Oscillation estimates for truncated singular Radon operators

In this paper we prove uniform oscillation estimates on $L^p$, with $p\in (1,\mathrm{\infty })$, 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.