Weighted $H^{p}$ spaces on a domain and singular integrals
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数理解析研究所講究録 1401 巻 2004 年 73-81
73 Weighted spaces on a domain and singular integrals $H^{p}$
東京女子大学文理学部
(Akihiko Miyachi)
宮地晶彦
Tokyo Woman’s Christian University
This note gives a survey of some recent results on weighted Hardy spaces and singular integral operators on a bounded domain. In Section 1, we shall review the works by Der-Chen Chang, S. Krantz, E. M. Stein, and G. Dafni concerning the Hardy spaces on a bounded domain of and the estimates of certain singular integral operators in those spaces. In Section 2, we review definitions of weighted Hardy spaces on a domain and some of their properties; here the results will be given in the most general setting, i.e., the results will be given for arbitrary open subsets of Rn, $n\geqq 1,$ which is equipped with a Borel measure satisfying certain doubling conditions. In Section 3, we consider the 1 dimensional case with a special weight and give some results on the estimates of singular integral operators in the weighted Hardy spaces on an interval. In the last section, Section 4, we give some remarks on the proofs of the results of Sections 2 and 3. We use the following notation. A ball is the subset of defined by $\mathbb{R}^{n}$
