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Dariusz Kosz

Publications and source records attributed to Dariusz Kosz.

At least 19 recordsLinked to original sources
arXiv2026

The multilinear circle method and a question of Bergelson

Let $k\in \mathbb Z_+$ and $(X, \mathcal B(X), μ)$ be a probability space equipped with a family of commuting invertible measure-preserving transformations $T_1,\ldots, T_k \colon X\to X$. Let $P_1,\ldots, P_k\in\mathbb Z[\rm n]$ be polynomials with integer coefficients and distinct degrees. We establish pointwise almost everywhere convergence of the multilinear polynomial ergodic averages \[ \frac{1}{N}\sum_{n=1}^Nf_1\big(T_1^{P_1(n)}x\big)\cdots f_k\big(T_k^{P_k(n)}x\big), \qquad x\in X, \] as $N\to\infty$ for any functions $f_1, \ldots, f_k\in L^{\infty}(X)$. Besides a couple of results in the bilinear setting $k=2$, and then only in the single transformation case $T_1 = T_2$, this is the first pointwise result for general polynomial multilinear ergodic averages in arbitrary measure-preserving systems. This answers a question of Bergelson from 1996 in the affirmative for any polynomials with distinct degrees, and makes progress on the Furstenberg--Bergelson--Leibman conjecture. In this paper, we build a versatile \emph{multilinear circle method} by developing the Ionescu--Wainger multiplier theorem for the set of canonical fractions, which gives a positive answer to a question of Ionescu and Wainger from 2005. We also establish multilinear $L^p$-improving bounds and an inverse theorem in higher order Fourier analysis for averages over polynomial corner configurations, which we use to establish a multilinear analogue of Weyl's inequality and its real counterpart, a Sobolev smoothing estimate.

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arXiv2026

Remarks on the Ionescu-Wainger multiplier theorem

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.

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arXiv2026

Uniform estimates for Delannoy numbers and dimension-free estimates for discrete maximal functions over cross-polytopes

We prove a uniform upper and lower bound for Delannoy numbers. This is achieved by using the representation of Delannoy numbers as the number of lattice points in high-dimensional cross-polytopes (also known as hyper-octahedrons or $\ell^1$ balls) and proving a uniform (dimension-free) count for these lattice points. Using this count, we establish dimension-free estimates for discrete maximal functions over cross-polytopes. By proving a comparison principle with the continuous setting, we obtain a dimension-free estimate on all $\ell^p(\mathbb{Z}^d)$ spaces for radii $R>C d^{3/2}.$ We also treat the full maximal function on $\ell^2(\mathbb{Z}^d)$ for small radii $R\le d^{1-\varepsilon}$ and the dyadic maximal function for any radii.

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arXiv2026

Polynomial ergodic theorems in the spirit of Dunford and Zygmund

The main goal of the paper is to prove convergence in norm and pointwise almost everywhere on $L^p$, $p\in (1,\infty)$, for certain multiparameter polynomial ergodic averages in the spirit of Dunford and Zygmund for continuous flows. We will pay special attention to quantitative aspects of pointwise convergence phenomena from the point of view of uniform oscillation estimates for multiparameter polynomial Radon averaging operators. In the proof of our main result we develop flexible Fourier methods that exhibit and handle the so-called "parameters-gluing'' phenomenon, an obstruction that arises in studying oscillation and variation inequalities for multiparameter polynomial Radon operators. We will also discuss connections of our main result with a multiparameter variant of the Bellow-Furstenberg problem.

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arXiv2025

Variation of the one-dimensional centered maximal operator on simple functions with gaps between pieces

Let $M$ denote the centered Hardy--Littlewood operator on $\mathbb{R}$. We prove that \[ {\rm Var} (Mf)\le {\rm Var} (f) - \frac12\big| |f(\infty)|-|f(-\infty)|\big| \] for piecewise constant functions $f$ with nonzero and zero values alternating. The above inequality strengthens a recent result of Bilz and Weigt \cite{BW} proved for indicator functions of bounded variation vanishing at $\pm\infty$. We conjecture that the inequality holds for all functions of bounded variation, representing a stronger version of the existing conjecture ${\rm Var} (Mf)\le {\rm Var} (f)$. We also obtain the discrete counterpart of our theorem, moreover proving a transference result on equivalency between both settings that is of independent interest.

