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Singular integrals along variable codimension one subspaces

Odysseas Bakas, Francesco Di Plinio, Ioannis Parissis, Luz Roncal

Abstract

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 $n$.

Singular integrals along variable codimension one subspaces

Abstract

This article deals with maximal operators on formed by taking arbitrary rotations of tensor products of a -dimensional Hörmander--Mihlin multiplier with the identity in coordinates, in the particular codimension 1 case . 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 -estimate on band-limited functions, leads to several corollaries. The first is a sharp 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 , , may be recovered from their averages along a measurable choice of codimension subspaces, a form of Zygmund's conjecture in general dimension .
Paper Structure (26 sections, 25 theorems, 293 equations, 4 figures)

This paper contains 26 sections, 25 theorems, 293 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Geometric preliminaries
  3. Smooth families of rotations
  4. Removing the SO(d) invariance
  5. Maximal codimension- multipliers and the Carleson--Sjölin theorem
  6. Maximal codimension-1 multipliers and the proof of Corollary
  7. The Carleson--Sjölin theorem
  8. Tiles, adapted functions, and model operator
  9. Preliminaries
  10. Grids and tiles
  11. Spatial grids
  12. Tiles
  13. Adapted classes and model sums
  14. A homogeneous partition of unity
  15. Discretization of the kernel
  16. ...and 11 more sections

Key Result

Theorem 1.1

There exists $A=A(d)$ such that the following holds. Suppose that the family ${\bf{m}}\coloneqq\{m_\sigma \in L^\infty( \varmathbb{R}^d):\sigma \in \mathrm{\bf{Gr}}(d,n)\}$ is such that Referring to e:thisismain, there holds

Figures (4)

  • Figure 1: A figure for the proof of Lemma \ref{['lem:rot']}
  • Figure 2: A figure for the proof of Lemma \ref{['l:derOsig']}.
  • Figure 3: The frequency component of a tile and its $\kappa$-children
  • Figure :

Theorems & Definitions (55)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 45 more