arXiv subjects

Suman Mukherjee

Publications and source records attributed to Suman Mukherjee.

7 recordsLinked to original sources

arXiv2026

Dimension free weak-type endpoint estimates for the vectors of the Dunkl--Riesz transform

Let $R\subset\mathbb{R}^d$ be a reduced root system, $G$ the associated finite reflection group, and $k\ge0$ a $G$-invariant multiplicity function. We develop a Dunkl analogue of the obstacle/partial-balayage method of Ouyang, Spector, and Stockdale (https://arxiv.org/abs/2608.18068) for the Euclidean fractional Laplacian. For $0 0$, we obtain a decomposition \[ f=\mu+(-\Delta_k)^{s/2}u, \qquad 0\le\mu\le\lambda, \] with $\mu=\lambda$ on $\Omega=\{u>0\}$ and \[ \lambda\nu_k(\Omega)\le\|f\|_{L^1(\nu_k)}. \] As an application, we prove that the vector Dunkl--Riesz transform $\mathcal R_k=\nabla_k(-\Delta_k)^{-1/2}$ is of weak type $(1,1)$ with constant at most $(M_k+2)$, where \[ M_k=\#\{\alpha\in R_+:k(\alpha)>0\}. \] For $G$-invariant functions, the reflection terms vanish and the same argument gives the universal constant $2$. We further establish a dimension-free weak-type $(1,1)$ estimate for the Dunkl--Schr\"odinger Riesz transform.

math.AP↗

arXiv2026

Weighted variational inequalities for the fractional Dunkl heat semigroup

We investigate the convergence properties of the family of operators $$ \mathcal{T}_{\bf N} f(x)=\sum_{j=N_1}^{N_2} v_j\Bigl( e^{-a_{j+1}(-\Delta_k)^s}f(x) - e^{-a_j(-\Delta_k)^s}f(x) \Bigr), \qquad x\in\mathbb{R}^d, $$ where $\{e^{-t(-\Delta_k)^s}\}_{t>0}$ denotes the fractional heat semigroup generated by the Dunkl Laplacian $\Delta_k$. Here ${\bf N}=(N_1,N_2)\in\mathbb{Z}^2$, $N_1<N_2$, the coefficients $\{v_j\}_{j\in\mathbb{Z}}$ form a bounded sequence of real numbers, and $\{a_j\}_{j\in\mathbb{Z}}$ is a monotone increasing sequence of reals. The primary objective of this work is to establish boundedness results for these differential transform operators on weighted $L^p(\mathbb{R}^d,d\mu_k)$ spaces as well as on Dunkl $\mathrm{BMO}(\mathbb{R}^d)$ spaces. We also establish analogous boundedness properties for the associated maximal operator $$ \mathcal{T}^* f(x)=\sup_{\bf N} |\mathcal{T}_{\bf N} f(x)| $$ and study the pointwise convergence of the corresponding series. In addition, we prove that, for compactly supported functions, the maximal differential transform operator $\mathcal{T}^*$ exhibits local behaviour comparable to that of classical singular integral operators.

math.FA↗

arXiv2026

Fractional Leibniz rules for the Dunkl Laplacian in Besov and Triebel--Lizorkin spaces

Let $L$ be the Dunkl Laplacian on the Euclidean space $\mathbb{R}^N$ associated with a normalized root system $R$ and a multiplicity function $k(\nu)\geq 0$, $\nu\in R$. We establish a Leibniz-type rule for the fractional powers of $L$ on Besov and Triebel--Lizorkin spaces in the Dunkl setting. Our approach exploits the interplay between spectral multipliers and the Dunkl transform, together with the support properties of the distributions associated with Dunkl translations. These results extend the corresponding Leibniz-type estimates previously established on $L^p$ spaces to the broader setting of Besov and Triebel--Lizorkin spaces.

math.AP↗

arXiv2026

Large time behaviour of the fractional heat equation associated with the Dunkl Laplacian

We consider the fractional heat equation associated with the Dunkl Laplacian and prove that the weak solutions to this equation converge to the fundamental solution as time becomes large, provided the initial data is an integrable function with respect to the associated measure. As an application, we also prove a similar result for the corresponding nonlinear equation.

math.AP↗

arXiv2025

Dunkl paraproducts and fractional Leibniz rules for the Dunkl Laplacian

We establish fractional Leibniz rules for the Dunkl Laplacian $\Delta_k$ of the form $$\|(-\Delta_k)^s(fg)\|_{L^p(d\mu_k)} \lesssim \|(-\Delta_k)^s f\|_{L^{p_1}(d\mu_k)} \|g\|_{L^{p_2}(d\mu_k)} + \|f\|_{L^{p_1}(d\mu_k)} \|(-\Delta_k)^s g\|_{L^{p_2}(d\mu_k)}.$$ Our approach relies on adapting the classical paraproduct decomposition to the Dunkl setting. In the process, we develop several new auxiliary results. Specifically, we show that for a Schwartz function $f$, the function $(-\Delta_k)^s f$ satisfies a pointwise decay estimate; we establish a version of almost orthogonality estimates adapted to the Dunkl framework; and we investigate the boundedness of Dunkl paraproduct operators on the Lebesgue spaces.

math.FA↗

arXiv2025

On the boundedness of Dunkl multipliers

In this article we use Littlewood-Paley-Stein theory to prove two versions of Dunkl multiplier theorem when the multiplier $ m $ satisfies a modified Hörmander condition. When $ m $ is radial we give a simple proof of a known result. For general $ m $ we prove that the Dunkl multiplier operator takes radial functions in $ L^p $ boundedly into $ L^p $ for all $ p \geq 2.$

math.FA↗

arXiv2024

Weighted Bilinear Multiplier Theorems in Dunkl Setting via Singular Integrals

The purpose of this article is to present one and two-weight inequalities for bilinear multiplier operators in Dunkl setting with multiple Muckenhoupt weights. In order to do so, new results regarding Littlewood-Paley type theorems and weighted inequalities for multilinear Calderón-Zygmund operators in Dunkl setting are also proved.

math.CA↗