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On trilinear singular Brascamp-Lieb integrals

Lars Becker, Polona Durcik, Fred Yu-Hsiang Lin

Abstract

We classify all trilinear singular Brascamp-Lieb forms, completing the classification in the two dimensional case by Demeter and Thiele in arXiv:0803.1268. We use known results in the representation theory of finite dimensional algebras, namely the classification of indecomposable representations of the four subspace quiver. Our classification lays out a roadmap for achieving bounds for all degenerate higher dimensional bilinear Hilbert transforms. As another step towards this goal, we prove new bounds for a particular class of forms that arises as a natural next candidate from our classification. We further prove conditional bounds for forms associated with mutually related representations. For this purpose we develop a method of rotations that allows us to decompose any homogeneous $d$-dimensional singular integral kernel into $(d-1)$-dimensional kernels on hyperplanes.

On trilinear singular Brascamp-Lieb integrals

Abstract

We classify all trilinear singular Brascamp-Lieb forms, completing the classification in the two dimensional case by Demeter and Thiele in arXiv:0803.1268. We use known results in the representation theory of finite dimensional algebras, namely the classification of indecomposable representations of the four subspace quiver. Our classification lays out a roadmap for achieving bounds for all degenerate higher dimensional bilinear Hilbert transforms. As another step towards this goal, we prove new bounds for a particular class of forms that arises as a natural next candidate from our classification. We further prove conditional bounds for forms associated with mutually related representations. For this purpose we develop a method of rotations that allows us to decompose any homogeneous -dimensional singular integral kernel into -dimensional kernels on hyperplanes.

Paper Structure

This paper contains 25 sections, 26 theorems, 294 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Singular Brascamp-Lieb forms
  3. The four subspace problem
  4. Classification of singular Brascamp-Lieb forms
  5. Projection results and method of rotations
  6. Positive boundedness results in the literature
  7. Coifman and Meyer
  8. Time-frequency analysis
  9. One and a half dimensional time-frequency analysis
  10. Twisted techniques
  11. The triangular Hilbert transform
  12. Further questions
  13. Comparison with the literature
  14. The classification: Proof of Theorem \ref{['thm equivalence']} and Theorem \ref{['thm Hclass']}
  15. Bounds for forms of non-Hölder type: Proof of Theorem \ref{['thm i)ii)']}
  16. ...and 10 more sections

Key Result

Lemma 1.6

Suppose that $\mathbf{H}$ and ${\mathbf{H}'}$ are equivalent singular Brascamp-Lieb data. Then for all $\mathbf{p}$, the form $\Lambda_\mathbf{H}$ is $\mathbf{p}$-bounded if and only $\Lambda_{\mathbf{H}'}$ is $\mathbf{p}$-bounded.

Theorems & Definitions (56)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Lemma 1.6
  • Remark 1.7
  • Definition 1.8
  • Definition 1.9
  • Definition 1.10
  • ...and 46 more