Singular integrals along variable codimension one subspaces

This article deals with maximal operators on \(ℝ^n\) 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 \(L^2(ℝ^n)\)-estimate on band-limited functions, leads to several corollaries. The first is a sharp \(L^2(ℝ^n)\) 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 \(B^0_{p,1}(ℝ^n)\), \(2≤p<∞\), may be recovered from their averages along a measurable choice of codimension 1 subspaces, a form of Zygmund’s conjecture in general dimension \(n\).