arXiv subjects
Publications and source records attributed to Bartosz Trojan.
We prove, in a unified way, $r$-variational estimates, $r>2$, on $\ell^{s}(\mathbb{Z})$ spaces, $s \in (1, \infty)$, for averages and truncated singular integrals along the set of prime numbers.
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 $V_r$ on $L^p$ spaces for all $1 \max\{p, p/(p-1)\}$. Moreover, we obtain the estimates which are uniform in the coefficients of a polynomial mapping of fixed degree.
The aim of this paper is to prove Cotlar's ergodic theorem modeled on the set of primes.
We obtain a global estimate of the transition density $p^n(0,x)$ associated to a nearest neighbor random walk, called here "simple", on affine buildings of type $\widetilde{A}_r$. Then we deduce a global estimate of the Green function. This is the analogue of a result on Riemannian symmetric spaces of the noncompact type.