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A note on $ L^p$-bounded point evaluations for polynomials


Author: Liming Yang
Journal: Proc. Amer. Math. Soc. 144 (2016), 4943-4948
MSC (2010): Primary 47B20, 30H50; Secondary 30H99, 47B38
DOI: https://doi.org/10.1090/proc/13119
Published electronically: April 19, 2016
MathSciNet review: 3544541
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Abstract: We construct a compact nowhere dense subset $ K$ of the closed unit disk $ \bar {\mathbb{D}}$ in the complex plane $ \mathbb{C}$ such that $ R(K) = C(K)$ and bounded point evaluations for $ P^t(dA \vert _K), ~ 1 \le t < \infty ,$ is the open unit disk $ \mathbb{D}.$ In fact, there exists $ C=C(t) > 0$ such that

$\displaystyle \ \int _{\mathbb{D}} \vert p\vert^t dA \le C \int _K \vert p\vert^t dA, $

for $ 1 \le t < \infty $ and all polynomials $ p.$

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Additional Information

Liming Yang
Affiliation: School of Mathematics, Fudan University, Shanghai, People’s Republic of China
Email: limingyang@fudan.edu.cn

DOI: https://doi.org/10.1090/proc/13119
Received by editor(s): November 18, 2015
Received by editor(s) in revised form: January 23, 2016
Published electronically: April 19, 2016
Communicated by: Pamela B. Gorkin
Article copyright: © Copyright 2016 American Mathematical Society