Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A counterexample to an F. and M. Riesz-type theorem

Author: Krzysztof Samotij
Journal: Proc. Amer. Math. Soc. 102 (1988), 337-340
MSC: Primary 30H05,; Secondary 28A12,46E99
MathSciNet review: 920996
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A premeasure is a finitely additive complex-valued function $ \mu $ defined on the semiring of all connected subsets of $ {\mathbf{T}}$, continuous at $ \emptyset $ and with $ \mu (\emptyset ) = \mu ({\mathbf{T}}) = 0$. Let $ \kappa $ be a continuous increasing concave function on $ [0,2\pi ]$ with $ \kappa (0) = 0$. A conjecture from [3] saying that if the Poisson integral of a premeasure $ \mu $ is holomorphic in the open unit disk and $ {\operatorname{Var}_\kappa}(\mu ) < \infty $ then $ {\lim _{\tau \to 0}}{\operatorname{Var}_\kappa }({\mu _\tau } - \mu ) = 0$ is disproved, where $ {\operatorname{Var} _\kappa }(\mu ) = \sup \sum\nolimits_j {\vert\mu ({I_j}} )\vert/\sum\nolimits_j {\kappa (\vert{I_j}\vert)} $ (the supremum is taken over all finite partitions of $ {\mathbf{T}}$ into connected subsets $ {I_j}$) and $ {\mu _\tau }$ denotes the $ \tau $-translation of $ \mu $.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 30H05,, 28A12,46E99

Retrieve articles in all journals with MSC: 30H05,, 28A12,46E99

Additional Information

Keywords: Poisson integral, analytic measures
Article copyright: © Copyright 1988 American Mathematical Society