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Vilenkin-Fourier series in variable Lebesgue spaces

Daviti Adamadze, Tengiz Kopaliani

TL;DR

This work characterizes, for the Vilenkin group $G$, exactly which variable exponents $p(\cdot)$ ensure that the partial sums $S_n f$ of the Vilenkin-Fourier series converge to $f$ in $L^{p(\cdot)}(G)$ whenever $f\in L^{p(\cdot)}(G)$. The authors introduce the $\mathcal{A}(G)$ condition on $p(\cdot)$, a dyadic-structured analogue of Muckenhoupt’s $A_p$ weights, and prove its equivalence to the boundedness of the Hardy-Littlewood maximal operator on $L^{p(\cdot)}(G)$ and to the boundedness/convergence of the averaging operators and Vilenkin partial sums via Calderón-Zygmund techniques and Rubio de Francia extrapolation. They establish a sharp trio of equivalent statements: $p(\cdot)\in\mathcal{A}(G)$, uniform boundedness of $S_n$ on $L^{p(\cdot)}(G)$, and convergence $S_n f \to f$ in $L^{p(\cdot)}(G)$. Additional results include a stability criterion for the maximal operator under exponent scaling (Theorem 1.8) and an extrapolation-based proof framework that extends constant-exponent harmonic analysis on Vilenkin groups to the variable-exponent setting.

Abstract

Let $S_{n}f$ denote the $n$th partial sum of the Vilenkin-Fourier series of a function $f \in L^{1}(G)$. For $1 < p_{-} \leq p_{+} < \infty$, we characterize all exponents $p(\cdot)$ for which the convergence of $S_{n}f$ to $f$ in $L^{p(\cdot)}(G)$ holds whenever $f \in L^{p(\cdot)}(G)$.

Vilenkin-Fourier series in variable Lebesgue spaces

TL;DR

This work characterizes, for the Vilenkin group , exactly which variable exponents ensure that the partial sums of the Vilenkin-Fourier series converge to in whenever . The authors introduce the condition on , a dyadic-structured analogue of Muckenhoupt’s weights, and prove its equivalence to the boundedness of the Hardy-Littlewood maximal operator on and to the boundedness/convergence of the averaging operators and Vilenkin partial sums via Calderón-Zygmund techniques and Rubio de Francia extrapolation. They establish a sharp trio of equivalent statements: , uniform boundedness of on , and convergence in . Additional results include a stability criterion for the maximal operator under exponent scaling (Theorem 1.8) and an extrapolation-based proof framework that extends constant-exponent harmonic analysis on Vilenkin groups to the variable-exponent setting.

Abstract

Let denote the th partial sum of the Vilenkin-Fourier series of a function . For , we characterize all exponents for which the convergence of to in holds whenever .
Paper Structure (5 sections, 17 theorems, 100 equations)

This paper contains 5 sections, 17 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['Muck']}
  4. Proof of Theorem 1.8
  5. Proof of Theorem 1.10

Key Result

Theorem 1.2

(You2) Let $w$ be a weight function on $G.$ For $1<p<\infty$, the following statements are equivalent: (i) $w\in A_{p}(G),$ (ii) There is a constant $C$, depending only on $w$ and $p$, such that for every $f\in L^{p}_{w}(G)$, we have (iii) For every $f\in L^{p}_{w}(G)$, we have

Theorems & Definitions (20)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 10 more