${\mathrm{\ell }}^p\left({\mathbb{Z}}^d\right)$-estimates for discrete operators of Radon type: Maximal functions and vector-valued estimates

We prove ${\mathrm{\ell }}^p\left({\mathbb{Z}}^d\right)$ bounds, for $p\in (1,\mathrm{\infty })$, of discrete maximal functions corresponding to averaging operators and truncated singular integrals of Radon type, and their applications to pointwise ergodic theory. Our new approach is based on a unified analysis of both types of operators, and also yields an extension to the vector-valued form of these results.