Searcharxiv⌕ Search

arXiv subjects

Jacek Dziubański

Publications and source records attributed to Jacek Dziubański.

At least 19 recordsLinked to original sources
arXiv2026

Poisson representations of Dunkl harmonic functions satisfying a Carleson condition

We prove that a smooth Dunkl harmonic function in the upper half-space whose Dunkl gradient satisfies a Carleson measure condition is the Poisson integral of a function in $\mathrm{BMO}(dw)$, defined using Euclidean balls. The result holds for arbitrary finite root systems and nonnegative multiplicities. The proof uses an estimate for increments from the origin, obtained from a finite-dimensional system indexed by the reflection group, together with the atomic characterization of the Dunkl Hardy space.

math.FA↗
arXiv2026

$H^p$ spaces in the Dunkl setting meet Coifman--Weiss--type atoms

Let $Δ_k$ be the Dunkl Laplacian associated with an arbitrary root system and a nonnegative multiplicity function. For every $0<p\leq1$, a Coifman-Weiss type atomic characterization of the Hardy space $H^p_{\mathrm{Dunkl}}$ is established. More precisely, it is proved that the space $H^p_{\mathrm{Dunkl}}$, which is originally defined by a relevant square function, coincides with the space generated by $({\rm CW},p,2)$-atoms, that is, atoms supported on Euclidean balls and satisfying cancellation conditions against all polynomials of degree $ \leq s_p$, where \[ s_p=\left\lfloor \mathbf N\left(\frac1p-1\right)\right\rfloor \] and $\mathbf N$ is the homogeneous dimension of the underlying Dunkl measure. The corresponding quasi-norms are equivalent. We also show that the same space is obtained when the $L^2$ size condition in the definition of atoms is replaced by the $L^\infty$ size condition. The strategy of the proof is to use an operator-type atomic decomposition associated with the Dunkl Laplacian, and then prove that each such operator atom can be written a linear combination of $({\rm CW},p,2)$-atoms.

math.FA↗
arXiv2026

A Remark on Hörmander Multipliers on Dunkl Hardy Spaces

Let $\mathcal R$ be a normalized root system in $\mathbb R^N$ with a nonnegative multiplicity function $k$, and let $\mathcal F$ be the associated Dunkl transform. We prove a Hörmander multiplier theorem on the Hardy spaces $H^p_{\mathrm{Dunkl}}$, $0 0} \|ψ(\cdot)m(t\cdot)\|_{W_2^s}<\infty \] for a nonzero radial cutoff $ψ\in C_c^\infty(\mathbb R^N\setminus\{0\})$ and \[ s>\mathbf N\left(\frac1p-\frac12\right), \] where $\mathbf N$ is the homogeneous dimension, then the Dunkl multiplier \[ \mathcal T_mf=\mathcal F^{-1}(m\mathcal Ff) \] extends to a bounded operator on $H^p_{\mathrm{Dunkl}}$.

math.FA↗
arXiv2026

Potential-free $L^1$-estimates for positivity-preserving Riesz transform related to Schrödinger operator in dimension one

Let $V\geq 0$ be a locally integrable function on $\mathbb R$. Consider the Schrödinger operator $L=-\frac{d^2}{dx^2} +V$. We prove that for all $0<a\leq 1$, there is a constant $C_a$, independent of $V$, such that the Riesz transform type operator $V^aL^{-a}$ is bounded on $L^1(\mathbb R)$ and $\| V^aL^{-a}f\|_{L^1(\mathbb R)}\leq C_a\|f\|_{L^1(\mathbb R)}$.

math.FA↗
arXiv2026

Hörmander's multiplier theorem on $H^p$-spaces in the rational Dunkl setting

On $\mathbb{R}^N$ equipped with a normalized root system $\mathcal R$ and a multiplicity function $k\geq 0$, let $dw(\mathbf x)=Π_{α\in \mathcal R}|\langle \mathbf x,α\rangle|^{k(α)}\, d\mathbf x$, $\mathbf{N}=N+\sum_{α\in \mathcal R}k(α)$ denote the associated measure and the homogeneous dimension of the system $(\mathcal R,k)$ respectively. Let $\mathcal F$ stand for the Dunkl transform. For $0 \mathbf{N}/p$. We show that the multiplier operator $\mathcal T_mf=\mathcal F^{-1}(m\mathcal Ff)$, initially defined on $H^p_{\mathrm{Dunkl}}\cap L^2(dw)$, has a unique extension to a bounded operator in $H^p_{\mathrm{Dunkl}}$, where the space $H^p_{\mathrm{Dunkl}}$ is defined by means of a Littlewood-Paley square function. To prove the theorem, we use special atomic and molecule characterizations of $H^p_{\mathrm{Dunkl}}$.

