Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

New Sparse Domination and Weighted Estimates for Fractional Operators Beyond Calderón-Zygmund Theory

The Anh Bui, Linfei Zheng

Abstract

Let $L$ be a closed, densely defined operator on $L^2(\mathbb{R}^n)$ satisfying suitable $L^p-L^q$ off-diagonal estimates of order $κ> 0$. This paper aims to investigate the two-weight estimate and the Bloom weighted estimate for the fractional operator $L^{-α/κ}$ with $0 < α< n$ through the method of sparse domination. Our assumptions on the operators are minimal, and our result applies to a wide range of differential operators. As a byproduct, we also establish a new sparse domination criterion for a general class of fractional operators, including the classical fractional integral.

New Sparse Domination and Weighted Estimates for Fractional Operators Beyond Calderón-Zygmund Theory

Abstract

Let be a closed, densely defined operator on satisfying suitable off-diagonal estimates of order . This paper aims to investigate the two-weight estimate and the Bloom weighted estimate for the fractional operator with through the method of sparse domination. Our assumptions on the operators are minimal, and our result applies to a wide range of differential operators. As a byproduct, we also establish a new sparse domination criterion for a general class of fractional operators, including the classical fractional integral.

Paper Structure

This paper contains 7 sections, 22 theorems, 165 equations.

Table of Contents

  1. Introduction and main results
  2. Statement of results
  3. Some applications
  4. A Sparse Domination for Commutators
  5. Two-weight estimates for general fractional sparse forms
  6. Bloom weighted estimates for fractional operators
  7. Applications to fractional powers of differential operators

Key Result

Theorem 1.2

Let $L$ satisfy (A1) and (A2) for some $1\le p_0<q_0\le \infty$, $\epsilon>0$ and $\kappa>0$. Let $p_0 <p <q< q_0$, $\alpha = n(\frac{1}{p}-\frac{1}{q})$. Suppose $\mu,\lambda$ are two weights, denote $u=\mu^{\frac{p_0 p}{p_0-p}}$, $v=\lambda^{\frac{q_0 q}{q_0 - q}}$ and suppose further that $u,v \i

Theorems & Definitions (39)

  • Remark 1.1
  • Theorem 1.2
  • Remark 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Definition 1.6
  • Theorem 1.7
  • Theorem 2.1
  • Theorem 2.2
  • Lemma 2.3
  • ...and 29 more