Weak type (1,1) estimates for inverses of discrete rough singular integral operators

TL;DR

This paper establishes weak type (1,1) bounds for the inverses of certain discrete rough Hilbert transforms, demonstrating sharpness and employing regularity estimates for convolutions with singular measures.

Contribution

It provides the first weak type (1,1) estimates for inverses of truncated discrete rough Hilbert transforms, including a sharpness example.

Findings

Weak type (1,1) estimates for inverse operators.

Sharpness of the main result demonstrated.

Regularity estimates for convolutions with singular measures.

Abstract

We obtain weak type (1,1) estimates for the inverses of truncated discrete rough Hilbert transform. We include an ex- ample showing that our result is sharp. One of the ingredients of the proof are regularity estimates for convolution of singular measure associated with the sequence $[m^\alpha]$.