arXiv subjects

Agnieszka Hejna-Łyżwa

Publications and source records attributed to Agnieszka Hejna-Łyżwa.

7 recordsLinked to original sources

arXiv2026

$L^p$ estimates for Dunkl--Schrödinger operators

We study the Dunkl--Schrödinger operator $L=-Δ_k+V$ on $\mathbb R^N$, $N>2$, where $V$ is a nonnegative potential in the reverse Hölder class ${\rm RH}^q(dw)$, $q>\mathbf N/2$, and $\mathbf N$ is the homogeneous dimension. We establish decay estimates for the resolvent kernel in terms of the critical radius and the orbit distance. For reflection-invariant potentials, we obtain $L^p(dw)$ bounds for $VL^{-1}$, second-order Dunkl derivatives of $L^{-1}$, and the Riesz transform $\nabla_kL^{-1/2}$, together with fractional potential and mixed estimates. The proofs combine local estimates for weak solutions, the Dunkl Fefferman--Phong inequality, and kernel estimates adapted to the Euclidean and orbit distances.

math.AP↗

arXiv2026

Riesz transform characterization of Hardy spaces associated with Dunkl--Schrödinger operators

Let $L=-Δ_k+V$ be a Schrödinger operator associated with the Dunkl Laplacian and a nonnegative potential. We characterize the Hardy space $H_L^1$, defined by the maximal operator of the semigroup generated by $-L$, in terms of the Riesz transforms associated with $L$. Under a local integral condition on the potential and a decay condition on the semigroup, formulated relative to an admissible covering by cubes, we prove that \[ \|f\|_{H_L^1} \asymp \|f\|_{L^1(dw)} +\sum_{j=1}^N \|\widetilde{R}_j f\|_{L^1(dw)}. \] The proof combines an atomic characterization of $H_L^1$ with local Hardy space theory and weighted integral estimates for Dunkl derivatives of the heat kernels. The result applies to arbitrary root systems and nonnegative multiplicity functions.

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

On Lipschitz Spaces, Semigroup Transference, and Heat Kernel Derivative Estimates in Dunkl--Schrödinger Analysis

In this paper, we study Lipschitz spaces adapted to Dunkl--Schrödinger operators $\mathcal{L} = -Δ_k + V$, where $Δ_k$ is the Dunkl Laplacian and $V \ge 0$ belongs to the reverse Hölder class $\mathrm{RH}^q(dw)$ with $q > \max(1, \mathbf{N}/2)$. For $0 < β< 2$, we establish the full equivalence between the heat semigroup-based Lipschitz spaces $\widetildeΛ_{\mathcal{L},k}^{β/2}(\mathbb{R}^N)$ and the pointwise weighted Zygmund classes $Λ_{\mathcal{L},k}^β(\mathbb{R}^N)$ defined via the critical radius function $m(\mathbf{x})$. Remarkably, this characterization shows that the spatial smoothness component decouples completely from the underlying reflection group symmetries and root systems. As an intermediate result, we establish new equivalences for inhomogeneous weighted Lipschitz spaces associated with the unperturbed Dunkl Laplacian $Δ_k$. A central difficulty in this context --- arising from the necessity to disentangle the Euclidean distance from the orbit metric $d(\mathbf{x},\mathbf{y})$ --- is resolved through a detailed geometric analysis. Finally, we provide pointwise Gaussian upper bounds for the time derivatives of the Dunkl--Schrödinger heat kernel, which are of independent interest for future work in harmonic analysis on Dunkl structures. Crucially, we do not assume the potential $V$ to be $G$-invariant under the action of the Weyl group, which introduces major geometric difficulties resolved here via a local analysis of the orbit metric.

math.FA↗

arXiv2026

Discrete analogues in harmonic analysis: Multi-parameter Radon averages

In this paper we study maximal and oscillation inequalities for multi-parameter discrete Radon averaging operators. We develop a robust variant of the multi-parameter circle method within the framework of Discrete Analogues in Harmonic Analysis. In particular, this gives quantitative estimates for these averages and their underlying Fourier multipliers which reveals an interesting major-arcs rigidity phenomenon. As a consequence, we completely resolve in the affirmative the multi-parameter Bellow--Furstenberg problem in pointwise ergodic theory.

math.CA↗

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↗