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

TL;DR

This paper establishes quantitative bounds and compactness results for commutators of area integral operators linked to self-adjoint operators on weighted $L^p$ and Morrey spaces, advancing understanding of their boundedness and compactness properties.

Contribution

It provides new quantitative bounds and proves compactness of these commutators on weighted Morrey spaces, extending previous results to more general operators and spaces.

Findings

Strong-type estimates for commutators on weighted $L^p$ spaces.

Verification of compactness of commutators on weighted Morrey spaces.

Extension of boundedness results to operators associated with heat and Poisson semigroups.

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.