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Littlewood, Paley and Almost-Orthogonality: a theory well ahead of its time

Anthony Carbery

TL;DR

Littlewood–Paley theory addresses the absence of orthogonality in $L^p$ by introducing square-function decompositions that mimic orthogonality. The paper surveys LP1 and traces its influence through real-variable harmonic analysis, vector-valued methods, wavelets, and geometric measure theory, highlighting quasi-abstract formulations and exotic decompositions. It emphasizes how almost-orthogonality principles underpin multiplier theory, PDE boundary-value problems, and modern developments like decoupling and Kakeya-type phenomena. The work positions LP theory as a unifying framework across analysis, geometry, and PDE with lasting impact.

Abstract

Littlewood--Paley theory began with the classic paper of Littlewood and Paley (J.\ E.\ Littlewood, R.\ E.\ A.\ C.\ Paley, {\em Theorems on Fourier Series and Power Series}. J. Lond. Math. Soc. (1), {\bf 6} (1931), 230--33). We discuss this paper and its impact from a historical perspective. We include an outline of the results in the paper and their subsequent significance in relation to developments over the last century, and set them into the context of the current state of the art in harmonic analysis and beyond.

Littlewood, Paley and Almost-Orthogonality: a theory well ahead of its time

TL;DR

Littlewood–Paley theory addresses the absence of orthogonality in by introducing square-function decompositions that mimic orthogonality. The paper surveys LP1 and traces its influence through real-variable harmonic analysis, vector-valued methods, wavelets, and geometric measure theory, highlighting quasi-abstract formulations and exotic decompositions. It emphasizes how almost-orthogonality principles underpin multiplier theory, PDE boundary-value problems, and modern developments like decoupling and Kakeya-type phenomena. The work positions LP theory as a unifying framework across analysis, geometry, and PDE with lasting impact.

Abstract

Littlewood--Paley theory began with the classic paper of Littlewood and Paley (J.\ E.\ Littlewood, R.\ E.\ A.\ C.\ Paley, {\em Theorems on Fourier Series and Power Series}. J. Lond. Math. Soc. (1), {\bf 6} (1931), 230--33). We discuss this paper and its impact from a historical perspective. We include an outline of the results in the paper and their subsequent significance in relation to developments over the last century, and set them into the context of the current state of the art in harmonic analysis and beyond.

Paper Structure

This paper contains 13 sections, 5 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. What's in the Littlewood--Paley paper (and what's not)
  3. What came next?
  4. Quasi-abstract Littlewood--Paley theories
  5. Vector-valued analysis and weighted inequalities
  6. Wavelets and the Calderón reproducing formula
  7. Almost-orthogonality principles beyond Littlewood--Paley
  8. Littlewood--Paley theory in Geometric Measure Theory and PDE
  9. Exotic Littlewood--Paley decompositions
  10. Equally-spaced intervals
  11. Arbitrary intervals -- the Rubio de Francia Littlewood--Paley inequality
  12. Angular decompositions -- reverse inequalities and curvature
  13. Reprise: Almost-everywhere Fourier convergence in $\mathbb{R}^n$ for $n \geq 2$

Key Result

Theorem 2.1

For each $1 < p < \infty$, there are constants $A_p$ and $B_p$ such that

Theorems & Definitions (6)

  • Theorem 2.1
  • Remark
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 7.1: Cotlar--Stein, see SteinBig
  • Theorem 10.1