$L^p$-estimates for singular integral operators along codimension one subspaces

n this talk, we will present recent results on $L^p$-estimates for maximal directional singular integral operators in $\mathbb{R}^n$. These operators are given by a Hörmander–Mihlin multiplier on an $(n-1)$-dimensional subspace and act trivially in the perpendicular direction. The subspace is allowed to depend measurably on the first $n-1$ variables of $\mathbb{R}^n$.

Assuming the subspace is non-degenerate (in the sense that it is away from a cone around $e_n$) and the function $f$ is frequency supported in a cone away from $\mathbb{R}^{n-1}$, we establish $L^p$-bounds for these operators for p > 3/2. If we additionally assume that $f$ is frequency supported in a single frequency band, we are able to extend the boundedness range to p > 1. We will also discuss why the non-degeneracy assumption cannot in general be removed, even in the band-limited case.