Oscillation estimates for truncated singular Radon operators

TL;DR

This paper establishes uniform oscillation estimates on L^p spaces for truncated singular Radon operators, extending previous results to both continuous and discrete cases with advanced harmonic analysis techniques.

Contribution

It provides the first unified approach to oscillation estimates for truncated singular Radon operators in both continuous and discrete settings, generalizing prior work.

Findings

Established L^p oscillation bounds for continuous Radon-type operators.

Extended oscillation estimates to discrete Radon operators using number-theoretic tools.

Unified approach applicable to Calderón-Zygmund kernels in both settings.

Abstract

In this paper we prove uniform oscillation estimates on $L^p$, with $p\in(1,\infty)$, for truncated singular integrals of the Radon type associated with Calder\'on-Zygmund kernel, both in continuous and discrete settings. In the discrete case we use the Ionescu-Wainger multiplier theorem and the Rademacher-Menshov inequality to handle the number-theoretic nature of the discrete singular integral. The result we obtained in the continuous setting can be seen as a generalisation of the results of Campbell, Jones, Reinhold and Wierdl for the continuous singular integrals of the Calder\'on-Zygmund type.