arXiv subjects
Publications and source records attributed to Wojciech Słomian.
In this short note, we show that the loss of order $N^{c_p/\log\log N}$ in the version of the Ionescu--Wainger multiplier theorem for canonical fractions, recently established by Kosz, Mirek, Słomian, and Zhang in arXiv:2606.12663, is optimal.
In this paper, we extend the recent Ionescu--Wainger multiplier theorem for the set of canonical fractions by Kosz, Mirek, Peluse, Wan, and Wright in several directions. First, we prove its weighted version, which allows us to combine a multifrequency setting with appropriate arithmetic weights. Second, we establish useful seminorm variants of the theorem. Third, we improve the norm upper bounds and, surprisingly, show that these bounds cannot be uniform in the size of the family of canonical fractions. Finally, we demonstrate how these refinements (especially handling arithmetic weights) can be applied by giving a short proof of Bourgain's pointwise ergodic theorem for polynomial iterates.
In this paper, we establish estimates for the oscillation seminorm for the so-called Carleson--Dunkl operator on weighted $L^p(\mathbb{R},w(x)|x|^{2α+1}{\rm d}x)$ spaces with power weights $w(x)=|x|^β$. As a result, we obtain oscillation estimates for the standard Carleson operator on $L_{\rm rad}^p(\mathbb{R}^n,|x|^β{\rm d}x)$. As a byproduct, we obtain a transference principle for radial multipliers on $L_{\rm rad}^p$ spaces, in the spirit of the Rubio de Francia transference principle.
We prove the uniform oscillation and jump inequalities for the polynomial ergodic averages modeled over multi-dimensional subset of primes. These inequalities provide endpoints for the $r$-variational estimates obtained by Trojan arXiv:1803.05406.
We give a relation between the radius and the dimension in which the asymptotic formula in the Waring problem holds in a multiplicative and dimension-free fashion.
In this paper we prove uniform oscillation estimates on $L^p$, with $p\in(1,\infty)$, for truncated singular integrals of the Radon type associated with Calderón-Zygmund kernel, both in continuous and discrete settings. In the discrete case we use the Ionescu-Wainger multiplier theorem and the Rademacher-Menshov inequality to handle the number-theoretic nature of the discrete singular integral. The result we obtained in the continuous setting can be seen as a generalisation of the results of Campbell, Jones, Reinhold and Wierdl for the continuous singular integrals of the Calderón-Zygmund type.
In this paper we establish uniform oscillation estimates on $L^p(X)$ with $p\in(1,\infty)$ for the polynomial ergodic averages. This result contributes to a certain problem about uniform oscillation bounds for ergodic averages formulated by Rosenblatt and Wierdl in the early 1990's. We also give a slightly different proof of the uniform oscillation inequality of Jones, Kaufman, Rosenblatt and Wierdl for bounded martingales. Finally, we show that oscillations, in contrast to jump inequalities, cannot be seen as an endpoint for $r$-variation inequalities.
The aim of this paper is to develop bootstrap arguments to establish maximal, oscillation, variational and jump inequalities for the discrete averaging Radon operators on $\ell^p(\mathbb Z^d)$.