Mathematics – Functional Analysis
Scientific paper
2010-09-07
Mathematics
Functional Analysis
37 pages; some minor changes; to appear in Journal of Geometric Analysis
Scientific paper
One defines a non-homogeneous space $(X, \mu)$ as a metric space equipped with 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 aim of this paper is to study the boundedness of a Calder\'on-Zygmund operator $T$ as well as the boundedness of certain related singular integrals associated with $T$ 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 work of X. Tolsa \cite{T1} on the non-homogeneous space $(\mathbb R^n, \mu)$ to the setting of a general non-homogeneous space $(X, \mu)$. While our framework is similar to that of \cite{H}, we are able to obtain quite a few properties similar to those of Calder\'on-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 Calder\'on-Zygmund decomposition on the non-homogeneous space $(X, \mu)$ 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 Calder\'on-Zygmund operators and BMO functions.
Bui The Anh
Duong Xuan Thinh
No associations
LandOfFree
Hardy spaces, Regularized BMO spaces and the boundedness of Calderón-Zygmund operators on non-homogeneous spaces does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Hardy spaces, Regularized BMO spaces and the boundedness of Calderón-Zygmund operators on non-homogeneous spaces, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Hardy spaces, Regularized BMO spaces and the boundedness of Calderón-Zygmund operators on non-homogeneous spaces will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-704625