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Agnieszka Hejna

Publications and source records attributed to Agnieszka Hejna.

17 recordsLinked to original sources
arXiv2025

Littlewood-Paley square functions and the local Hardy space for Multi-Norm Structures on $\mathbb{R}^{d}$

Multi-norm singular integrals and Fourier multipliers were introduced in [29], and one application of these notions was a precise description of the composition of convolution operators with Calderón-Zygmund kernels adapted to $n$ different families of dilations. The description of the resulting operators was given in terms of differential inequalities specified by a matrix $\mathbf E$, and in terms of dyadic decompositions of the kernels and multipliers. In this paper we extend the analysis of multi-norm structures on $\mathbb{R}^d$ by studying the induced Littlewood-Paley decomposition of the frequency space and various associated square functions. After establishing their $L^1$-equivalence, we use these square functions to define a local multi-norm Hardy space $\mathbf{h}^{1}_{\mathbf{E}}(\mathbb{R}^d)$. We give several equivalent descriptions of this space, including an atomic characterization. There has been recent work, limited to the $2$-dilation case, by other authors. The general $n$-dilation case treated here is considerably harder and requires new ideas and a more systematic approach.

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↗
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↗
arXiv2020

Behavior of eigenvalues of certain Schrödinger operators in the rational Dunkl setting

For a normalized root system $R$ in $\mathbb R^N$ and a multiplicity function $k\geq 0$ let $\mathbf N=N+\sum_{α\in R} k(α)$. We denote by $dw(\mathbf{x})=Π_{α\in R}|\langle \mathbf{x},α\rangle|^{k(α)}\,d\mathbf{x}$ the associated measure in $\mathbb{R}^N$. Let $L=-Δ+V$, $V\geq 0$, be the Dunkl--Schrödinger operator on $\mathbb R^N$. Assume that there exists $q >\max(1,\frac{\mathbf{N}}{2})$ such that $V$ belongs to the reverse Hölder class ${\rm{RH}}^{q}(dw)$. For $λ>0$ we provide upper and lower estimates for the number of eigenvalues of $L$ which are less or equal to $λ$. Our main tool in the Fefferman--Phong type inequality in the rational Dunkl setting.

math.CA↗
arXiv2019

Schrödinger operators with reverse Hölder class potentials in the Dunkl setting and their Hardy spaces

For a normalized root system $R$ in $\mathbb R^N$ and a multiplicity function $k\geq 0$ let $\mathbf N=N+\sum_{α\in R} k(α)$. Let $L=-Δ+V$, $V\geq 0$, be the Dunkl--Schrödinger operator on $\mathbb R^N$. Assume that there exists $q >\max(1,\frac{\mathbf{N}}{2})$ such that $V$ belongs to the reverse Hölder class $\text{RH}^q(dw)$. We prove the Fefferman--Phong inequality for $L$. As an application, we conclude that the Hardy space $H^1_{L}$, which is originally defined by means of the maximal function associated with the semigroup $e^{tL}$, admits an atomic decomposition with local atoms in the sense of Goldberg, where their localization are adapted to $V$.

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

Hardy spaces for the Dunkl harmonic oscillator

Let $Δ$ and $L=Δ-\|\mathbf x\|^2$ be the Dunkl Laplacian and the Dunkl harmonic oscillator respectively. We define the Hardy space $\mathcal H^1$ associated with the Dunkl harmonic oscillator by means of the nontangential maximal function with respect to the semigroup $e^{tL}$. We prove that the space $\mathcal H^1$ admits characterizations by relevant Riesz transforms and atomic decompositions. The atoms which occur in the atomic decompositions are of local type.

math.FA↗
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

Hörmander's multiplier theorem for the Dunkl transform

For a normalized root system $R$ in $\mathbb R^N$ and a multiplicity function $k\geq 0$ let $\mathbf N=N+\sum_{α\in R} k(α)$. Denote by $dw(\mathbf x)=\prod_{α\in R}|\langle \mathbf x,α\rangle|^{k(α)}\, d\mathbf x $ the associated measure in $\mathbb R^N$. Let $\mathcal F$ stands for the Dunkl transform. Given a bounded function $m$ on $\mathbb R^N$, we prove that if there is $s>\mathbf N$ such that $m$ satisfies the classical Hörmander condition with the smoothness $s$, then the multiplier operator $\mathcal T_mf=\mathcal F^{-1}(m\mathcal Ff)$ is of weak type $(1,1)$, strong type $(p,p)$ for $1<p<\infty$, and bounded on a relevant Hardy space $H^1$. To this end we study the Dunkl translations and the Dunkl convolution operators and prove that if $F$ is sufficiently regular, for example its certain Schwartz class seminorm is finite, then the Dunkl convolution operator with the function $F$ is bounded on $L^p(dw)$ for $1\leq p\leq \infty$. We also consider boundedness of maximal operators associated with the Dunkl convolutions with Schwartz class functions.

math.FA↗
arXiv2018

Harmonic functions, conjugate harmonic functions and the Hardy space $H^1$ in the rational Dunkl setting

In this work we extend the theory of the classical Hardy space $H^1$ to the rational Dunkl setting. Specifically, let $Δ$ be the Dunkl Laplacian on a Euclidean space $\mathbb{R}^N$. On the half-space $\mathbb{R}_+\times\mathbb{R}^N$, we consider systems of conjugate $(\partial_t^2+Δ_{\mathbf{x}})$-harmonic functions satisfying an appropriate uniform $L^1$ condition. We prove that the boundary values of such harmonic functions, which constitute the real Hardy space $H^1$, can be characterized in several different ways, namely by means of atoms, Riesz transforms, maximal functions or Littlewood-Paley square functions.

math.FA↗