EXISTENCE OF PRINCIPAL VALUES OF SOME SINGULAR INTEGRALS ON CANTOR SETS, AND HAUSDORFF DIMENSION

Joan Verdera; Julia Cufi; Juan Jesus Donaire; Pertti Mattila

EXISTENCE OF PRINCIPAL VALUES OF SOME SINGULAR INTEGRALS ON CANTOR SETS, AND HAUSDORFF DIMENSION

Joan Verdera ^*, Julia Cufi , Juan Jesus Donaire , Pertti Mattila

^*Autor corresponent d’aquest treball

Producció científica: Contribució a revista › Article › Recerca › Avaluat per experts

1 Descàrregues (Pure)

Resum

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 R d 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.

Idioma original	Anglès
Nombre de pàgines	14
Revista	Pacific Journal of Mathematics
Estat de la publicació	Publicada - 1 de des. 2023

Accés al document

https://ddd.uab.cat/record/287011

Com citar-ho

@article{da6ad61a049f43d883dfd7082b3c13cc,

title = "EXISTENCE OF PRINCIPAL VALUES OF SOME SINGULAR INTEGRALS ON CANTOR SETS, AND HAUSDORFF DIMENSION",

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 R d 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. ",

author = "Joan Verdera and Julia Cufi and Donaire, \{Juan Jesus\} and Pertti Mattila",

year = "2023",

month = dec,

day = "1",

language = "English",

journal = "Pacific Journal of Mathematics",

issn = "0030-8730",

publisher = "Mathematical Sciences Publishers",

}

TY - JOUR

T1 - EXISTENCE OF PRINCIPAL VALUES OF SOME SINGULAR INTEGRALS ON CANTOR SETS, AND HAUSDORFF DIMENSION

AU - Verdera , Joan

AU - Cufi , Julia

AU - Donaire , Juan Jesus

AU - Mattila, Pertti

PY - 2023/12/1

Y1 - 2023/12/1

N2 - 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 R d 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.

AB - 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 R d 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.

M3 - Article

SN - 0030-8730

JO - Pacific Journal of Mathematics

JF - Pacific Journal of Mathematics

ER -

EXISTENCE OF PRINCIPAL VALUES OF SOME SINGULAR INTEGRALS ON CANTOR SETS, AND HAUSDORFF DIMENSION

Resum

Accés al document

Fingerprint

Com citar-ho