$L^2\times L^2 \to L^1$ boundedness criteria

TL;DR

This paper establishes a precise $L^2\times L^2 \to L^1$ boundedness criterion for certain bilinear operators based on the $L^q$ integrability of their multipliers, with a sharp threshold at $q<4$, and explores applications to rough singular integrals and maximal functions.

Contribution

It provides the first sharp $L^2\times L^2 \to L^1$ boundedness criterion for a class of bilinear operators, linking boundedness to the $L^q$ integrability of multipliers.

Findings

Boundedness holds if and only if $q<4$ for the class of operators studied.

Established an optimal $L^2\times L^2 \to L^1$ criterion for multipliers with $L^\infty$ derivatives.

Applied results to bilinear rough singular integrals and dyadic spherical maximal functions.

Abstract

We obtain a sharp $L^2\times L^2 \to L^1$ boundedness criterion for a class of bilinear operators associated with a multiplier given by a signed sum of dyadic dilations of a given function, in terms of the $L^q$ integrability of this function; precisely we show that boundedness holds if and only if $q<4$. We discuss applications of this result concerning bilinear rough singular integrals and bilinear dyadic spherical maximal functions. Our second result is an optimal $L^2\times L^2\to L^1$ boundedness criterion for bilinear operators associated with multipliers with $L^\infty$ derivatives. This result provides the main tool in the proof of the first theorem and is also manifested in terms of the $L^q$ integrability of the multiplier. The optimal range is $q<4$ which, in the absence of Plancherel's identity on $L^1$, should be compared to $q=\infty$ in the classical $L^2\to L^2$ boundedness for linear multiplier operators.