Hardy spaces, Regularized BMO spaces and the boundedness of Calder\'on-Zygmund operators on non-homogeneous spaces

摘要

One defines a non-homogeneous space $(X, \mu)$ as a metric space equippedwith a non-doubling measure $\mu$ 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 aimof this paper is to study the boundedness of a Calder\'on-Zygmund operator $T$as well as the boundedness of certain related singular integrals associatedwith $T$ on various function spaces on $(X, \mu)$ such as the Hardy spaces, the$L^p$ spaces and the regularized BMO spaces. This article thus extends the workof X. Tolsa \cite{T1} on the non-homogeneous space $(\mathbb R^n, \mu)$ to thesetting of a general non-homogeneous space $(X, \mu)$. While our framework issimilar to that of \cite{H}, we are able to obtain quite a few propertiessimilar to those of Calder\'on-Zygmund operators on doubling spaces, includingthe following for such an operator $T$: weak type $(1,1)$ estimate, boundednessfrom Hardy space into $L^1$, boundedness from $L^{\infty}$ into the regularizedBMO and an interpolation theorem. We also prove that the dual space of theHardy space is the regularized BMO space, obtain a Calder\'on-Zygmunddecomposition on the non-homogeneous space $(X, \mu)$ and use thisdecomposition to show the boundedness of the maximal operators in the form ofCotlar inequality as well as the boundedness of commutators ofCalder\'on-Zygmund operators and BMO functions.