Rough Bilinear Singular Integrals

TL;DR

This paper investigates the boundedness of rough bilinear singular integrals with functions in L^q spaces, introducing a new wavelet-based technique to establish boundedness results across various L^p spaces.

Contribution

The paper introduces a novel bilinear wavelet decomposition method to analyze rough bilinear singular integrals with minimal regularity assumptions.

Findings

Boundedness for q=∞ in L^{p_1}×L^{p_2} to L^p spaces.

Boundedness for q=2 from L^2×L^2 to L^1.

Extended boundedness results for intermediate q values around (1/2,1/2,1).

Abstract

We study the rough bilinear singular integral, introduced by Coifman and Meyer , $$ T_\Omega(f,g)(x)=p.v. \! \int_{\mathbb R^{n}}\! \int_{\mathbb R^{n}}\! |(y,z)|^{-2n} \Omega((y,z)/|(y,z)|)f(x-y)g(x-z) dydz, $$ when $\Omega $ is a function in $L^q(\mathbb S^{2n-1})$ with vanishing integral and $2\le q\le \infty$. When $q=\infty$ we obtain boundedness for $T_\Omega$ from $L^{p_1}(\mathbb R^n)\times L^{p_2}(\mathbb R^n)$ to $ L^p(\mathbb R^n) $ when $1<p_1, p_2<\infty$ and $1/p=1/p_1+1/p_2$. For $q=2$ we obtain that $T_\Omega$ is bounded from $L^{2}(\mathbb R^n)\times L^{ 2}(\mathbb R^n)$ to $ L^1(\mathbb R^n) $. For $q$ between $2$ and infinity we obtain the analogous boundedness on a set of indices around the point $(1/2,1/2,1)$. To obtain our results we introduce a new bilinear technique based on tensor-type wavelet decompositions.