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 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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.$

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

  • [1] Alexandru Aleman, Stefan Richter, and Carl Sundberg, Nontangential limits in $ \mathcal {P}^t(\mu )$-spaces and the index of invariant subspaces, Ann. of Math. (2) 169 (2009), no. 2, 449-490. MR 2480609, https://doi.org/10.4007/annals.2009.169.449
  • [2] J. E. Brennan and E. R. Militzer, $ L^p$-bounded point evaluations for polynomials and uniform rational approximation, Algebra i Analiz 22 (2010), no. 1, 57-74; English transl., St. Petersburg Math. J. 22 (2011), no. 1, 41-53. MR 2641080, https://doi.org/10.1090/S1061-0022-2010-01131-2
  • [3] Theodore W. Gamelin, Uniform algebras, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1969. MR 0410387
  • [4] Kristian Seip, Beurling type density theorems in the unit disk, Invent. Math. 113 (1993), no. 1, 21-39. MR 1223222, https://doi.org/10.1007/BF01244300
  • [5] James E. Thomson, Approximation in the mean by polynomials, Ann. of Math. (2) 133 (1991), no. 3, 477-507. MR 1109351, https://doi.org/10.2307/2944317
  • [6] James E. Thomson, Bounded point evaluations and polynomial approximation, Proc. Amer. Math. Soc. 123 (1995), no. 6, 1757-1761. MR 1242106, https://doi.org/10.2307/2160988
  • [7] Xavier Tolsa, Painlevé's problem and the semiadditivity of analytic capacity, Acta Math. 190 (2003), no. 1, 105-149. MR 1982794, https://doi.org/10.1007/BF02393237

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 47B20, 30H50, 30H99, 47B38

Retrieve articles in all journals with MSC (2010): 47B20, 30H50, 30H99, 47B38


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

American Mathematical Society