric evolution equations with broad applicability.

Abstract

For any non-Archimedean local field and any integer , we show that the Taibleson operator admits a bounded functional calculus on the Bochner space for any Banach function space and any angle , where and . Moreover, we prove that it even admits a bounded Hörmander functional calculus of order . In our study, we explore harmonic analysis on locally compact Spector-Vilenkin groups establishing the -boundedness of a family of convolution operators. Our results enhance the understanding of functional calculi of operators acting on vector-valued -spaces associated with totally disconnected spaces. As an application, we obtain maximal regularity results and well-posedness for a class of evolution equations driven by the Taibleson operator.
Paper Structure (28 sections, 25 theorems, 145 equations, 1 figure)

This paper contains 28 sections, 25 theorems, 145 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Application to maximal regularity of evolution equations
  3. Application to local and global existence of solutions of semilinear evolution equations
  4. Approach of the proof of Theorem \ref{['Main-intro']}
  5. Locally compact Vilenkin groups
  6. Structure of the paper
  7. Preliminaries
  8. Local fields
  9. Pontryagin duality
  10. Fourier transform
  11. Improper integrals
  12. $R$-boundedness
  13. $\mathrm{H}^\infty$ functional calculus
  14. Hörmander functional calculus
  15. Conditional expectations
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $\mathbb{K}$ be a non-Archimedean local field and $n \geqslant 1$ be an integer. Consider a $\mathrm{UMD}$ Banach function space $Y$. Suppose that $1 < p < \infty$. The Taibleson operator $D^\alpha \otimes \mathrm{Id}_Y$ admits a bounded $\mathrm{H}^\infty(\Sigma_\theta)$ functional calculus for

Figures (1)

  • Figure :

Theorems & Definitions (50)

  • Theorem 1.1
  • Corollary 1.2
  • Example 2.1: fields of $q$-adic numbers
  • Example 2.2: fields of formal Laurent series over finite fields
  • Example 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Proposition 3.1
  • ...and 40 more