Linear truncations of the Hilbert transform

Abstract

We study \(L^p\) to \(L^q\) mapping properties of nonconvolution singular integral operators on \(\mathbb{R}^1\), whose kernels are obtained by truncating the Hilbert kernel \(1/x\) in ways that depend linearly on the input variable. Some of these operators arise as special cases of the bilinear Hilbert transform and they are shown to map \(L^p\) to \(L^q\) for \(q \lt p\).