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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Characterization and semiadditivity of the $ \mathcal C^1$-harmonic capacity


Authors: Aleix Ruiz de Villa and Xavier Tolsa
Journal: Trans. Amer. Math. Soc. 362 (2010), 3641-3675
MSC (2000): Primary 31A15, 31C05
Published electronically: February 17, 2010
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Abstract: The $ \mathcal C^1$-harmonic capacity $ \kappa^c$ plays a central role in problems of approximation by harmonic functions in the $ \mathcal{C}^1$-norm in $ \mathbb{R}^{n+1}$. In this paper we prove the comparability between the capacity $ \kappa^c$ and its positive version $ \kappa^c_+$. As a corollary, we deduce the semiadditivity of $ \kappa^c$. This capacity can be considered as a generalization in $ \mathbb{R}^{n+1}$ of the continuous analytic capacity $ \alpha$ in $ \mathbb{C}$. Moreover, we also show that the so-called inner boundary conjecture fails for dimensions $ n>1$, unlike in the case $ n=1$.


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Additional Information

Aleix Ruiz de Villa
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bella- terra (Barcelona), Catalonia
Email: aleixrv@mat.uab.cat

Xavier Tolsa
Affiliation: Institució Catalana de Recerca i Estudis Avançats (ICREA) and Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Catalonia
Email: xtolsa@mat.uab.cat

DOI: http://dx.doi.org/10.1090/S0002-9947-10-05105-6
PII: S 0002-9947(10)05105-6
Received by editor(s): May 14, 2008
Published electronically: February 17, 2010
Additional Notes: The first author was supported by grant AP-2004-5141. Also, both authors were partially supported by grant MTM2007-62817 (Spain).
Article copyright: © Copyright 2010 American Mathematical Society