Hardy spaces, Regularized BMO spaces and the boundedness of Calderón-Zygmund operators on non-homogeneous spaces

Abstract

One defines a non-homogeneous space $(X, μ)$ as a metric space equipped with a non-doubling measure $μ$ so that the volume of the ball with center $x$, radius $r$ has an upper bound of the form $r^n$ for some $n> 0$. The aim of this paper is to study the boundedness of a Calderón-Zygmund operator $T$ as well as the boundedness of certain related singular integrals associated with $T$ on various function spaces on $(X, μ)$ such as the Hardy spaces, the $L^p$ spaces and the regularized BMO spaces. This article thus extends the work of X. Tolsa \cite{T1} on the non-homogeneous space $(\mathbb R^n, μ)$ to the setting of a general non-homogeneous space $(X, μ)$. While our framework is similar to that of \cite{H}, we are able to obtain quite a few properties similar to those of Calderón-Zygmund operators on doubling spaces, including the following for such an operator $T$: weak type $(1,1)$ estimate, boundedness from Hardy space into $L^1$, boundedness from $L^{\infty}$ into the regularized BMO and an interpolation theorem. We also prove that the dual space of the Hardy space is the regularized BMO space, obtain a Calderón-Zygmund decomposition on the non-homogeneous space $(X, μ)$ and use this decomposition to show the boundedness of the maximal operators in the form of Cotlar inequality as well as the boundedness of commutators of Calderón-Zygmund operators and BMO functions.