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arXiv2025

On differentiation of integrals in Lebesgue spaces

We study the problem of differentiation of integrals for certain bases in the infinite-dimensional torus $\mathbb{T}^ω$. In particular, for every $p_0 \in [1,\infty)$, we construct a basis $\mathcal{B}$ which differentiates $L^p(\mathbb{T}^ω)$ if and only if $p \geq p_0$, thus reproving classical theorems of Hayes in $\mathbb{R}$. The main novelty is that our $\mathcal{B}$ is a Busemann--Feller basis consisting of rectangles (of arbitrarily large dimensions) with sides parallel to the coordinate axes. Our construction gives us the opportunity to classify all possible ranges of differentiation for general complete spaces. Namely, let $\mathcal{B}$ be a basis in a metric measure space $\mathcal{X}$. If $\mathcal{X}$ is complete, then the set $\{ p \in [1,\infty] : \mathcal{B} \text{ differentiates } L^p(\mathcal{X}) \}$ takes one of the six forms\[ \emptyset, \, \{\infty\}, \, [p_0,\infty], \, (p_0,\infty], \, [p_0,\infty), \, (p_0,\infty) \quad \text{for some} \quad p_0 \in [1,\infty). \] Conversely, for every $p_0 \in [1,\infty)$ and each of the six cases above, we construct a complete space $\mathcal{X}$ and a basis $\mathcal{B}$ illustrating the corresponding range of differentiation.

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arXiv2023

Sharp constants in inequalities admitting the Calderón transference principle

The aim of this note is twofold. Firstly, we prove an abstract version of the Calderón transference principle for inequalities of admissible type in the general commutative multilinear and multiparameter setting. Such an operation does not increase the constants in the transferred inequalities. Secondly, we use the last information to study a certain dichotomy arising in problems of finding the best constants in the weak type $(1,1)$ and strong type $(p,p)$ inequalities for one-parameter ergodic maximal operators.

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arXiv2023

Sharp estimates for Jacobi heat kernels in conic domains

We prove genuinely sharp estimates for the Jacobi heat kernels introduced in the context of the multidimensional cone $\mathbb{V}^{d+1}$ and its surface $\mathbb{V}^{d+1}_0$. To do so, we combine the theory of Jacobi polynomials on the cone explored by Xu with the recent techniques by Nowak, Sjögren, and Szarek, developed to find genuinely sharp estimates for the spherical heat kernel.

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arXiv2023

Weak-type maximal function estimates on the infinite-dimensional torus

We prove necessary and sufficient conditions for the weak-$L^p$ boundedness, for $p \in (1,\infty)$, of a maximal operator on the infinite-dimensional torus. In the endpoint case $p=1$ we obtain the same weak-type inequality enjoyed by the strong maximal function in dimension two. Our results are quantitatively sharp.

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arXiv2022

On the doubling condition in the infinite-dimensional setting

We present a systematic approach to the problem whether a topologically infinite-dimensional space can be made homogeneous in the Coifman--Weiss sense. The answer to the examined question is negative, as expected. Our leading representative of spaces with this property is $\mathbb{T}^ω = \mathbb{T} \times \mathbb{T} \times \cdots$ with the natural product topology.

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arXiv2022

The maximal function of the Devil's staircase is absolutely continuous

We study the problem of whether the centered Hardy--Littlewood maximal function of a singular function is absolutely continuous. For a parameter $d \in (0,1)$ and a closed set $E\subset [0,1]$, let $μ$ be a $d$-Ahlfors regular measure associated with $E$. We prove that for the cumulative distribution function $f(x)=μ([0,x])$ its maximal function $Mf$ is absolutely continuous. We then adapt our method to the multiparameter case and show that the same is true in the positive cone defined by these functions, i.e., for functions of the form $f(x)=\sum_{i=1}^{n}μ_i([0,x])$ where $\{μ_i\}_{i=1}^{n}$ is any collection of $d_i$-Ahlfors regular measures, $d_i \in (0,1)$, associated with closed sets $E_i\subset [0,1]$. This provides the first improvement of regularity for the classical centered maximal operator, and can be seen as a partial analogue of the result of Aldaz and Pérez Lázaro about the uncentered maximal operator.

