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

Abstract

We prove ${\ell }^p\left({\mathbb{Z}}^d\right)$ bounds for $p\in (1,\infty )$, of $r$-variations $r\in (2,\infty )$, 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.

42 pages, no figures. arXiv admin note: text overlap with arXiv:1512.07518

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