Restricted weighted weak boundedness for product type operators

TL;DR

This paper establishes restricted weighted weak type inequalities for product operators formed by singular integrals satisfying Dini conditions, advancing understanding of their boundedness properties in weighted Lebesgue spaces.

Contribution

It introduces new restricted weighted weak type bounds for product of singular integrals and applies these results to bilinear Fourier multipliers of bounded variation.

Findings

Proved weighted weak inequalities for product singular integrals.

Established boundedness of certain bilinear Fourier multipliers.

Extended previous results to operators with Dini condition assumptions.

Abstract

Given a bilinear (or sub-bilinear) operator $B$, we prove restricted weighted weak type inequalities of the form $$ ||B(f_1, f_2)||_{L^{p, \infty}(w_1^{p/p_1}w_2^{p/p_2})}\lesssim ||f_1||_{L^{p_1, 1}(w_1)}||f_2||_{L^{p_2, 1}(w_2)}, $$ whenever $B(f_1, f_2)= (T_1f_1) (T_2 f_2)$ is the product of two singular integral operators satisfying Dini conditions. Additionally, we also establish, as an application, the boundedness of a certain class of bounded variation bilinear Fourier multipliers solving a question posted in [Bilinear Fourier multipliers of bounded variation; Int. Math. Res. Not. (2023), no.24, 21943--21975 by Baena-Miret, Carro, Luque and Sanchez-Pascuala].