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 $n$.