We prove ℓ p Z d bounds for p ∈ (1, ∞), of r-variations r ∈ (2, ∞), for discrete averaging operators and truncated singular integrals of Radon type. We shall present a new powerful method which allows us to deal with these operators in a unified way and obtain the range of parameters of p and r which coincide with the ranges of their continuous counterparts.where B t = {x ∈ Z k : |x| ≤ t} and t > 0. We will be also interested in discrete truncated singular integrals.Assume that K ∈ C 1 R k \ {0} is a Calderón-Zygmund kernel satisfying the differential inequality |y| k |K(y)| + |y| k+1 |∇K(y)| ≤ 1 for all y ∈ R k with |y| ≥ 1. We also impose the following cancellation condition B λ 2 \B λ 1 K(y) dy = 0 (1.2) for every 0 < λ 1 ≤ λ 2 where B λ is the Euclidean ball in R k centered at the origin with radius λ > 0. Define, for a finitely supported function f : Z d0 → C, the truncated singular Radon transformsThe basic aim of this paper is to strengthen the ℓ p Z d0 boundedness, p ∈ (1, ∞), of maximal functions corresponding to operators (1.1) and (1.3), which have been recently proven in [15], and provide sharp r-variational bounds in the full range of exponents.Recall that for any r ∈ [1, ∞) the r-variational seminorm V r of a sequence a n (x) : n ∈ N of complexvalued functions is defined byThe main results of this article are the following theorems.Theorem A. For every p ∈ (1, ∞) and r ∈ (2, ∞) there is C p,r > 0 such that for all f ∈ ℓ p Z d0Moreover, the constant C p,r ≤ C p r r−2 for some C p > 0 which is independent of the coefficients of the polynomial mapping P.[1], he was able to circumvent this issue for the operators A P N , in the one dimensional case k = d 0 = 1, by controlling their oscillation seminorm. Given a lacunary sequence (n j : j ∈ N), the oscillation seminorm for a sequence a n : n ∈ N of complex numbers is defined by O J a n : n ∈ N = J j=1 sup nj 0 and c < 1/2 such that for all J ∈ NVariational estimates have been the subject of many papers, see [9,10,11,16,22] and the references therein. Our motivation to study r-variational seminorms is threefold. Firstly, for any sequence of functions a n (x) : n ∈ N , if for some 1 ≤ r < ∞ V r a n (x) : n ∈ N < ∞

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.

The authors dedicate this work to the memory of Ante Mimica.Abstract. We prove the asymptotic formulas for the transition densities of isotropic unimodal convolution semigroups of probability measures on R d under the assumption that its Lévy-Khintchine exponent varies slowly. We also derive some new estimates of the transition densities and Green functions.2010 Mathematics Subject Classification. Primary 47D06, 60J75; Secondary: 44A10, 46F12.

Abstract. We establish a higher dimensional counterpart of Bourgain's pointwise ergodic theorem along an arbitrary integer-valued polynomial mapping. We achieve this by proving variational estimates Vr on L p spaces for all 1 < p < ∞ and r > max{p, p/(p − 1)}. Moreover, we obtain the estimates which are uniform in the coefficients of a polynomial mapping of fixed degree.

Abstract. We study asymptotics of generalized eigenvectors associated with Jacobi matrices. Under weak conditions on the coefficients we identify when the matrices are self-adjoint and we show that they satisfy strong non-subordinacy condition.