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The missing proof of Paley's theorem about lacunary coefficients

John J. F. Fournier

Abstract

We modify the standard proof of Paley's theorem about lacunary coefficients of functions in $H^1$ to work without analytic factorization. This leads to the first direct proof of the extension of Paley's theorem that we applied to the former Littlewood conjecture about $L^1$ norms of exponential sums.

The missing proof of Paley's theorem about lacunary coefficients

Abstract

We modify the standard proof of Paley's theorem about lacunary coefficients of functions in to work without analytic factorization. This leads to the first direct proof of the extension of Paley's theorem that we applied to the former Littlewood conjecture about norms of exponential sums.

Paper Structure

This paper contains 7 sections, 6 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Pairs of nested projections
  3. Partially ordered dual groups
  4. Finite Riesz products
  5. Analysing our method
  6. Direct Proof of Theorem \ref{['th:S_1(K)']}
  7. Other nestings

Key Result

Theorem 1.1

There is a constant $C$ so that if $K$ is strongly lacunary, and if $\hat{f}(n) = 0$ when $n<0$, then

Theorems & Definitions (26)

  • Theorem 1.1
  • Remark 1.2
  • proof : Proof of Paley's theorem
  • Remark 2.1
  • Remark 2.2
  • Theorem 3.1
  • Theorem 3.2
  • proof
  • Remark 3.3
  • Remark 3.4
  • ...and 16 more