Abstract
We prove the extensions of Birkhoff’s and Cotlar’s ergodic theorems to multi-dimensional polynomial subsets of prime numbers mathbb {P}^k. We deduce them from ell ^pbig (mathbb {Z}^dbig )-boundedness of r-variational seminorms for the corresponding discrete operators of Radon type, where p > 1 and r > 2.
Highlights
Let (X, B, μ) be a σ -finite measure space with d0 invertible commuting and measure preserving transformations T1, . . . , Td0 : X → X
In [26], Zorin-Kranich has proved (1.4) for the averaging operators modeled on prime numbers, that is when d0 = k = k = 1 with a polynomial P(n) = n
The oscillating part is controlled by a multi-dimensional version of Weyl–Vinogradov’s inequality with a logarithmic loss together with p Zd estimates for multipliers of Ionescu–Wainger type
Summary
Let (X , B, μ) be a σ -finite measure space with d0 invertible commuting and measure preserving transformations T1, . . . , Td0 : X → X. In [26], Zorin-Kranich has proved (1.4) for the averaging operators modeled on prime numbers, that is when d0 = k = k = 1 with a polynomial P(n) = n. The variational estimates for discrete singular operators have been studied in [3, 12,13,16]. Concerning pointwise ergodic theorems over prime numbers, there are some results using oscillation seminorms. The oscillating part is controlled by a multi-dimensional version of Weyl–Vinogradov’s inequality with a logarithmic loss together with p Zd estimates for multipliers of Ionescu–Wainger type. We get completely unified approach to the variational estimates for the averaging operators and the truncated discrete singular operators. To get completely unified approach to the variational estimates for the averaging operators and truncated singular operators, at the beginning of Sect.
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