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Pritam Ganguly

Publications and source records attributed to Pritam Ganguly.

16 recordsLinked to original sources
arXiv2026

On Rellich-type asymptotics for eigenfunctions on rank one symmetric spaces of noncompact type

We study eigenfunctions of the Laplace--Beltrami operator \(Δ_X\) in exterior domains \(Ω\) of rank-one Riemannian symmetric spaces of noncompact type \(X\), a class that includes all hyperbolic spaces. Extending the classical \(L^2\) Rellich theorem for the Euclidean Laplacian, we analyze the asymptotic behaviour and \(L^p\)-integrability of solutions to the Helmholtz equation \[ Δ_X f + (λ^2 + ρ^2) f = 0 \quad \text{in } Ω, \] where \(λ\in \mathbb{C}\setminus i\mathbb{Z}\) and \(ρ\) denotes the half-sum of positive roots. We establish sharp Rellich-type quantitative \(L^p\)-growth estimates in geodesic annuli, which yield the nonexistence of nontrivial \(L^p(Ω)\)-solutions in the optimal range \(1 \leq p \leq 2\) for spectral parameters satisfying \(|\Im(λ)| \leq (2/p - 1)ρ\). For non-real spectral parameters, we further obtain refined Rellich-type uniqueness results under weak \(L^p\)-assumptions. As a by-product, we also prove a Rellich-type uniqueness theorem in terms of Hardy-type norms. Our results provide a geometric extension of the Euclidean Rellich theorem, highlighting the role of exponential volume growth and the \(p\)-dependence of the \(L^p\)-spectrum of \(Δ_X\) in producing genuinely non-Euclidean spectral phenomena.

math.AP↗
arXiv2026

$L^p$- Heisenberg--Pauli--Weyl uncertainty inequalities on certain two-step nilpotent Lie groups

This article presents the $L^p$-Heisenberg--Pauli--Weyl uncertainty inequality for the group Fourier transform on a class of two-step nilpotent Lie groups, specifically the Métivier groups. This inequality quantitatively demonstrates that on Métivier groups, a nonzero function and its group Fourier transform cannot both be sharply localized. The proof primarily relies on utilizing the dilation structure inherent to two-step nilpotent Lie groups and estimating the Schatten class norms of the group Fourier transform. The inequality we establish is new, even in the simplest case of Heisenberg groups. Our result significantly sharpens all previously known $L^p$-Heisenberg--Pauli--Weyl uncertainty inequalities for $1 \leq p < 2$ on Métivier groups.

math.FA↗
arXiv2025

On norms on Harish-Chandra modules

The Casselman-Wallach theorem is a foundational result in the theory of representations of real reductive groups connecting algebraic representations to topological representations. We provide a quantitative version of this theorem. For that we introduce the notion of {\it Sobolev gap} for a Harish-Chandra module. This is a new invariant whose finiteness is highly non-trivial. We determine the Sobolev gap for representations in the unitary dual of the group $\SL(2,\R)$ and establish uniform finiteness results in general for representations of the discrete series and the minimal principal series. We use these notions to reformulate and extend classical results of Bernstein and Reznikov concerning automorphic functionals with respect to cocompact lattices. In particular, we prove an abstract convexity bound which applies to automorphic functionals with respect to general lattices in $\SL(2,\R)$ and is independent of the type of unitarizable irreducible Harish-Chandra module. Finally, we offer an extensive list of open problems.

math.RT↗
arXiv2025

Dimension free estimates for the vector-valued Hardy--Littlewood maximal function on the Heisenberg group

In this article, we establish dimension-free Fefferman-Stein inequalities for the Hardy-Littlewood maximal function associated with averages over Korányi balls in the Heisenberg group. We also generalize the result to more general UMD lattices. As a key stepping stone, we establish the $L^p$- boundedness of the vector-valued Nevo-Thangavelu spherical maximal function, which plays a crucial role in our proofs of the main theorems.

math.CA↗
arXiv2023

Weighted estimates for Hardy-Littlewood maximal functions on Harmonic $NA$ groups

Our aim in this article is to study the weighted boundedness of the centered Hardy-Littlewood maximal operator in Harmonic $NA$ groups. Following Ombrosi et al. \cite{ORR}, we define a suitable notion of $A_p$ weights, and for such weights, we prove the weighted $L^p$-boundedness of the maximal operator. Furthermore, as an endpoint case, we prove a variant of the Fefferman-Stein inequality, from which vector-valued maximal inequality has been established. We also provide various examples of weights to substantiate many aspects of our results. In particular, we have shown certain spherical functions of the Harmonic $NA$ group constitute examples of $A_p$ weights. The purely exponential volume growth property of the Harmonic $NA$ group has played a crucial role in our proofs.

math.CA↗
arXiv2022

Strong unique continuation for variable coefficient parabolic operators with Hardy type potential

