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A New Approach to $(1,1)-$Weak-type Estimate for the Littlewood-Paley Operators $S_{α,ψ}$ and $g^{*}_{λ,ψ}$

Arash Ghorbanalizadeh, Monire Mikaeili Nia

Abstract

In this study, $(1,1)-$weak type boundedness of square function $S_{α,ψ}$ is obtained by using Nazarov-Treil and Volberg technique. Also using this result, the $(1,1)-$ weak type boundedness of $g^{*}_{λ,ψ}$ operator is investigated.

A New Approach to $(1,1)-$Weak-type Estimate for the Littlewood-Paley Operators $S_{α,ψ}$ and $g^{*}_{λ,ψ}$

Abstract

In this study, weak type boundedness of square function is obtained by using Nazarov-Treil and Volberg technique. Also using this result, the weak type boundedness of operator is investigated.
Paper Structure (5 sections, 8 theorems, 87 equations)

This paper contains 5 sections, 8 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Auxiliary lemmas
  4. $(1,1)-$ weak type boundedness of square function
  5. $(1,1)-$ Weak type boundedness of $g^{*}_{\lambda,\psi}$ operator

Key Result

Proposition 2.1

grafakous Let $\emptyset\neq\varOmega \subsetneq \mathbb{R}^{n}$ be open. Then there exists a family of closed cubes $\{Q_{i}\}_{i}$ such that Here, $r_{i}$ describes the cube side length of $Q_{i}.$

Theorems & Definitions (15)

  • Proposition 2.1
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Remark 3.4
  • Lemma 3.5
  • proof
  • Lemma 3.6
  • ...and 5 more