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

Abstract

In this paper we study maximal directional singular integral operators in $ \mathbb{R}^n $ given by a H\"ormander--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 > 3/2 $. 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.