Singular integrals along variable codimension one subspaces

This article deals with maximal operators on $\mathbb{R}^n$ formed by taking arbitrary rotations of tensor products of a $d$-dimensional H\"ormander--Mihlin multiplier with the identity in $n-d$ coordinates, in the particular \emph{codimension 1} case $d=n-1$. These maximal operators are natura...

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Detalles Bibliográficos
Autores: Bakas, O., Di Plinio, F., Parissis, I., Roncal, L.
Tipo de recurso: artículo
Estado:Versión publicada
Fecha de publicación:2024
País:España
Institución:Basque Center for Applied Mathematics (BCAM)
Repositorio:BIRD. BCAM's Institutional Repository Data
OAI Identifier:oai:bird.bcamath.org:20.500.11824/2130
Acceso en línea:http://hdl.handle.net/20.500.11824/2130
https://doi.org/10.15781/md28-ws10
Access Level:acceso abierto
Palabra clave:Directional operators
Zygmund's conjecture
Stein's conjecture
maximally rotated singular integrals
time-frequency analysis
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spelling Singular integrals along variable codimension one subspacesBakas, O.Di Plinio, F.Parissis, I.Roncal, L.Directional operatorsZygmund's conjectureStein's conjecturemaximally rotated singular integralstime-frequency analysisThis article deals with maximal operators on $\mathbb{R}^n$ formed by taking arbitrary rotations of tensor products of a $d$-dimensional H\"ormander--Mihlin multiplier with the identity in $n-d$ coordinates, in the particular \emph{codimension 1} case $d=n-1$. These maximal operators are naturally connected to differentiation problems and maximally modulated singular integrals such as Sj\"olin's generalization of Carleson's maximal operator. Our main result, a weak-type $L^{2}(\mathbb{R}^n)$-estimate on band-limited functions, leads to several corollaries. The first is a sharp $L^2(\mathbb{R}^n)$ estimate for the maximal operator restricted to a finite set of rotations in terms of the cardinality of the finite set. The second is a version of the Carleson--Sj\"olin theorem. In addition, we obtain that functions in the Besov space $B_{p,1}^0(\mathbb{R}^n)$, $2\le p <\infty$, may be recovered from their averages along a measurable choice of codimension $1$ subspaces, a form of Zygmund's conjecture in general dimension $n$.RYC2018-025477-I IKERBASQUE202520252024info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionapplication/pdfhttp://hdl.handle.net/20.500.11824/2130https://doi.org/10.15781/md28-ws10reponame:BIRD. BCAM's Institutional Repository Datainstname:Basque Center for Applied Mathematics (BCAM)Ingléshttps://ars-ojs-utexas.tdl.org/ars/article/view/25info:eu-repo/grantAgreement/MINECO//SEV-2017-0718info:eu-repo/grantAgreement/AEI/Plan Estatal de Investigación Científica y Técnica y de Innovación 2017-2020/PID2020-113156GB-I00info:eu-repo/grantAgreement/AEI/Plan Estatal de Investigación Científica y Técnica y de Innovación 2021-2023/PID2021-122156NB-I00info:eu-repo/grantAgreement/Gobierno Vasco/BERC/BERC.2022-2025Reconocimiento-NoComercial-CompartirIgual 3.0 Españahttp://creativecommons.org/licenses/by-nc-sa/3.0/es/info:eu-repo/semantics/openAccessoai:bird.bcamath.org:20.500.11824/21302026-06-19T12:47:47Z
dc.title.none.fl_str_mv Singular integrals along variable codimension one subspaces
title Singular integrals along variable codimension one subspaces
spellingShingle Singular integrals along variable codimension one subspaces
Bakas, O.
Directional operators
Zygmund's conjecture
Stein's conjecture
maximally rotated singular integrals
time-frequency analysis
title_short Singular integrals along variable codimension one subspaces
title_full Singular integrals along variable codimension one subspaces
title_fullStr Singular integrals along variable codimension one subspaces
title_full_unstemmed Singular integrals along variable codimension one subspaces
title_sort Singular integrals along variable codimension one subspaces
dc.creator.none.fl_str_mv Bakas, O.
Di Plinio, F.
Parissis, I.
Roncal, L.
author Bakas, O.
author_facet Bakas, O.
Di Plinio, F.
Parissis, I.
Roncal, L.
author_role author
author2 Di Plinio, F.
Parissis, I.
Roncal, L.
author2_role author
author
author
dc.subject.none.fl_str_mv Directional operators
Zygmund's conjecture
Stein's conjecture
maximally rotated singular integrals
time-frequency analysis
topic Directional operators
Zygmund's conjecture
Stein's conjecture
maximally rotated singular integrals
time-frequency analysis
description This article deals with maximal operators on $\mathbb{R}^n$ formed by taking arbitrary rotations of tensor products of a $d$-dimensional H\"ormander--Mihlin multiplier with the identity in $n-d$ coordinates, in the particular \emph{codimension 1} case $d=n-1$. These maximal operators are naturally connected to differentiation problems and maximally modulated singular integrals such as Sj\"olin's generalization of Carleson's maximal operator. Our main result, a weak-type $L^{2}(\mathbb{R}^n)$-estimate on band-limited functions, leads to several corollaries. The first is a sharp $L^2(\mathbb{R}^n)$ estimate for the maximal operator restricted to a finite set of rotations in terms of the cardinality of the finite set. The second is a version of the Carleson--Sj\"olin theorem. In addition, we obtain that functions in the Besov space $B_{p,1}^0(\mathbb{R}^n)$, $2\le p <\infty$, may be recovered from their averages along a measurable choice of codimension $1$ subspaces, a form of Zygmund's conjecture in general dimension $n$.
publishDate 2024
dc.date.none.fl_str_mv 2024
2025
2025
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/20.500.11824/2130
https://doi.org/10.15781/md28-ws10
url http://hdl.handle.net/20.500.11824/2130
https://doi.org/10.15781/md28-ws10
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv https://ars-ojs-utexas.tdl.org/ars/article/view/25
info:eu-repo/grantAgreement/MINECO//SEV-2017-0718
info:eu-repo/grantAgreement/AEI/Plan Estatal de Investigación Científica y Técnica y de Innovación 2017-2020/PID2020-113156GB-I00
info:eu-repo/grantAgreement/AEI/Plan Estatal de Investigación Científica y Técnica y de Innovación 2021-2023/PID2021-122156NB-I00
info:eu-repo/grantAgreement/Gobierno Vasco/BERC/BERC.2022-2025
dc.rights.none.fl_str_mv Reconocimiento-NoComercial-CompartirIgual 3.0 España
http://creativecommons.org/licenses/by-nc-sa/3.0/es/
info:eu-repo/semantics/openAccess
rights_invalid_str_mv Reconocimiento-NoComercial-CompartirIgual 3.0 España
http://creativecommons.org/licenses/by-nc-sa/3.0/es/
eu_rights_str_mv openAccess
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dc.source.none.fl_str_mv reponame:BIRD. BCAM's Institutional Repository Data
instname:Basque Center for Applied Mathematics (BCAM)
instname_str Basque Center for Applied Mathematics (BCAM)
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collection BIRD. BCAM's Institutional Repository Data
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