### 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 $CMO_{L^2}^{p(\cdot)}(X)$.

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