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

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.