Littlewood-Paley and Hardy space theory on spaces of homogeneous type in the sense of Coifman and Weiss via orthonormal wavelet bases

Abstract

Spaces of homogeneous type were introduced by Coifman and Weiss in the early 1970s. They include many special spaces in analysis and have many applications in the theory of singular integrals and function spaces. For instance, Coifman and Weiss introduced the atomic Hardy space $H_{CW}^p\left(X\right)$ on $\left(X,d,\mu \right)$ and proved that if $T$ is a Calderón-Zygmund singular integral operator that is bounded on $L_2\left(X\right)$, then $T$ is bounded from $H_{CW}^p\left(X\right)$ to $L^p\left(X\right)$ for some $p\le 1$. However, for some applications, additional assumptions were imposed on these general spaces of homogeneous type, because the quasi-metric $d$ may have no regularity and quasi-metric balls, even Borel sets, may not be open.

Using the remarkable orthonormal wavelet basis constructed recently by Auscher and Hytönen, we establish the theory of product Hardy spaces on spaces $X=X_1\times X_2\times \cdots \times X_n$, where each factor $X_i$ is a space of homogeneous type in the sense of Coifman and Weiss. The main tool we develop is the Littlewood-Paley theory on $X$, which in turn is a consequence of a corresponding theory on each factor space.

We make no additional assumptions on the quasi-metric or the doubling measure for each factor space, and thus we extend to the full generality of product spaces of homogeneous type the aspects of both one-parameter and multiparameter theory involving the Littlewood-Paley theory and function spaces. Moreover, our methods would be expected to be a powerful tool for developing function spaces and the boundedness of singular integrals on spaces of homogeneous type.

This is joint work with Yongsheng Han and Lesley Ward.