Singular integrals along variable codimension one subspaces

Ars Inveniendi Analytica (2024), Paper No. 2, 53 pp.

Abstract

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This article deals with maximal operators on ℝⁿ 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 L²(ℝⁿ)-estimate on band-limited functions, leads to several corollaries. The first is a sharp L²(ℝⁿ) 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 B⁰_p,₁(ℝⁿ), 2 ≤ p < ∞, may be recovered from their averages along a measurable choice of codimension 1 subspaces, a form of Zygmund’s conjecture in general dimension n.

DOI: https://doi.org/10.15781/md28-ws10