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Maximal multilinear operators

Ciprian Demeter, Terence Tao, Christoph Thiele

Abstract

We establish multilinear $L^p$ bounds for a class of maximal multilinear averages of functions on one variable, reproving and generalizing the bilinear maximal function bounds of Lacey. As an application we obtain almost everywhere convergence results for these averages, and in some cases we also obtain almost everywhere convergence for their ergodic counterparts on a dynamical system.

Maximal multilinear operators

Abstract

We establish multilinear bounds for a class of maximal multilinear averages of functions on one variable, reproving and generalizing the bilinear maximal function bounds of Lacey. As an application we obtain almost everywhere convergence results for these averages, and in some cases we also obtain almost everywhere convergence for their ergodic counterparts on a dynamical system.

Paper Structure

This paper contains 19 sections, 41 theorems, 285 equations.

Table of Contents

  1. Introduction
  2. Linear averages
  3. Bilinear averages
  4. Furstenberg averages
  5. Averages along cubes
  6. Main results
  7. Interpolation reductions
  8. Fourier representation
  9. Discretization
  10. Trees
  11. High-level overview of proof
  12. Single tree size estimate
  13. Reduction to Bessel inequality
  14. Good-$\lambda$ reduction
  15. Tileset refinements
  16. ...and 4 more sections

Key Result

Theorem 1.1

Assume $n\ge 3$ and let $A$ be a matrix as above. Let $(p_1,\ldots,p_{n-1})$ be a tuple of real numbers with for $1\le i\le n-1$ and set If then the operator $T_{A,{\mathbf R}}^{*}$ is bounded.

Theorems & Definitions (98)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Corollary 1.2
  • proof
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • ...and 88 more