On BMO functions on Riemann surface
On BMO functions on Riemann surface
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黎曼曲面上的 BMO 函数
DOI：
10.1215/kjm/1250521112
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发表时间：
1985
影响因子：
--
通讯作者：
Yasuhiro Gotoh
中科院分区：
文献类型：
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作者：
Yasuhiro Gotoh
设D是单位圆盘，aD是单位圆盘的边界，d2是D上的双曲测度，dm是D上的二维Lebesgue测度。对于D上的调和函数，我们可以自然地考虑三种不同的BMO性质：（a）边界函数的BMO性质。(b)D关于di的BMO性质（c）D关于dm的B MO性质.性质（a）已从各种观点得到广泛研究（见[2 ]，[5 ]）。另一方面，对BMO的其他两个性质似乎了解不多。在§ 1中，我们研究了D上解析函数和调和函数的这些B MO性质。我们将证明性质（a）和（B）是等价的，性质（c）等价于Bloch性质.在§ 2中，我们研究了n个曲面上的B MO调和函数. Metzger [7 ]证明了任意黎曼曲面R的A D（R）CBM 0 A（R）。本文证明了这一对应关系对有限型黎曼曲面上的调和函数成立，而对一般黎曼曲面上的调和函数不成立.事实上，我们可以证明存在一个无限连通平面区域R，对于该区域，HD（R）V 3 M0 H（R）。本文的写作是受Kusunoki-Taniguchi [6 ]的近期工作的启发，作者希望感谢Y. J.教授。Kusunoki提供有用的建议。
L et D be the un it disk, aD its boundary, d2 th e hyperbolic measure on D, and dm be the 2-dimensional Lebesgue measure o n D . For harmonic functions on D, we can consider three types of different BMO properties naturally : (a) BMO property o f their boundary functions. (b) BMO property o n D with respect to d i (c) B M O property o n D with respect to dm. T h e property (a) has been extensively studied from various points of view (for in s tan ce [2 ], [5 ]) . O n th e other hand, it seems to be not well known about other two BMO properties. In § 1 w e study th e r e la t io n o f these B M O properties for analytic and harmonic functions o n D . We shall show that properties (a) and (b) are equivalent, and property (c) is equivalent to Bloch property. Next in § 2 w e stu d y B M O harmonic functions o n R ie m a n n surfaces. Metzger [7 ] has shown A D(R)CBM 0A (R) f o r arbitrary R iem ann surface R. W e shall sh o w that th e corresponding relation is valid fo r harmonic functions on a Riem ann surface of finite type , and on the other hand, it is not valid for harmonic functions o n a general R iem ann surface. Indeed, we can show that there exists a infinitely connected plane domain R for which HD(R)V 3M0H(R). T h e present paper was motivated by the recent work of Kusunoki-Taniguchi [6 ] and the author wishes to thank Professer Y. Kusunoki for helpful advices.