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

TL;DR

This paper establishes $L^p$ bounds for maximal directional singular integral operators in $R^n$ associated with variable subspaces, extending known results to a broader class of operators with non-degenerate geometric conditions.

Contribution

The paper introduces new $L^p$ estimates for singular integrals along variable codimension-one subspaces, including band-limited functions, under non-degeneracy assumptions.

Findings

Proves $L^p$ bounds for $p > 3/2$ under non-degeneracy.

Extends bounds to $p > 1$ for band-limited functions.

Shows non-degeneracy cannot be removed in general.

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.