ℓp(Zd)-estimates for discrete operators of Radon type: Maximal functions and vector-valued estimates

Abstract

We prove ℓ p Z d bounds, for p ∈ (1, ∞), 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.

Jump inequalities for translation-invariant operators of Radon type on Zd

“…Statement of the main results. We recall that in [MST15;MST17], strong maximal and r-variational estimates on p (Z Γ ) were obtained for the operators M t and H t with the sharp range of exponents p ∈ (1, ∞) and r ∈ (2, ∞). The main aim of this paper is to strengthen these results and provide strong uniform p (Z Γ ) bounds for λ-jumps that are a substitute for the r-variational estimates at the r = 2 endpoint.…”

“…Estimates in the full range of p ∈ (1, ∞) for (1.19) were first obtained by Ionescu and Wainger [IW06], see also [Mir18] for a different approach. Their ideas were taken up in [MST15;MST17] in order to prove sharp variational estimates for M t and H t . In this article we further developed their ideas.…”

“…, N ), and it thus becomes difficult to retain control of the p operator norms for p far away from 2. A different combinatorial organization of family of rational fractions with small denominators was introduced by Ionescu and Wainger in [IW06, Theorem 1.5], (see also [MST15] and [Mir18]). It makes possible to exploit a strong orthogonality in p when p is an even integer, and thus their ideas allow one to control (up to a logarithmic loss) multipliers concentrated at rational frequencies with denominators 1, .…”

“…Finally, Appendix A is devoted to a tool needed for the above: control of multidimensional Weyl sums with an appropriate logarithmic loss. This appeared first in [MST15] and was in turn based on the corresponding approach in [SW99] that allowed a power loss. Here we give a more precise formulation and proof of the result, which implies the corresponding results in [SW99] and [MST15].…”

Averages Along the Primes: Improving and Sparse Bounds

“…Bourgain's work [3] gave a comprehensive approach to the ℓ p theory of arithmetic averages. The subject continues to be under development, with important contributions by [8,16,17]. We point to the work of Mirek-Trojan and Trojan [18,19] also focused on the primes.…”