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...
| Authors: | ; ; ; |
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| Format: | Article |
| Status: | Published version |
| Publication date: | 2024 |
| Country: | España |
| Institution: | Basque Center for Applied Mathematics (BCAM) |
| Repository: | BIRD. BCAM's Institutional Repository Data |
| OAI Identifier: | oai:bird.bcamath.org:20.500.11824/2130 |
| Online Access: | http://hdl.handle.net/20.500.11824/2130 https://doi.org/10.15781/md28-ws10 |
| Access Level: | Open access |
| Keyword: | Directional operators Zygmund's conjecture Stein's conjecture maximally rotated singular integrals time-frequency analysis |
| Summary: | 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 $n$. |
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