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The shifted bilinear Hilbert transform

Lars Becker, Polona Durcik

Abstract

We prove $L^p$ estimates for the shifted bilinear Hilbert transform, with a polylogarithmic bound in the size of the shift. As applications, we obtain $r$-variation estimates for bilinear ergodic averages in the sharp range $r > 2$, a sharp bilinear Hörmander multiplier theorem, and a $\log$-Dini theorem for bilinear singular integrals.

The shifted bilinear Hilbert transform

Abstract

We prove estimates for the shifted bilinear Hilbert transform, with a polylogarithmic bound in the size of the shift. As applications, we obtain -variation estimates for bilinear ergodic averages in the sharp range , a sharp bilinear Hörmander multiplier theorem, and a -Dini theorem for bilinear singular integrals.
Paper Structure (28 sections, 36 theorems, 284 equations, 1 figure)

This paper contains 28 sections, 36 theorems, 284 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The shifted bilinear Hilbert transform
  3. Sharp variation estimates for ergodic averages
  4. Dini kernel regularity for the bilinear Hilbert transform
  5. The Hörmander multiplier theorem for the bilinear Hilbert transform
  6. The shifted Carleson theorem
  7. Notation
  8. Outline of the proof of boundedness of the shifted BHT
  9. Discretization
  10. The dyadic model operator
  11. Interpolation
  12. Trees and sizes
  13. A weak Bessel inequality
  14. The tree selection algorithm
  15. Control of sizes
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $p, p_1, p_2$ be exponents satisfying e:p_aspt. There exists $C > 0$ such that the following holds. Let $m \ge 0$ and let $(\tau_s)_{s \in \mathbb{Z}}$ be a sequence of integers with Let $(\psi_s)_{s \in \mathbb{Z}}$ be a sequence with $\psi_s \in \mathcal{S}_0^{\tau_s}$. Then for all Schwartz functions $f_1,f_2:{\mathbb R}\to {\mathbb C}$,

Figures (1)

  • Figure 1: Green intervals $I_{p_2}$ of an $(i,3)$-tree with dark-green top interval, for $m = 2$.

Theorems & Definitions (56)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 46 more