math.FA↗
arXiv2025

Characterizations of $H^1$ and Fefferman-Stein decompositions of ${\rm BMO}$ functions by systems of singular integrals in the Dunkl setting

We extend the classical theorem of Uchiyama about constructive Fefferman-Stein decompositions of ${\rm BMO}$ functions by systems of singular integrals to the rational Dunkl setting. On $\mathbb{R}^N$ equipped with a root system $R$ and a multiplicity function $k \geq 0$, let \[ dw(\mathbf{x}) = \prod_{α\in R} |\langle α, \mathbf{x} \rangle|^{k(α)} \, d\mathbf{x} \] denote the associated measure, and let $\mathcal{F}$ stand for the Dunkl transform. Consider a system $(θ_0, θ_1, θ_2, \dots, θ_d)$ of functions on $\mathbb{R}^N$ that are smooth away from the origin and homogeneous of degree zero, with $θ_0(ξ) \equiv 1$. We prove that if \[ \text{rank} \left( \begin{array}{ccccc} 1 & θ_1(ξ) & θ_2(ξ) & \ldots & θ_d(ξ) \\ 1 & θ_1(-ξ) & θ_2(-ξ) & \ldots & θ_d(-ξ) \end{array} \right) = 2 \quad \text{for all } ξ\in \mathbb{R}^N \text{ with } \|ξ\| = 1, \] then any compactly supported ${\rm BMO}(\mathbb{R}^N, \|\mathbf{x} - \mathbf{y}\|, dw)$ function $f$ can be decomposed into \[ f = g_0 + \sum_{j=1}^d \mathbf{S}^{\{j\}} g_j, \quad \left\| \sum_{j=0}^d g_j \right\|_{L^\infty} \leq C \|f\|_{\rm BMO}, \] where $\mathbf{S}^{\{j\}} g = \mathcal{F}^{-1}(θ_j \mathcal{F}g)$. As a corollary, we obtain characterizations of the Hardy space $H^1_{\rm Dunkl}$ by the system of singular integral operators $({\rm Id}, \mathbf{S}^{\{1\}}, \mathbf{S}^{\{2\}}, \dots, \mathbf{S}^{\{d\}})$.

math.FA↗
arXiv2025

On dimension-free and potential-free estimates for Riesz transforms associated with Schrödinger operators

Let $L=-Δ+ V(x)$ be a Schrödinger operator on $\mathbb R^d$, where $V(x)\geq 0$, $V\in L^2_{\rm loc} (\mathbb R^d)$. We give a short proof of dimension free $L^p(\mathbb R^d)$ estimates, $1<p\leq 2$, for the vector of the Riesz transforms $$\big(\frac{\partial}{\partial x_1}L^{-1/2}, \frac{\partial}{\partial x_2}L^{-1/2},\dots,\frac{\partial}{\partial x_d}L^{-1/2}\Big).$$ The constant in the estimates does not depend on the potential $V$. We simultaneously provide a short proof of the weak type $(1,1)$ estimates for $\frac{\partial}{\partial x_j}L^{-1/2}$.

math.FA↗
arXiv2024

On Lipschitz spaces in the Dunkl setting -- semigroup approach

Let $\{P_t\}_{t>0}$ be the Dunkl-Poisson semigroup associated with a root system $R\subset \mathbb R^N$ and a multiplicity function $k\geq 0$. Analogously to the classical theory, we say that a bounded measurable function $f$ defined on $\mathbb R^N$ belongs to the inhomogeneous Lipschitz space $Λ_k^β$, $β>0$, if $$\sup_{t>0} t^{m-β} \Big\|\frac{d^m}{dt^m} P_tf\Big\|_{L^\infty}<\infty,$$ where $m=[β]+1$. We prove that the spaces $Λ^β_k$ coincide with the classical Lipschitz spaces.

math.FA↗
arXiv2023

A note on commutators of singular integrals with ${\rm BMO}$ and ${\rm VMO}$ functions in the Dunkl setting

On $\mathbb R^N$ equipped with a root system $R$, multiplicity function $k \geq 0$, and the associated measure $dw(\mathbf{x})=\prod_{α\in R}|\langle \mathbf{x},α\rangle|^{k(α)}\,d\mathbf{x}$, we consider a (non-radial) kernel ${K}(\mathbf{x})$ which has properties similar to those from the classical theory of singular integrals and the Dunkl convolution operator $\mathbf{T}f=f*K$ associated with ${K}$. Assuming that $b$ belongs to the ${\rm BMO}$ space on the space of homogeneous type $X=(\mathbb{R}^N,\|\cdot\|,dw)$, we prove that the commutator $[b,\mathbf{T}]f(\mathbf{x})=b(\mathbf{x})\mathbf{T}f(\mathbf{x})-\mathbf{T}(bf)(\mathbf{x})$ is a bounded operator on $L^p(dw)$ for all $1<p<\infty$. Moreover, $[b,\mathbf T]$ is compact on $L^p(dw)$, provided $b\in {\rm VMO} (X)$. The paper extents results of Han, Lee, Li and Wick.

