Spaces of Goldberg type on certain measured metric spaces

TL;DR

This paper introduces a Hardy-Goldberg space on measured metric spaces, characterizes its dual as a BMO-type space, and explores interpolation and applications to singular integrals on manifolds.

Contribution

It extends classical harmonic analysis results to measured metric spaces, defining new function spaces and establishing duality and interpolation properties.

Findings

Dual of $hg{M}$ is $mo{M}$

$ ext{Lp}(M)$ interpolates between $hg{M}$ and $ ext{L}^2(M)$ for $p ext{ in }(1,2)$

Applications to singular integral operators on Riemannian manifolds

Abstract

In this paper we define a space $\ghu{M}$ of Hardy--Goldberg type on a measured metric space satisfying some mild conditions. We prove that the dual of $\ghu{M}$ may be identified with $\gbmo{M}$, a space of functions with "local" bounded mean oscillation, and that if $p$ is in $(1,2)$, then $\lp{M}$ is a complex interpolation space between $\ghu{M}$ and $\ld{M}$. This extends previous results of Strichartz, Carbonaro, Mauceri and Meda, and Taylor. Applications to singular integral operators on Riemannian manifolds are given.