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A sharp Hörmander condition for bilinear Fourier multipliers with Lipschitz singularities

Jiao Chen, Martin Hsu, Fred Yu-Hsiang Lin

Abstract

This paper studies the $L^{p}$ boundedness of bilinear Fourier multipliers in the local $L^{2}$ range. We assume a Hörmander condition relative to a singular set that is a finite union of Lipschitz curves. The Hörmander condition is sharp with respect to the Sobolev exponent. Our setup generalizes the non-degenerate bilinear Hilbert transform but avoids issues of uniform bounds near degeneracy.

A sharp Hörmander condition for bilinear Fourier multipliers with Lipschitz singularities

Abstract

This paper studies the boundedness of bilinear Fourier multipliers in the local range. We assume a Hörmander condition relative to a singular set that is a finite union of Lipschitz curves. The Hörmander condition is sharp with respect to the Sobolev exponent. Our setup generalizes the non-degenerate bilinear Hilbert transform but avoids issues of uniform bounds near degeneracy.
Paper Structure (2 sections, 1 theorem, 21 equations, 1 figure)

This paper contains 2 sections, 1 theorem, 21 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Overview of the proof

Key Result

Theorem 1.1

Let $n=2$. Let $2<p_{1},p_{2},p_{3}<\infty$ with $\frac{1}{p_{1}}+\frac{1}{p_{2}}+\frac{1}{p_{3}}=1$. Let $0\leq \theta_0<\frac{\pi}{6}.$ Let $s>1$. There is a constant $C(p_1,p_2,p_3,\theta_{0},s,N)$ such that the following holds. For every $1\le \iota\le N$, let $\Gamma_\iota\subset V$ be a closed Then we have for the form bound goalbdd the inequality

Figures (1)

  • Figure 1: We may view \ref{['Hormcond']} as testing Sobolev norm of $m$ on scaled Whitney bumps.

Theorems & Definitions (1)

  • Theorem 1.1