In this paper, we prove the strong unique continuation property at the origin for solutions of the following scaling critical parabolic differential inequality \[ |\operatorname{div} (A(x,t) \nabla u) - u_t| \leq \frac{M}{|x|^{2}} |u|,\ \ \ \ \] where the coefficient matrix $A$ is Lipschitz continuous in $x$ and $t$. Our main result sharpens a previous one of Vessella concerned with the subcritical case as well as extends a recent result of one of us with Garofalo and Manna for the heat operator.

math.AP↗
arXiv2022

An analogue of Ingham's theorem on the Heisenberg group

We prove an exact analogue of Ingham's uncertainty principle for the group Fourier transform on the Heisenberg group. This is accomplished by explicitly constructing compactly supported functions on the Heisenberg group whose operator-valued Fourier transforms have suitable Ingham type decay and proving an analogue of Chernoff's theorem for the family of special Hermite operators.

math.CA↗
arXiv2021

Analogues of theorems of Chernoff and Ingham on the Heisenberg group

We prove an analogue of Chernoff's theorem for the Laplacian $ Δ_{\mathbb{H}} $ on the Heisenberg group $ \mathbb{H}^n.$ As an application, we prove Ingham type theorems for the group Fourier transform on $ \mathbb{H}^n $ and also for the spectral projections associated to the sublaplacian.

math.CA↗
arXiv2021

Theorems of Chernoff and Ingham for certain eigenfunction expansions

We prove an uncertainty principle for certain eigenfunction expansions on $ L^2(\mathbb{R}^+,w(r)dr) $ and use it to prove analogues of theorems of Chernoff and Ingham for Laplace-Beltrami operators on compact symmetric spaces, special Hermite operator on $ \mathbb{C}^n $ and Hermite operator on $ \mathbb{R}^n.$

math.FA↗
arXiv2021

An uncertainty principle for spectral projections on rank one symmetric spaces of noncompact type

Let $G $ be a noncompact semisimple Lie group with finite centre. Let $X=G/K$ be the associated Riemannian symmetric space and assume that $X$ is of rank one. The spectral projections associated to the Laplace-Beltrami operator are given by $P_λf =f\ast Φ_λ$, where $Φ_λ$ are the elementary spherical functions on $X$. In this paper, we prove an Ingham type uncertainty principle for $P_λf$. Moreover, similar results are obtained in the case of spectral projections associated to Dunkl Laplacian.

math.FA↗
arXiv2021

On a theorem of Chernoff on rank one Riemannian symmetric spaces

In 1975, P.R. Chernoff used iterates of the Laplacian on $\mathbb{R}^n$ to prove an $L^2$ version of the Denjoy-Carleman theorem which provides a sufficient condition for a smooth function on $\mathbb{R}^n$ to be quasi-analytic. In this paper, we prove an exact analogue of Chernoff's theorem for all rank one Riemannian symmetric spaces (of noncompact and compact types) using iterates of the associated Laplace-Beltrami operators.

math.FA↗
arXiv2020

An extension problem, trace Hardy and Hardy's inequalities for Ornstein-Uhlenbeck operator

In this paper, we study an extension problem for the Ornstein-Uhlenbeck operator $L=-Δ+2x\cdot\nabla +n$ and we obtain various characterisations of the solution of the same. We use a particular solution of that extension problem to prove a trace Hardy inequality for $L$ from which Hardy's inequality for fractional powers of $L$ is obtained. We also prove an isometry property of the solution operator associated to the extension problem. Moreover, new $L^p-L^q$ estimates are obtained for the fractional powers of the Hermite operator.

math.AP↗
arXiv2019

On the lacunary spherical maximal function on the Heisenberg group

In this paper we investigate the $L^p$ boundedness of the lacunary maximal function $ M_{\Ha}^{lac} $ associated to the spherical means $ A_r f$ taken over Koranyi spheres on the Heisenberg group. Closely following an approach used by M. Lacey in the Euclidean case, we obtain sparse bounds for these maximal functions leading to new unweighted and weighted estimates. The key ingredients in the proof are the $L^p$ improving property of the operator $A_rf$ and a continuity property of the difference $A_rf-τ_y A_rf$, where $τ_yf(x)=f(xy^{-1})$ is the right translation operator.

math.CA↗
arXiv2004

An algorithm for two-dimensional mesh generation based on the pinwheel tiling

We propose a new two-dimensional meshing algorithm called PINW able to generate meshes that accurately approximate the distance between any two domain points by paths composed only of cell edges. This technique is based on an extension of pinwheel tilings proposed by Radin and Conway. We prove that the algorithm produces triangles of bounded aspect ratio. This kind of mesh would be useful in cohesive interface finite element modeling when the crack propagation pathis an outcome of a simulation process.

cs.CG↗