Singular integrals along variable codimension one subspaces
Fuente:
arXiv
Salvato in:
| Autori principali: | , , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2022
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866910339082223616 |
|---|---|
| author | Bakas, Odysseas Di Plinio, Francesco Parissis, Ioannis Roncal, Luz |
| author_facet | Bakas, Odysseas Di Plinio, Francesco Parissis, Ioannis Roncal, Luz |
| contents | This article deals with maximal operators on ${\mathbb R}^n$ formed by taking arbitrary rotations of tensor products of a $d$-dimensional Hörmander--Mihlin multiplier with the identity in $n-d$ coordinates, in the particular codimension 1 case $d=n-1$. These maximal operators are naturally connected to differentiation problems and maximally modulated singular integrals such as Sjölin'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--Sjölin 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 $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_13646 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Singular integrals along variable codimension one subspaces Bakas, Odysseas Di Plinio, Francesco Parissis, Ioannis Roncal, Luz Classical Analysis and ODEs 42B20 This article deals with maximal operators on ${\mathbb R}^n$ formed by taking arbitrary rotations of tensor products of a $d$-dimensional Hörmander--Mihlin multiplier with the identity in $n-d$ coordinates, in the particular codimension 1 case $d=n-1$. These maximal operators are naturally connected to differentiation problems and maximally modulated singular integrals such as Sjölin'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--Sjölin 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 $n$. |
| title | Singular integrals along variable codimension one subspaces |
| topic | Classical Analysis and ODEs 42B20 |
| url | https://arxiv.org/abs/2211.13646 |