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Littlewood-type theorems for Hardy spaces in infinitely many variables

Jiaqi Ni

Abstract

Littlewood's theorem is one of the pioneering results in random analytic functions over the open unit disk. In this paper, we prove some analogues of this theorem for Hardy spaces in infinitely many variables. Our results not only cover finite-variable setting, but also apply in cases of Dirichlet series.

Littlewood-type theorems for Hardy spaces in infinitely many variables

Abstract

Littlewood's theorem is one of the pioneering results in random analytic functions over the open unit disk. In this paper, we prove some analogues of this theorem for Hardy spaces in infinitely many variables. Our results not only cover finite-variable setting, but also apply in cases of Dirichlet series.
Paper Structure (6 sections, 20 theorems, 82 equations)

This paper contains 6 sections, 20 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Littlewood-type theorems for standard random sequences
  3. Littlewood-type theorems for Gaussian processes
  4. Cases associated with centered Gaussian processes
  5. Coefficient multipliers and cases associated with general Gaussian processes
  6. Applications to Dirichlet series

Key Result

Theorem A

Let $\{X_n\}_{n=0}^\infty$ be a standard Bernoulli sequence, that is, a sequence of independent, identically distributed random variables with $\mathbb{P}(X_n=1)=\mathbb{P}(X_n=-1)=\frac{1}{2}$. Suppose that $f$ is analytic on $\mathbb{D}$. (1) If $f\in H^2(\mathbb{D})$, then for all $1\leq p<\infty

Theorems & Definitions (29)

  • Theorem A: Littlewood's theorem, Li2
  • Theorem B: A Gaussian version of Littlewood's theorem, CFGL
  • Theorem 2.1
  • Proposition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Proposition 2.5
  • proof
  • Theorem 3.1
  • ...and 19 more