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

In this paper we study maximal directional singular integral operators in ${\mathbb{R}}^n$ given by a Hörmander–Mihlin multiplier on an $(n-1)$-dimensional subspace and acting 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 to be non degenerate in the sense that it is away from a cone around $e_n$ and the function $f$ to be frequency supported in a cone away from ${\mathbb{R}}^{n-1}$, we prove $L^p$-bounds for these operators for $p>3\mathrm{/}2$. If we assume, additionally, that $\widehat{f}$ is supported in a single frequency band, we are able to extend the boundedness range to $p>1$. The non-degeneracy assumption cannot in general be removed, even in the band-limited case.