math.FA↗
arXiv2022

Remarks on Dunkl translations of non-radial kernels

On $\mathbb R^N$ equipped with a root system $R$ and a multiplicity function $k>0$, we study the generalized (Dunkl) translations $τ_{\mathbf x}g(-\mathbf y)$ of not necessarily radial kernels $g$. Under certain regularity assumptions on $g$, we derive bounds for $τ_{\mathbf x}g(-\mathbf y)$ by means the Euclidean distance $\|\mathbf x-\mathbf y\|$ and the distance $d(\mathbf x,\mathbf y)=\min_{σ\in G} \| \mathbf x-σ(\mathbf y)\|$, where $G$ is the reflection group associated with $R$. Moreover, we prove that $τ$ does not preserve positivity, that is, there is a non-negative Schwartz class function $φ$, such that $τ_{\mathbf x}φ(-\mathbf y)<0$ for some points $\mathbf x,\mathbf y\in\mathbb R^N$.

math.FA↗
arXiv2022

On Dunkl Schrödinger semigroups with Green bounded potentials

On $\mathbb R^N$ equipped with a normalized root system $R$, a multiplicity function $k(α) > 0$, and the associated measure $$ dw(\mathbf x)=\prod_{α\in R}|\langle \mathbf x,α\rangle|^{k(α)}\, d\mathbf x, $$ we consider a Dunkl Schrödinger operator $L=-Δ_k+V$, where $Δ_k$ is the Dunkl Laplace operator and $V\in L^1_{\rm loc} (dw)$ is a non-negative potential. Let $h_t(\mathbf x,\mathbf y)$ and $k^{\{V\}}_t(\mathbf x,\mathbf y)$ denote the Dunkl heat kernel and the integral kernel of the semigroup generated by $-L$ respectively. We prove that $k^{\{V\}}_t(\mathbf x,\mathbf y)$ satisfies the following heat kernel lower bounds: there are constants $C, c>0$ such that $$ h_{ct}(\mathbf x,\mathbf y)\leq C k^{\{V\}}_t(\mathbf x,\mathbf y)$$ if and only if $$ \sup_{\mathbf x\in\mathbb R^N} \int_0^\infty \int_{\mathbb R^N} V(\mathbf y)w(B(\mathbf x,\sqrt{t}))^{-1}e^{-\|\mathbf x-\mathbf y\|^2/t}\, dw(\mathbf y)\, dt<\infty, $$ where $B(\mathbf x,\sqrt{t})$ stands for the Euclidean ball centered at $\mathbf x \in \mathbb{R}^N$ and radius $\sqrt{t}$.

math.FA↗
arXiv2021

Upper and lower bounds for Dunkl heat kernel

On $\mathbb R^N$ equipped with a normalized root system $R$, a multiplicity function $k(α) > 0$, and the associated measure $$ dw(\mathbf x)=\prod_{α\in R}|\langle \mathbf x,α\rangle|^{k(α)}\, d\mathbf x, $$ let $h_t(\mathbf x,\mathbf y)$ denote the heat kernel of the semigroup generated by the Dunkl Laplace operator $Δ_k$. Let $d(\mathbf x,\mathbf y)=\min_{σ\in G} \| \mathbf x-σ(\mathbf y)\|$, where $G$ is the reflection group associated with $R$. We derive the following upper and lower bounds for $h_t(\mathbf x,\mathbf y)$: for all $c_l>1/4$ and $0 0$ such that $$ C_{l}w(B(\mathbf{x},\sqrt{t}))^{-1}e^{-c_{l}\frac{d(\mathbf{x},\mathbf{y})^2}{t}} Λ(\mathbf x,\mathbf y,t) \leq h_t(\mathbf{x},\mathbf{y}) \leq C_{u}w(B(\mathbf{x},\sqrt{t}))^{-1}e^{-c_{u}\frac{d(\mathbf{x},\mathbf{y})^2}{t}} Λ(\mathbf x,\mathbf y,t), $$ where $Λ(\mathbf x,\mathbf y,t)$ can be expressed by means of some rational functions of $\| \mathbf x-σ(\mathbf y)\|/\sqrt{t}$. An exact formula for $Λ(\mathbf x,\mathbf y,t)$ is provided.