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arXiv2021

Maximal operators in nondoubling metric measure spaces

This is a revised version of the doctoral dissertation of the same title, written under the supervision of Professor Krzysztof Stempak in 2019. For general (possibly nondoubling) metric measure spaces various properties of the associated maximal operators, centered $\mathcal{M}^{\rm c}$ and noncentered $\mathcal{M}$, are investigated. Chapter 1 is the introduction to the topic. In Chapter 2 the classification of possible interrelations between the occurrences of strong, weak, and restricted weak type inequalities for both $\mathcal{M}^{\rm c}$ and $\mathcal{M}$ simultaneously is given. In Chapter 3 a similar analysis for the so-called modified maximal operators is performed. Chapter 4 is devoted to studying the boundedness of $\mathcal{M}^{\rm c}$ from $L^{p,q}$ to $L^{p,r}$. In particular, for each fixed $p \in (1, \infty)$ the classification of possible shapes of the sets \[ \Big\{ \Big( \frac{1}{q},\frac{1}{r} \Big) \in [0,1] \times [0,1] : \mathcal{M}^{\rm c} \text{ is bounded from } L^{p,q} \text{ to } L^{p,r} \Big\} \] is given for the class of spaces satisfying a mild support assumption $μ(X \setminus {\rm supp}(μ)) = 0$. The main result of Chapter 5 is the classification of possible interrelations between the spaces ${\rm BMO}^p$, $p \in [1,\infty)$. In Chapter 6 a dichotomy regarding the finiteness of maximal functions associated with doubling spaces is tested in general setting. As a result, each of the four configurations regarding its occurrence or not for $\mathcal{M}^{\rm c}$ and $\mathcal{M}$ is illustrated with a suitably chosen nondoubling space. Finally, Appendix contains a new elementary proof of the interpolation theorem for Lorentz spaces with the first parameter fixed and the second parameter varying among its natural range of admissibility.

math.CA↗
arXiv2021

Maximal operators on the infinite-dimensional torus

We study maximal operators related to bases on the infinite-dimensional torus $\mathbb{T}^ω$. {For the normalized Haar measure $dx$ on $\mathbb{T}^ω$ it is known that $M^{\mathcal{R}_0}$, the maximal operator associated with the dyadic basis $\mathcal{R}_0$, is of weak type $(1,1)$, but $M^{\mathcal{R}}$, the operator associated with the natural general basis $\mathcal{R}$, is not. We extend the latter result to all $q \in [1,\infty)$. Then we find a wide class of intermediate bases $\mathcal{R}_0 \subset \mathcal{R}' \subset \mathcal{R}$, for which maximal functions have controlled, but sometimes very peculiar behavior.} Precisely, for given $q_0 \in [1, \infty)$ we construct $\mathcal{R}'$ such that $M^{\mathcal{R}'}$ is of restricted weak type $(q,q)$ if and only if $q$ belongs to a predetermined range of the form $(q_0, \infty]$ or $[q_0, \infty]$. Finally, we study the weighted setting, considering the Muckenhoupt $A_p^\mathcal{R}(\mathbb{T}^ω)$ and reverse Hölder $\mathrm{RH}_r^\mathcal{R}(\mathbb{T}^ω)$ classes of weights associated with $\mathcal{R}$. For each $p \in (1, \infty)$ and each $w \in A_p^\mathcal{R}(\mathbb{T}^ω)$ we obtain that $M^{\mathcal{R}}$ is not bounded on $L^q(w)$ in the whole range $q \in [1,\infty)$. Since we are able to show that \[ \bigcup_{p \in (1, \infty)}A_p^\mathcal{R}(\mathbb{T}^ω) = \bigcup_{r \in (1, \infty)} \mathrm{RH}_r^\mathcal{R}(\mathbb{T}^ω), \] the unboundedness result applies also to all reverse Hölder weights.

