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A short proof on the boundedness of triangular Hilbert transform along curves

Martin Hsu, Fred Yu-Hsiang Lin

Abstract

We give a short and elementary proof of the boundedness of triangular Hilbert transform along non-flat curves definable in a polynomially bounded o-minimal structure. We also provide a criterion on the multiplier to determine whether the associated fiber-wisely defined bilinear operator admits a smoothing inequality.

A short proof on the boundedness of triangular Hilbert transform along curves

Abstract

We give a short and elementary proof of the boundedness of triangular Hilbert transform along non-flat curves definable in a polynomially bounded o-minimal structure. We also provide a criterion on the multiplier to determine whether the associated fiber-wisely defined bilinear operator admits a smoothing inequality.

Paper Structure

This paper contains 7 sections, 6 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Proof of Proposition \ref{['propsmoothingmultiplier']}
  3. Proof of Proposition \ref{['mulcheck']}
  4. Proof of Theorem \ref{['mainthm']}
  5. Low Frequency
  6. Mix Frequency
  7. High Frequency

Key Result

Theorem 1.1

Let $\gamma:\mathbb{R}\setminus\left\{ 0 \right\}\to\mathbb{R}$ be a definable function satisfying eq_cond_on_gamma. Let $p_{1},p_{2}\in (1,\infty)$, $p\in [1,2)$ satisfy $\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}$. Then $T_\gamma$ extends to a bounded operator $L^{p_{1}}\times L^{p_{2}}\rightarro

Theorems & Definitions (6)

  • Theorem 1.1
  • Proposition 1.2
  • Lemma 1.3
  • Lemma 1.4
  • Lemma 1.5
  • Proposition 1.6