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Quantitative Bounds and Compactness for the Commutators of Area Integrals Associated with Self-adjoint Operators on Weighted $L^p$ and Morrey Spaces

Chunmei Zhang, Xiangxing Tao

Abstract

Let $L$ be a non-negative self-adjoint operator, we consider some commutators generated by the BMO function $b$ and the area integral operator $S_H$ associated with the heat semigroup $\{e^{-tL}\}_{t>0}$ or the area integral operator $S_P$ associated with the Poisson semigroup $\{e^{-t\sqrt{L}}\}_{t>0}$. The strong-type estimates of these commutators on weighted $L^p$ spaces and weighted Morrey spaces are established. At the same time, we verified that these commutators are compact operators on weighted Morrey spaces.

Quantitative Bounds and Compactness for the Commutators of Area Integrals Associated with Self-adjoint Operators on Weighted $L^p$ and Morrey Spaces

Abstract

Let be a non-negative self-adjoint operator, we consider some commutators generated by the BMO function and the area integral operator associated with the heat semigroup or the area integral operator associated with the Poisson semigroup . The strong-type estimates of these commutators on weighted spaces and weighted Morrey spaces are established. At the same time, we verified that these commutators are compact operators on weighted Morrey spaces.
Paper Structure (3 sections, 17 theorems, 110 equations)

This paper contains 3 sections, 17 theorems, 110 equations.

Table of Contents

  1. Introduction and main results
  2. Definitions and some lemmas
  3. Proof of Theorems

Key Result

Theorem 1.1

Let $L$ be a non-negative self-adjoint operator and the corresponding heat kernel satisfies Gaussian estimates (GE). Assume that $1<p<\infty$ and $w \in A_p$, $b\in \operatorname{BMO}(\mathbb{R}^n)$, then there exists a constant $C$, such that for all $f \in L^{p}(w)$, where $\alpha_p=\operatorname{max}\{\frac{1}{2},\frac{1}{p-1}\}$. Also, estimate lp holds for the commutator $S_{H,b}$.

Theorems & Definitions (23)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Lemma 2.1: GX
  • Definition 2.2: $A_p$ weights
  • Lemma 2.3: XZ
  • Lemma 2.4: P13
  • Lemma 2.5: CPP12
  • ...and 13 more