Bounded point derivations on certain function spaces

Brennan, J.

doi:10.1090/spmj/1598

Bounded point derivations on certain function spaces
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by J. E. Brennan

St. Petersburg Math. J. 31 (2020), 313-323

DOI: https://doi.org/10.1090/spmj/1598

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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