dc.contributor.author
Cufí, J.
dc.contributor.author
Donaire, J.J.
dc.contributor.author
Mattila, P.
dc.contributor.author
Verdera, J.
dc.date.accessioned
2024-05-08T13:27:50Z
dc.date.accessioned
2024-09-19T14:29:48Z
dc.date.available
2024-05-08T13:27:50Z
dc.date.available
2024-09-19T14:29:48Z
dc.date.issued
2024-01-09
dc.identifier.uri
http://hdl.handle.net/2072/537583
dc.description.abstract
Consider a standard Cantor set in the plane of Hausdorff dimension 1. If the linear density of the associated measure µ vanishes, then the set of points where the principal value of the Cauchy singular integral of µ exists has Hausdorff dimension 1. The result is extended to Cantor sets in Rd of Hausdorff dimension α and Riesz singular integrals of homogeneity −α, 0 < α < d: the set of points where the principal value of the Riesz singular integral of µ exists has Hausdorff dimension α. A martingale associated with the singular integral is introduced to support the proof. © 2023 The Authors, under license to MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). Open Access made possible by subscribing institutions via Subscribe to Open.
eng
dc.description.sponsorship
The authors are grateful to X. Tolsa for various conversations on the subject and tothe referee for many suggestions that have improved considerably the exposition.Pertti Mattila is grateful for the invitation to visit the Department of Mathematicsat Universitat Autònoma de Barcelona. J. Verdera acknowledges support from thegrants 2021-SGR-00071 (Generalitat de Catalunya), PID2020-112881GB-I00 andSevero Ochoa and Maria de Maeztu CEX2020-001084-M (Ministerio de Ciencia eInnovación). J. J. Donaire acknowledges support from the grant MTM2017-85666-P(Ministerio de Ciencia e Innovación)
dc.format.extent
20 p.
cat
dc.publisher
Mathematical Sciences Publishers
cat
dc.relation.ispartof
Pacific Journal of Mathematics
cat
dc.rights
L'accés als continguts d'aquest document queda condicionat a l'acceptació de les condicions d'ús establertes per la següent llicència Creative Commons: https://creativecommons.org/licenses/by/4.0/
dc.source
RECERCAT (Dipòsit de la Recerca de Catalunya)
dc.subject.other
Cantor set; Cauchy singular integral; Hausdorff dimension; martingale; Riesz singular integral
cat
dc.title
EXISTENCE OF PRINCIPAL VALUES OF SOME SINGULAR INTEGRALS ON CANTOR SETS, AND HAUSDORFF DIMENSION
cat
dc.type
info:eu-repo/semantics/article
cat
dc.type
info:eu-repo/semantics/publishedVersion
cat
dc.identifier.doi
10.2140/pjm.2023.326.285
cat
dc.rights.accessLevel
info:eu-repo/semantics/openAccess
