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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Bounded point derivations on certain function spaces

Author: J. E. Brennan
Original publication: Algebra i Analiz, tom 31 (2019), nomer 2.
Journal: St. Petersburg Math. J. 31 (2020), 313-323
MSC (2010): Primary 41A15; Secondary 30H10
Published electronically: February 4, 2020
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Abstract: Let $ X$ be a compact nowhere dense subset of the complex plane $ \mathbb{C}$, and let $ dA$ denote two-dimensional Lebesgue (or area) measure in $ \mathbb{C}$. Denote by $ \mathcal {R}(X)$ the set of all rational functions having no poles on $ X$, and by $ R^p(X)$ the closure of $ \mathcal {R}(X)$ in $ L^p(X,dA)$ whenever $ 1\leq p<\infty $. The purpose of this paper is to study the relationship between bounded derivations on $ R^p(X)$ and the existence of approximate derivatives provided $ 2<p<\infty $, and to draw attention to an anomaly that occurs when $ p=2$.

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Additional Information

J. E. Brennan
Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506

Keywords: Point derivation, approximate derivative, monogeneity, capacity
Received by editor(s): November 13, 2018
Published electronically: February 4, 2020
Dedicated: Dedicated to V. G. Maz’ya on the occasion of his $80$th birthday and $50$ years of friendship
Article copyright: © Copyright 2020 American Mathematical Society