Sparse Operators and their boundedness on Morrey-type Spaces: An Expository Note

Abstract

Sparse domination is a central tool in modern harmonic analysis, offering a unified approach to weighted inequalities for Calderón--Zygmund operators and related operators such as commutator operators, rough singular integrals, square functions etc. In this expository note, we briefly survey the main ideas behind sparse bounds on $L^p$ and weighted $L^p$ spaces, and discuss boundedness results for sparse operators on Morrey and generalized Morrey spaces. We establish a model result on the boundedness of sparse operators on Komori--Shirai type weighted Morrey spaces $L^{p,\kappa }(w)$, $w\in A_p$. These bounds offer a simpler framework for deriving Morrey-space estimates for Calderón--Zygmund operators and related operators via sparse domination. We conclude with remarks about extensions and connections to other Morrey space settings.

Accepted to Contemporary Mathematics special volume