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Hypercontractive inequalities and Nikol'skiĭ-type inequalities on weighted Bergman spaces

Zipeng Wang, Kenan Zhang

TL;DR

The paper addresses sharp hypercontractive inequalities between weighted Bergman spaces $A_\alpha^p(\mathbb{D})$ and $A_\beta^q(\mathbb{D})$ under the dilation operator $T_r f(z)=f(rz)$. It proves a precise criterion $r^2\le \dfrac{\beta p}{\alpha q}$ (with $0<p\le q$, $q\ge 2$, and $\beta p\le \alpha q$) for the contraction $\|T_r f\|_{A_\beta^q}\le \|f\|_{A_\alpha^p}$, and extends this to higher dimensions via a coordinatewise dilation on the polydisk, establishing a dimension-free Nikol'skii-type inequality with optimal constant $C(\alpha,\beta,p,q)=\sqrt{\dfrac{\alpha q}{\beta p}}$. The proofs combine one-variable hypercontractivity results (inspired by Weissler and Kulikov), a convexity-based reduction, and a probabilistic limiting argument to confirm sharpness. The results generalize known Hardy-space bounds and provide tight estimates for analytic function spaces, with potential applications in operator theory and several complex variables.

Abstract

In this note, we obtianed hypercontractive inequalities between different weighted Bergman spaces. In addition, we establish Nikol'skiĭ-type inequalities for weighted Bergman spaces with optimal constants.

Hypercontractive inequalities and Nikol'skiĭ-type inequalities on weighted Bergman spaces

TL;DR

The paper addresses sharp hypercontractive inequalities between weighted Bergman spaces and under the dilation operator . It proves a precise criterion (with , , and ) for the contraction , and extends this to higher dimensions via a coordinatewise dilation on the polydisk, establishing a dimension-free Nikol'skii-type inequality with optimal constant . The proofs combine one-variable hypercontractivity results (inspired by Weissler and Kulikov), a convexity-based reduction, and a probabilistic limiting argument to confirm sharpness. The results generalize known Hardy-space bounds and provide tight estimates for analytic function spaces, with potential applications in operator theory and several complex variables.

Abstract

In this note, we obtianed hypercontractive inequalities between different weighted Bergman spaces. In addition, we establish Nikol'skiĭ-type inequalities for weighted Bergman spaces with optimal constants.
Paper Structure (5 sections, 5 theorems, 82 equations)

This paper contains 5 sections, 5 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. The proof of Theorem \ref{['main01']}
  3. The Necessity
  4. The Sufficienty
  5. Proof of Theorem \ref{['main02']}

Key Result

Theorem 1.1

For $1<\alpha,\beta<\infty$ and $0<p\leqslant q<\infty$, let $q\geqslant 2$ and $\beta p\leqslant\alpha q$. Then we have for any $f\in A_\alpha^p(\mathbb{D})$ if and only if $r^2\leqslant\beta p/\alpha q$.

Theorems & Definitions (9)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof