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

Abstract

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

31 pages. Major revision. To appear in the Journal of Functional Analysis

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• Jump inequalities for translation-invariant operators of Radon type on ${\mathbb{Z}}^d$
• Averages along the Square Integers: ${\ell }^p$ improving and Sparse Inequalities
• Discrete Analogoues in Harmonic Analysis: Maximally Monomially Modulated Singular Integrals Related to Carleson's Theorem
• A Discrete Quadratic Carleson Theorem on ${\ell }^2$ with a Restricted Supremum
• Discrete Analogues in Harmonic Analysis: Directional Maximal Functions in ${\mathbb{Z}}^2$
• Discrete Fractional Integration Operators Along the Primes