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 naturally c...
| Autores: | ; ; ; |
|---|---|
| Tipo de documento: | Artículo |
| Estado: | Published version |
| 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 |
| Nivel de acceso: | Acceso abierto |
| Palabra clave: | Directional operators Zygmund's conjecture Stein's conjecture maximally rotated singular integrals time-frequency analysis |
| _version_ | 1878742287682371584 |
|---|---|
| author | Bakas, O. |
| author2 | Di Plinio, F. Parissis, I. Roncal, L. |
| author2_role | author author author |
| author_browse | Bakas, O. Di Plinio, F. Parissis, I. Roncal, L. |
| author_facet | Bakas, O. Di Plinio, F. Parissis, I. Roncal, L. |
| author_role | author |
| collection | BIRD. BCAM's Institutional Repository Data |
| 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 $n$. |
| eu_rights_str_mv | openAccess |
| format | article |
| id | ES_f33ca7d9937907ef88cd9f55e5c8e46d |
| institution | Basque Center for Applied Mathematics (BCAM) |
| instname_str | Basque Center for Applied Mathematics (BCAM) |
| network_acronym_str | ES |
| network_name_str | España |
| oai_identifier_str | oai:bird.bcamath.org:20.500.11824/2130 |
| publishDate | 2024 |
| publishDateSort | 2024 |
| reponame_str | BIRD. BCAM's Institutional Repository Data |
| spelling | Singular integrals along variable codimension one subspaces Bakas, O. Di Plinio, F. Parissis, I. Roncal, L. Directional operators Zygmund's conjecture Stein's conjecture maximally rotated singular integrals time-frequency analysis 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 $n$. RYC2018-025477-I IKERBASQUE http://hdl.handle.net/20.500.11824/2130 https://doi.org/10.15781/md28-ws10 |
| 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 |
| status_str | publishedVersion |
| title | 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_short | Singular integrals along variable codimension one subspaces |
| title_sort | Singular integrals along variable codimension one subspaces |
| topic | Directional operators Zygmund's conjecture Stein's conjecture maximally rotated singular integrals time-frequency analysis |
| topic_facet | Directional operators Zygmund's conjecture Stein's conjecture maximally rotated singular integrals time-frequency analysis |
| url | http://hdl.handle.net/20.500.11824/2130 https://doi.org/10.15781/md28-ws10 |