Weighted Bourgain-Morrey-Besov type and Triebel-Lizorkin type spaces associated with operators

TL;DR

This paper introduces new weighted function spaces related to a self-adjoint operator on a space of homogeneous type, providing their characterizations, decompositions, and boundedness of associated operators.

Contribution

It develops weighted Bourgain-Morrey-Besov and Triebel-Lizorkin spaces linked to an operator with Gaussian heat kernel bounds, including their characterizations and atomic decompositions.

Findings

Established continuous characterizations via Peetre maximal functions

Provided atomic and molecular decompositions of the spaces

Proved boundedness of fractional powers and spectral multipliers of the operator

Abstract

Let $(X,\mu)$ be a space of homogeneous type satisfying $\mu(X) =\infty$, the doubling property and the reverse doubling condition. Let $L$ be a nonnegative self-adjoint operator on $L^2(X)$ whose heat kernel enjoys a Gaussian upper bound. We introduce the weighted homogeneous Bourgain-Morrey-Besov type spaces and Triebel-Lizorkin type spaces associated with the operator $L$. We obtain their continuous characterizations in terms of Peetre maximal functions, noncompactly supported functional calculus, heat kernel. Atomic and molecular decompositions of weighted homogeneous Bourgain-Morrey-Besov type spaces and Triebel-Lizorkin type spaces are also given. As an application, we obtain the boundedness of the fractional power of $L$, the spectral multiplier of $L$ on Bourgain-Morrey-Besov type spaces and Triebel-Lizorkin type spaces.