L^p(Z^d)-estimates for discrete operators of Radon type: Variational estimates

We prove ${\mathrm{\ell }}^p\left({\mathbb{Z}}^d\right)$ bounds for $p\in (1,\mathrm{\infty })$, of $r$-variations $r\in (2,\mathrm{\infty })$, for discrete averaging operators and truncated singular integrals of Radon type. We shall present a new powerful method which allows us to deal with these operators in a unified way and obtain the range of parameters of $p$ and $r$ which coincide with the ranges of their continuous counterparts.