Riesz transforms of non-integer homogeneity on uniformly disconnected sets
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- by Maria Carmen Reguera and Xavier Tolsa PDF
- Trans. Amer. Math. Soc. 368 (2016), 7045-7095 Request permission
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 $x/|x|^{s+1}$ is comparable to the capacity $\dot {C}_{\frac {2}{3}(d-s),\frac {3}{2}}$ from non-linear potential theory.References
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Additional Information
- Maria Carmen Reguera
- Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, Catalonia, Spain – and – School of Mathematics, University of Birmingham, Birmingham, United Kingdom
- MR Author ID: 902705
- 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
- MR Author ID: 639506
- ORCID: 0000-0001-7976-5433
- Email: xtolsa@mat.uab.cat
- 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).
- © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 7045-7095
- MSC (2010): Primary 42B20; Secondary 31C45
- DOI: https://doi.org/10.1090/tran/6587
- MathSciNet review: 3471085