Seminar on theory of function spaces

Singular integrals along variable codimension one subspaces

In this talk we will consider maximal operators on $\mathbb{R}^n$ formed by taking arbitrary rotations of tensor products of a $n−1$-dimensional Hörmander--Mihlin multiplier with the identity in 1 coordinate. 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 is a weak-type $L^2(\mathbb{R}^n)$-estimate on band-limited functions. As corollaries, we obtain:

1. 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.
2. 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 111 subspaces, a form of the so-called Zygmund's conjecture in general dimension nnn.
This is joint work with Odysseas Bakas, Francesco Di Plinio, and Ioannis Parissis.