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Sandeep Kumar Verma

Publications and source records attributed to Sandeep Kumar Verma.

7 recordsLinked to original sources
arXiv2026

Pseudo-differential operator associated with the linear canonical Dunkl transform

In this paper, we introduce a pseudo-differential operator associated with the linear canonical Dunkl transform. For a particular class of symbols, we prove that this operator defines a continuous linear mapping from the Schwartz space into itself. We further establish an amplitude integral and kernel representation of the operator and investigate the smoothness and decay properties of the associated kernel. Subsequently, we establish an $L^1$-norm inequality for the pseudo-differential operator on the linear canonical Dunkl Sobolev spaces. Moreover, we extend the pseudo-differential operator to tempered distributions, proving its continuity. We then investigate the $L^2$-boundedness of the operator for the symbol class $S^m_0$. Finally, as an application, we employ the pseudo-differential operator to study non-homogeneous and nonlinear parabolic partial differential equations.

math.FA↗
arXiv2025

Caccioppoli-type inequalities for the Dunkl-$A$-Laplacian and their application to nonexistence result

For a suitable function $A:\mathbb{R}^n\to \mathbb{R}^n$, we introduce the $A$-Laplacian in the Dunkl framework as $\Delta_{k,A}(u) =\text{div}_k(A(\nabla_ku))$, where $\nabla_k$ is the Dunkl-gradient operator associated with the multiplicity function $k$ and the root system $\mathcal{R}$. We derive the local and global Caccioppoli-type inequality for an element $u$ in the Dunkl-Orlicz-Sobolev space, satisfying the Dunkl-differential inequality $$ -\Delta_{k, A}(u) \geq b\Phi(u)\chi_{\{u>0\}}. $$ Using the Caccioppoli inequality, we establish a sufficient condition for the nonexistence of a nonzero solution $u$ to the Dunkl-differential inequality.

math.AP↗
arXiv2025

Wavelet-based inversion and analysis of Flett, Riesz and bi-parametric potentials in $(k,1)$ generalized Fourier framework

In this paper, we construct and analyze Bessel and Flett potentials associated with the heat and Poisson semigroups in the framework of the $(k,1)$-generalized Fourier transform. We establish fundamental properties of these potentials and derive an explicit inversion formula for the Flett potential using a wavelet-like transform. Furthermore, we introduce a $\beta$-semigroup $\mathcal{B}_k^{(\beta,t)}$, defined via $W_k^{(\beta, t)}$, which enables the formulation of an inversion formula for the Riesz potential. As a unifying extension, we define and investigate bi-parametric potentials $\mathfrak{J}_k^{(\alpha,\beta)}$, which generalize both the Bessel potential and the Flett potential. In addition, we define the associated function spaces.

math.FA↗
arXiv2025

Characterization of bi-parametric potentials and rate of convergence of truncated hypersingular integrals in the Dunkl setting

In this work, we introduce the $\beta$-semigroup for $\beta > 0$, which unifies and extends the classical Poisson (for $\beta=1$) and heat (for $\beta=2$) semigroups within the Dunkl analysis framework. Leveraging this semigroup, we derive an explicit representation for the inverse of the Dunkl-Riesz potential and characterize the image of the function space $L_k^p(\mathbb{R}^n)$ for $1 \leq p < \frac{n + 2\gamma}{\alpha}$. We further define the bi-parametric potential of order $\alpha$ by $$\mathfrak{S}_k^{(\alpha,\beta)} = \left(I + (-\Delta_k)^{\beta/2}\right)^{-\alpha/\beta}$$ and establish its inverse along with a detailed description of the associated range space. Our approach employs a wavelet-based method that represents the inverse as the limit of truncated hypersingular integrals parameterized by $\epsilon > 0$. To analyze the convergence of these approximations, we introduce the concept of $\eta$-smoothness at a point $x_0$ in the Dunkl setting. We show that if a function $f \in L_k^p(\mathbb{R}^n) \cap L_k^2(\mathbb{R}^n)$, for $1 \leq p \leq \infty$, possesses $\eta$-smoothness at $x_0$, then the truncated hypersingular approximations converge to $f(x_0)$ as $\epsilon \to 0^+$.

math.FA↗
arXiv2025

Calderon's reproducing formula and extremal functions associated with the linear canonical Dunkl wavelet transform

In this article, we undertake a two-fold investigation. First, we establish Calderons reproducing formula for the linear canonical Dunkl continuous wavelet transform. Further, we define the reproducing kernel linear canonical Dunkl Sobolev space and introduce a novel inner product associated with the continuous wavelet transform in this space. We then derive explicit formulas for the reproducing kernels and present several related results. In the second part, we investigate extremal functions associated with both the continuous wavelet and linear canonical Dunkl transform. In particular, we characterize the extremal functions, represent them in terms of the corresponding reproducing kernels, and establish structural properties relevant to their formulation.

math.FA↗
arXiv2025

Real Paley-Wiener theorems for the linear canonical Dunkl transform

We examine the Sobolev space associated with the linear canonical Dunkl transform and explore some properties of the linear canonical Dunkl operators. Building on these results, we establish a real Paley-Wiener theorem for the linear canonical Dunkl transform. Further, we characterize the square-integrable function f whose linear canonical Dunkl transform of the function is supported in the polynomial domain. Finally, we develop the Boas-type Paley-Wiener theorem for the linear canonical Dunkl transform.

math.CA↗