Tame measures on certain compact sets
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- by Hsuan Pei Lee PDF
- Proc. Amer. Math. Soc. 80 (1980), 61-67 Request permission
Abstract:
A finite complex Borel measure $\mu$ on a compact subset $X \subset {{\mathbf {C}}^n}$ is called tame if there exist finite measures ${\sigma _1}, \ldots ,{\sigma _n}$ on X with \[ \int _X \phi d\mu = \int _X {\sum \limits _1^n {\frac {{\partial \phi }}{{\partial {{\bar z}_j}}}d{\sigma _j}} } \] for every $\phi \in C_0^\infty ({{\mathbf {C}}^n})$. We define ${X_T} = \{ ({z_1},{z_2}):|{z_1}{|^2} + |{z_2}{|^2} = 1$ and ${z_1} \in T\}$, where T is a compact subset of $\{ |{z_1}| < 1\}$ in ${{\mathbf {C}}^1}$. It is shown in this paper that tame measures form a weak-$^ \ast$ dense subset of $R{({X_T})^ \bot }$. It follows then, with the help of a theorem by Weinstock, that $R({X_T})$ is a local algebra.References
- Richard F. Basener, On rationally convex hulls, Trans. Amer. Math. Soc. 182 (1973), 353β381. MR 379899, DOI 10.1090/S0002-9947-1973-0379899-1
- Barnet M. Weinstock, Approximation by holomorphic functions on certain product sets in $C^{n}$, Pacific J. Math. 43 (1972), 811β822. MR 344523
Additional Information
- © Copyright 1980 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 80 (1980), 61-67
- MSC: Primary 46J10; Secondary 46E27
- DOI: https://doi.org/10.1090/S0002-9939-1980-0574509-1
- MathSciNet review: 574509