math.FA↗
arXiv2019

Singular integrals in the rational Dunkl setting

On $\mathbb R^N$ equipped with a normalized root system $R$ and a multiplicity function $k\geq 0$ let us consider a (non-radial) kernel $K(\mathbf x)$ which has properties similar to those from the classical theory. We prove that a singular integral Dunkl convolution operator associated with the kernel $K$ is bounded on $L^p$ for $1<p<\infty$ and of weak-type (1,1). Further we study a maximal function related to the Dunkl convolutions with truncation of $K$.

math.FA↗
arXiv2019

On semigroups generated by sums of even powers of Dunkl operators

On the Euclidean space $\mathbb R^N$ equipped with a normalized root system $R$, a multiplicity function $k\geq 0$, and the associated measure $dw(\mathbf x)=\prod_{α\in R} |\langle \mathbf x,α\rangle|^{k(α)}d\mathbf x$ we consider the differential-difference operator $$L=(-1)^{\ell+1} \sum_{j=1}^m T_{ζ_j}^{2\ell},$$ where $ζ_1,...,ζ_m$ are nonzero vectors in $\mathbb R^N$, which span $\mathbb R^N$, and $T_{ζ_j}$ are the Dunkl operators. The operator $L$ is essentially self-adjoint on $L^2(dw)$ and generates a semigroup $\{S_t\}_{t \geq 0}$ of linear self-adjoint contractions, which has the form $S_tf(\mathbf x)=f*q_t(\mathbf{x})$, $q_t(\mathbf x)=t^{-\mathbf N/ (2\ell)}q(\mathbf x/ t^{1/ (2\ell)})$, where $q(\mathbf x)$ is the Dunkl transform of the function $ \exp(-\sum_{j=1}^m \langle ζ_j,ξ\rangle^{2\ell})$. We prove that $q(\mathbf x)$ satisfies the following exponential decay: $$ |q(\mathbf x)| \lesssim \exp(-c \| \mathbf x\|^{2\ell/ (2\ell-1)})$$ for a certain constant $c>0$. Moreover, if $q(\mathbf x,\mathbf y)=τ_{\mathbf x}q(-\mathbf y)$, then $|q(\mathbf x,\mathbf y)|\lesssim w(B(\mathbf x,1))^{-1} \exp(-c d(\mathbf x,\mathbf y)^{2\ell / (2\ell-1)})$, where $d(\mathbf x,\mathbf y)=\min_{σ\in G}\| \mathbf x- σ(\mathbf y)\| $, $G$~is the reflection group for $R$, and $τ_{\mathbf x}$ denotes the Dunkl translation.

math.FA↗
arXiv2019

Lie group approach to Grushin operators

We consider a finite system $\{X_1, X_2, \ldots, X_n\}$ of complete vector fields acting on smooth manifolds $M$ equipped with a smooth positive measure. We assume that the system satisfies Hörmander's condition and generates a finite dimensional Lie algebra of type (R). We investigate the sum of squares of the vector fields operator corresponding to this system which can be viewed as a generalisation of the notion of Grushin operators. In this setting we prove the Poincaré inequality and Li-Yau estimates for the corresponding heat kernel as well as the doubling condition for the optimal control metrics defined by the system. We discuss a surprisingly broad class of examples of described setting.

math.AP↗
arXiv2019

Remark on atomic decompositions for Hardy space $H^1$ in the rational Dunkl setting

Let $Δ$ be the Dunkl Laplacian on $\mathbb R^N$ associated with a normalized root system $R$ and a multiplicity function $k(α)\geq 0$. We say that a function $f$ belongs to the Hardy space $H^1_Δ$ if the nontangential maximal function $\mathcal M_H f(\mathbf x)=\sup_{\| \mathbf x-\mathbf y\|<t} |\exp(t^2Δ)f(\mathbf x)|$ belongs to $L^1(w(\mathbf x)\, d\mathbf x)$, where $w(\mathbf x)=\prod_{α\in R} |\langle α,\mathbf x\rangle|^{k(α)}$. We prove that $H^1_Δ$ coincides with the space $H^1_{\rm atom}(\mathbb R^N, \| \mathbf x-\mathbf y\|, w(\mathbf x)d\mathbf x)$ understood as the atomic Hardy space on the space of homogeneous type in the sense of Coifman--Weiss. To this end we improve estimates for the heat kernel of $e^{tΔ}$.

math.FA↗
arXiv2018

Sharp Gaussian estimates for heat kernels of Schrödinger operators

We characterize functions $V\le 0$ for which the heat kernel of the Schrö\-dinger operator $Δ+V$ is comparable with the Gauss-Weierstrass kernel uniformly in space and time. In dimension $4$ and higher the condition turns out to be more restrictive than the condition of the boundedness of the Newtonian potential of $V$. This resolves the question of V.~Liskevich and Y.~Semenov posed in 1998. We also give specialized sufficient conditions for the comparability, showing that local $L^p$ integrability of $V$ for $p>1$ is not necessary for the comparability.

math.FA↗