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arXiv2021

Some remarks on dimension-free estimates for the discrete Hardy-Littlewood maximal functions

Dependencies of the optimal constants in strong and weak type bounds will be studied between maximal functions corresponding to the Hardy--Littlewood averaging operators over convex symmetric bodies acting on $\mathbb R^d$ and $\mathbb Z^d$. Firstly, we show, in the full range of $p\in[1,\infty]$, that these optimal constants in $L^p(\mathbb R^d)$ are always not larger than their discrete analogues in $\ell^p(\mathbb Z^d)$; and we also show that the equality holds for the cubes in the case of $p=1$. This in particular implies that the best constant in the weak type $(1,1)$ inequality for the discrete Hardy--Littlewood maximal function associated with centered cubes in $\mathbb Z^d$ grows to infinity as $d\to\infty$, and if $d=1$ it is equal to the largest root of the quadratic equation $12C^2-22C+5=0$. Secondly, we prove dimension-free estimates for the $\ell^p(\mathbb Z^d)$ norms, $p\in(1,\infty]$, of the discrete Hardy--Littlewood maximal operators with the restricted range of scales $t\geq C_q d$ corresponding to $q$-balls, $q\in[2,\infty)$. Finally, we extend the latter result on $\ell^2(\mathbb Z^d)$ for the maximal operators restricted to dyadic scales $2^n\ge C_q d^{1/q}$.

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arXiv2021

$A_\infty$ condition for general bases revisited: complete classification of definitions

We refer to the discussion on different characterizations of the $A_\infty$ class of weights, initiated by Duoandikoetxea, Martín-Reyes, and Ombrosi. Twelve definitions of the $A_\infty$ condition are considered. For cubes in $\mathbb{R}^d$ every two conditions are known to be equivalent, while for general bases we have a trichotomy: equivalence, one-way implication, or no dependency may occur. In most cases the relations between different conditions have already been established. Here all the unsolved cases are treated and, as a result, a full diagram of the said relations is presented.

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arXiv2020

Boundedness properties of maximal operators on Lorentz spaces

We study mapping properties of the centered Hardy--Littlewood maximal operator $\mathcal{M}$ acting on Lorentz spaces. Given $p \in (1,\infty)$ and a metric measure space $\mathfrak{X}$ we let $Ω^p_{\rm HL}(\mathfrak{X}) \subset [0,1]^2$ be the set of all pairs $(\frac{1}{q},\frac{1}{r})$ such that $\mathcal{M}$ is bounded from $L^{p,q}(\mathfrak{X})$ to $L^{p,r}(\mathfrak{X})$. For each fixed $p$ all possible shapes of $Ω^p_{\rm HL}(\mathfrak{X})$ are characterized. Namely, we show that the boundary of $Ω^p_{\rm HL}(\mathfrak{X})$ either is empty or takes the form $$\{ δ\} \times [0, \lim_{u \rightarrow δ} F(u)] \ \cup \ \{(u, F(u)) : u \in (δ, 1] \},$$ where $δ\in [0,1]$ and $F \colon [δ, 1] \rightarrow [0,1]$ is concave, non-decreasing, and satisfying $F(u) \leq u$. Conversely, for each such $F$ we find $\mathfrak{X}$ such that $\mathcal{M}$ is bounded from $L^{p,q}(\mathfrak{X})$ to $L^{p,r}(\mathfrak{X})$ if and only if the point $(\frac{1}{q}, \frac{1}{r})$ lies on or under the graph of $F$, that is, $\frac{1}{q} \geq δ$ and $\frac{1}{r} \leq F\big(\frac{1}{q}\big)$.

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arXiv2020

Maximal operators on Lorentz spaces in non-doubling setting

We study mapping properties of the centered Hardy--Littlewood maximal operator $\mathcal{M}$ acting on Lorentz spaces $L^{p,q}(\mathfrak{X})$ in the context of certain non-doubling metric measure spaces $\mathfrak{X}$. The special class of spaces for which these properties are very peculiar is introduced and many examples are given. In particular, for each $p_0, q_0, r_0 \in (1, \infty)$ with $r_0 \geq q_0$ we construct a space $\mathfrak{X}$ for which the associated operator $\mathcal{M}$ is bounded from $L^{p_0,q_0}(\mathfrak{X})$ to $L^{p_0,r}(\mathfrak{X})$ if and only if $r \geq r_0$.

math.CA↗
arXiv2020

On differentiation of integrals in the infinite-dimensional torus

We answer the recently posed questions regarding the problem of differentiation of integrals for the Rubio de Francia basis $\mathcal{R}$ in the infinite torus $\mathbb{T}^ω$. In particular, we prove that $\mathcal{R}$ does not differentiate $L^\infty(\mathbb{T}^ω)$. Some remarks about differentiation in the context of arbitrary bases are also included.

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