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# A counterexample in $L^p$ approximation by harmonic functions

Published:1997-06-01
Printed: Jun 1997
• Joan Mateu
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## Abstract

For ${n \over {n-2}}\leq p<\infty$ we show that the conditions $C_{2,q}(G\setminus \dox)=C_{2,q}(G \setminus X)$ for all open sets $G$, $C_{2,q}$ denoting Bessel capacity, are not sufficient to characterize the compact sets $X$ with the property that each function harmonic on $\dox$ and in $L^p(X)$ is the limit in the $L^p$ norm of a sequence of functions which are harmonic on neighbourhoods of $X$.
 MSC Classifications: 41A30 - Approximation by other special function classes 31C15 - Potentials and capacities