2010/09/07 by Bui, The Anh, Duong, Xuan Thinh
#42B20 #42B35 #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.1009.1274
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 rn 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 Lp spaces and the regularized BMO spaces. This article thus extends the work of X. Tolsa \citeT1 on the non-homogeneous space (\mathbb Rn, μ) to the setting of a general non-homogeneous space (X, μ). While our framework is similar to that of \citeH, 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 L1, boundedness from L∞ 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.