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Restricted weighted weak boundedness for product type operators

María Jesús Carro, Sheldy Ombrosi

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].

Restricted weighted weak boundedness for product type operators

Abstract

Given a bilinear (or sub-bilinear) operator , we prove restricted weighted weak type inequalities of the form whenever 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].

Paper Structure

This paper contains 4 sections, 10 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. preliminaries
  3. Proofs of Theorems
  4. application to bounded variation fourier multipliers

Key Result

Theorem 1.1

For every $(p_1, p_2; p)$ and every $w_j\in A_{p_j}^\mathcal{R}$ ($j=1,2$),

Theorems & Definitions (19)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Definition 1.1
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 2.1
  • Remark 2.2
  • Lemma 2.3
  • ...and 9 more