Littlewood--Paley characterizations of $\mathrm{BMO}$ and Triebel--Lizorkin spaces

Abstract

We prove one generalization of the Littlewood--Paley characterization of the $\mathrm{BMO}$ space where the dilations of a Schwartz function are replaced by a family of functions with suitable conditions imposed on them. We also prove that a certain family of Triebel--Lizorkin spaces can be characterized in a similar way.