Variational estimates for operators over some thin subsets of primes

We establish $\ell^p (\mathbb{Z})$ boundedness of $r$-variational seminorm for operators of Radon type along subsets of prime numbers of the form $\lbrace p \in \mathbb{P} : \lbrace \varphi_1 (p) \rbrace \lt \psi(p) \rbrace$ As an application we obtain the corresponding pointwise ergodic theorems.

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Received 4 June 2018

Accepted 13 February 2019

Published 8 June 2020