On polynomials in primes and J. Bourgain's circle method approach to ergodic theorems II

• Autorzy: R. Nair
• Języki publikacji: EN

Abstrakty

EN

We show that if q is greater than one, T is a measure preserving transformation of the measure space (X,β,μ) and f is in $L^{q}(X,β,μ)$ then if ϕ is a non-constant polynomial mapping the natural numbers to themselves, the averages $π_{N}^{-1} ∑_{1≤p≤N} f(T^{ϕ(p)} x) (N = 1, 2, ...) converge μ almost everywhere. Here p runs over the primes and $π_N$ denotes their number in [1, N].

Kategorie tematyczne

• 42A45: Multipliers
• 47A35: Ergodic theory
• 42B25: Maximal functions, Littlewood-Paley theory
• 28D05: Measure-preserving transformations
• 11L20: Sums over primes
• 11P55: Applications of the Hardy-Littlewood method

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1993
• Tom: 105
• Numer: 3
• Strony: 207-233

Daty

wydano

1993

otrzymano

1991-09-11

poprawiono

1993-01-26

Twórcy

autor

R. Nair

• Department of Pure Mathematics, University of Liverpool, P.O. Box 147, Liverpool L69, 3BX, U.K

Bibliografia

• [1] A. Bellow and V. Losert, On bad universal sequences in ergodic theory (II), in: Proc. Sherbrooke Workshop on Measure Theory, Lecture Notes in Math. 1033, Springer, 1984, 74-78.
• [2] J. Bourgain, On the maximal ergodic theorem for certain subsets of the integers, Israel J. Math. 61 (1988), 39-72.
• [3] J. Bourgain, Pointwise ergodic theorems for arithmetic sets, Publ. I.H.E.S. 69 (1989), 5-45.
• [4] A. P. Calderón, Ergodic theory and translation-invariant operators, Proc. Nat. Acad. Sci. U.S.A. 59 (1968), 349-353.
• [5] R. Nair, On polynomials in primes and J. Bourgain's circle method approach to ergodic theorems, Ergodic Theory Dynamical Systems 11 (1991), 485-499.
• [6] E. Stein, On limits of sequences of operators, Ann. of Math. 74 (1961), 140-170.
• [7] I. M. Vinogradov, Selected Works, Springer, 1985.
• [8] M. Wierdl, Pointwise ergodic theorem along the prime numbers, Israel J. Math. 64 (1988), 315-336.