69 citations · 257 across the 37 of their papers we have counts for
41 papers
Polynomial sequences in discrete nilpotent groups of step 2
Alexandru D. Ionescu, Akos Magyar, Mariusz Mirek +1
We discuss some of our work on averages along polynomial sequences in nilpotent groups of step 2. Our main results include boundedness of associated maximal functions and singular…
Oscillation inequalities in ergodic theory and analysis: one-parameter and multi-parameter perspectives
Mariusz Mirek, Tomasz Z. Szarek, James Wright
In this survey we review useful tools that naturally arise in the study of pointwise convergence problems in analysis, ergodic theory and probability. We will pay special attention…
Endpoint estimates and optimality for the generalized spherical maximal operator on radial functions
Adam Nowak, Luz Roncal, Tomasz Z. Szarek
We find sharp conditions for the maximal operator associated with generalized spherical mean Radon transform on radial functions $M^{\a,\b}_t$ to be bounded on power weighted Lebes…
Polynomial averages and pointwise ergodic theorems on nilpotent groups
Alexandru D. Ionescu, Ákos Magyar, Mariusz Mirek +1
We establish pointwise almost everywhere convergence for ergodic averages along polynomial sequences in nilpotent groups of step two of measure-preserving transformations on -fi…
Some remarks on oscillation inequalities
Mariusz Mirek, Wojciech Słomian, Tomasz Z. Szarek
In this paper we establish uniform oscillation estimates on with for the polynomial ergodic averages. This result contributes to a certain problem about u…
Lattice points problem, equidistribution and ergodic theorems for certain arithmetic spheres
Alex Iosevich, Bartosz Langowski, Mariusz Mirek +1
We establish an asymptotic formula for the number of lattice points in the sets \[ \mathbf S_{h_1, h_2, h_3}(λ): =\{x\in\mathbb Z_+^3:\lfloor h_1(x_1)\rfloor+\lfloor h_2(x_2)\rfloo…
Distortion in the group of circle homeomorphisms
Juliusz Banecki, Tomasz Szarek
Let be the group of piecewise--affine circle homeomorphisms or the group ${\Diff}^{\infty}(\mathbb R/\mathbb Z)$ of smooth circle diffeomorphisms. A c…
Dimension-free estimates for the discrete spherical maximal functions
Mariusz Mirek, Tomasz Z. Szarek, Błażej Wróbel
We prove that the discrete spherical maximal functions (in the spirit of Magyar, Stein and Wainger) corresponding to the Euclidean spheres in with dyadic radii have $…
On uniqueness of invariant measures for random walks on HOMEO(R)
Sara Brofferio, Dariusz Buraczewski, Tomasz Szarek
We consider random walks on the group of orientation-preserving homeomorphisms of the real line . In particular, the fundamental question of uniqueness of an invariant…
Ergodicity and central limit theorem for random interval homeomorphisms
Klaudiusz Czudek, Tomasz Szarek
The central limit theorem for Markov chains generated by iterated function systems consisting of orientation preserving homeomorphisms of the interval is proved. We study also ergo…
Generic invariant measures for iterated systems of interval homeomorphisms
Wojciech Czernous, Tomasz Szarek
It is known that Iterated Function Systems generated by orientation preserving homeomorphisms of the unit interval admit a unique invariant measure on . The setup for this r…
Genuinely sharp heat kernel estimates on compact rank-one symmetric spaces, for Jacobi expansions, on a ball and on a simplex
Adam Nowak, Peter Sjögren, Tomasz Z. Szarek
We prove genuinely sharp two-sided global estimates for heat kernels on all compact rank-one symmetric spaces. This generalizes the authors' recent result obtained for a Euclidean…
Analysis in the multi-dimensional ball
Peter Sjögren, Tomasz Z. Szarek
We study the heat semigroup maximal operator associated with a well-known orthonormal system in the d-dimensional ball. The corresponding heat kernel is shown to satisfy Gaussian b…
Sharp estimates of the spherical heat kernel
Adam Nowak, Peter Sjögren, Tomasz Z. Szarek
We prove sharp two-sided global estimates for the heat kernel associated with a Euclidean sphere of arbitrary dimension. This solves a long-standing open problem.
Equicontinuous Families of Markov Operators in View of Asymptotic Stability
Sander C. Hille, Tomasz Szarek, Maria A. Ziemlanska
Relation between equicontinuity, the so called e property and stability of Markov operators is studied. In particular, it is shown that any asymptotically stable Markov operator wi…
The Central Limit Theorem for function systems on the circle
Tomasz Szarek, Anna Zdunik
The Central Limit Theorem for Iterated Functions Systems on the circle is proved. We study also ergodicity of such systems.
On a Schur-like property for spaces of measures
Sander C. Hille, Tomasz Szarek, Daniel T. H. Worm +1
A Banach space has the Schur property when every weakly convergent sequence converges in norm. We prove a Schur-like property for measures: if a sequence of finite signed Borel mea…
Stability of iterated function systems on the circle
Tomasz Szarek, Anna Zdunik
We prove that any Iterated Function System of circle homeomorphisms with at least one of them having dense orbit, is asymptotically stable. The corresponding Perron-Frobenius opera…
Lusin area integrals related to Jacobi expansions
Tomasz Z. Szarek
We investigate mixed Lusin area integrals associated with Jacobi trigonometric polynomial expansions. We prove that these operators can be viewed as vector-valued Calderón-Zygmund…
Convex set of quantum states with positive partial transpose analysed by hit and run algorithm
Konrad Szymański, Benoît Collins, Tomasz Szarek +1
The convex set of quantum states of a composite system with positive partial transpose is analysed. A version of the hit and run algorithm is used to generate a sequen…
Maximal operators of exotic and non-exotic Laguerre and other semigroups associated with classical orthogonal expansions
Adam Nowak, Peter Sjögren, Tomasz Z. Szarek
Classical settings of discrete and continuous orthogonal expansions, like Laguerre, Bessel and Jacobi, are associated with second order differential operators playing the role of t…
On Harmonic Analysis Operators in Laguerre-Dunkl and Laguerre-Symmetrized Settings
Adam Nowak, Krzysztof Stempak, Tomasz Z. Szarek
We study several fundamental harmonic analysis operators in the multi-dimensional context of the Dunkl harmonic oscillator and the underlying group of reflections isomorphic to $\m…
Law of the Iterated Logarithm for some Markov operators
Sander C. Hille, Katarzyna Horbacz, Tomasz Szarek +1
The Law of the Iterated Logarithm for some Markov operators, which converge exponentially to the invariant measure, is established. The operators correspond to iterated function sy…
Limit Theorems for Some Markov Operators
Sander Hille, Katarzyna Horbacz, Tomasz Szarek +1
The exponential rate of convergence and the Central Limit Theorem for some Markov operators are established. The operators correspond to iterated function systems which, for exampl…