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Transactions of the American Mathematical Society

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Riesz transforms of non-integer homogeneity on uniformly disconnected sets


Authors: Maria Carmen Reguera and Xavier Tolsa
Journal: Trans. Amer. Math. Soc. 368 (2016), 7045-7095
MSC (2010): Primary 42B20; Secondary 31C45
DOI: https://doi.org/10.1090/tran/6587
Published electronically: February 10, 2016
MathSciNet review: 3471085
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Abstract: In this paper we obtain precise estimates for the $ L^2$ norm of the $ s$-dimensional Riesz transforms on very general measures supported on Cantor sets in $ \mathbb{R}^d$, with $ d-1<s<d$. From these estimates we infer that, for the so-called uniformly disconnected compact sets, the capacity $ \gamma _s$ associated with the Riesz kernel $ x/\vert x\vert^{s+1}$ is comparable to the capacity $ \dot {C}_{\frac {2}{3}(d-s),\frac {3}{2}}$ from non-linear potential theory.


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Maria Carmen Reguera
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, Catalonia, Spain – and – School of Mathematics, University of Birmingham, Birmingham, United Kingdom
Email: m.reguera@bham.ac.uk

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

DOI: https://doi.org/10.1090/tran/6587
Received by editor(s): February 21, 2014
Received by editor(s) in revised form: September 4, 2014
Published electronically: February 10, 2016
Additional Notes: The authors were partially supported by grants 2009SGR-000420 (Catalonia), and MTM-2010-16232 and MTM2013-44304-P (Spain). The first author was also supported by the Juan de la Cierva programme 2011. The second author was also funded by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreement 320501 and by 2014 SGR 75 (Catalonia).
Article copyright: © Copyright 2016 American Mathematical Society

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