A new approach for Hardy spaces with variable exponents on spaces of homogeneous type

Abstract

In the paper, we establish and study Hardy spaces with variable exponents on spaces of homogeneous type $(X,d,\mu)$ in the sense of Coifman and Weiss, where $d$ may have no any regularity property and $\mu$ fulfills the doubling property only. First we introduce the Hardy spaces with variable exponents $H^{p(\cdot)}(X)$ by using the wavelet Littlewood--Paley square functions and give their equivalent characterizations. Then we establish the atomic characterization theory for $H^{p(\cdot)}(X)$ via the new Calder\'on-type reproducing identity and the Littlewood--Paley--Stein theory. Finally, we give a unified method for defining these variable Hardy spaces $H^{p(\cdot)}(X)$ in terms of the same spaces of test functions and distributions. More precisely, we introduce the variable Carleson measure spaces $CMO_{L^2}^{p(\cdot)}(X)$ and characterize the variable Hardy spaces via the distributions of $CMO_{L^2}^{p(\cdot)}(X)$.