Restricted weighted weak boundedness for product type operators

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 $B(f_1,f_2)=(T_1{f}_1)(T_2{f}_2)$ 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].

14 pages