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Sundaram Thangavelu

Publications and source records attributed to Sundaram Thangavelu.

At least 19 recordsLinked to original sources
arXiv2026

Weighted Norm Inequalities for the Strichartz Fourier transform on the Heisenberg Group

In this article, we establish an analogue of Pitt's inequality for the Strichartz Fourier transform on the Heisenberg group $\mathbb{H}^n$. By exploiting the scalar-valued formulation of the transform and the framework of decreasing rearrangements, we derive weighted $L^p$-$L^q$ estimates of Pitt type. In particular, we obtain sufficient conditions for the validity of such inequalities via weighted Hardy inequalities and Calderón's interpolation method, and we also prove necessary conditions in the case of radial weights, using structural properties of Laguerre functions and zeros of Bessel function. As an application, we deduce an uncertainty principle of Heisenberg-Pauli-Weyl type in this setting and establish a Paley inequality for the Strichartz Fourier transform. We also derive Pitt's inequality using Hardy's inequality for the case $p=q=2$. These results extend the classical Euclidean theory of Pitt's inequality to the non-commutative, nilpotent setting of $\mathbb{H}^n$ for the sub-Laplacian and conformal Laplacian. Here we highlight the role of Laguerre functions in harmonic analysis on the Heisenberg group.

math.FA↗
arXiv2026

Quantum Hermite functions and Fourier Transform of operators

We construct operator analogues of Hermite functions which form an orthonormal basis for the Hilbert space $ \mathcal{S}_2$ of Hilbert-Schmidt operators on $ L^2(\R^n).$ We use this orthonormal basis to define Fourier transform on $ \mathcal{S}_2 $ and study some of its basic properties.

math.FA↗
arXiv2025

Some Versions of Beurling's Theorem on H-type Groups

We prove an analogue of Beurling's theorem on the H-type groups of certain dimensions after establishing the Gutzmer's formula for the H-type groups. We also obtain some other versions of the theorem using the modified Radon transform.

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↗
arXiv2025

Modulation spaces on the Heisenberg group

In this article we show how certain irreducible unitary representation $ Π_λ$ of the twisted Heisenberg group $ \He_λ^n(\C)$ leads to the twisted modulation spaces $ M_λ^{p,q}(\R^{2n}).$ These $ Π_λ$ also turn out to be irreducible unitary representations of another nilpotent Lie group $ G_n $ which contains two copies of the Heisenberg group $ \He^n.$ By lifting $ Π_λ$ we obtain another unitary representation $ Π$ of $ G_n $ acting on $ L^2(\He^n).$ We define our modulation spaces $ M^{p,q}(\He^n) $ in terms of the matrix coefficients associated to $ Π.$ These spaces are shown to be invariant under Heisenberg translations and Heisenberg modulations which are different from euclidean modulations. We also establish some of the basic properties of $ M_λ^{p,q}(\R^{2n})$ and $ M^{p,q}(\He^n) $ such as completeness and invariance under suitable Fourier transforms.

math.FA↗
arXiv2024

Poisson transform and unipotent complex geometry

Our concern is with Riemannian symmetric spaces $Z=G/K$ of the non-compact type and more precisely with the Poisson transform $\mathcal{P}_λ$ which maps generalized functions on the boundary $\partial Z$ to $λ$-eigenfunctions on $Z$. Special emphasis is given to a maximal unipotent group $N 0})^r<G$ (Iwasawa) and letting the level $a\in A$ tend to $0$ on a ray we retrieve $N$ via $\lim_{a\to 0} Na$ as an open dense orbit in $\partial Z$ (Bruhat). For positive parameters $λ$ the Poisson transform $\mathcal{P}_λ$ is defined an injective for functions $f\in L^2(N)$ and we give a novel characterization of $\mathcal{P}_λ(L^2(N))$ in terms of complex analysis. For that we view eigenfunctions $ϕ= \mathcal{P}_λ(f)$ as families $(ϕ_a)_{a\in A}$ of functions on the $N$-orbits, i.e. $ϕ_a(n)= ϕ(na)$ for $n\in N$. The general theory then tells us that there is a tube domain $\mathcal{T}=N\exp(iΛ)\subset N_\mathbb{C}$ such that each $ϕ_a$ extends to a holomorphic function on the scaled tube $\mathcal{T}_a=N\exp(i\operatorname{Ad}(a)Λ)$. We define a class of $N$-invariant weight functions ${\bf w}_λ$ on the tube $\mathcal{T}$, rescale them for every $a\in A$ to a weight ${\bf w}_{λ, a}$ on $\mathcal{T}_a$, and show that each $ϕ_a$ lies in the $L^2$-weighted Bergman space $\mathcal{B}(\mathcal{T}_a, {\bf w}_{λ, a}):=\mathcal{O}(\mathcal{T}_a)\cap L^2(\mathcal{T}_a, {\bf w}_{λ, a})$. The main result of the article then describes $\mathcal{P}_λ(L^2(N))$ as those eigenfunctions $ϕ$ for which $ϕ_a\in \mathcal{B}(\mathcal{T}_a, {\bf w}_{λ, a})$ and $$\|ϕ\|:=\sup_{a\in A} a^{\operatorname{Re}λ-2ρ} \|ϕ_a\|_{\mathcal{B}_{a,λ}}<\infty$$ holds.

math.RT↗
arXiv2023

Algebras of entire functions and representations of the twisted Heisenberg group

On the twisted Fock spaces $ \mathcal{F}^λ(\C^{2n}) $ we consider a family of unitary operators $ρ_λ(a,b) $ indexed by $ (a,b) \in \C^n \times \C^n.$ The composition formula for $ ρ_λ(a,b) \circ ρ_λ(a^\prime,b^\prime) $ leads us to a group $ \He^n_λ(\C) $ which contains two copies of the Heisenberg group $ \He^n.$ The operators $ ρ_λ(a,b) $ lift to $ \He_λ^n(\C) $ providing an irreducible unitary representation. However, its restriction to $ \He^n_λ(\R) $ is not irreducible.

math.FA↗
arXiv2023

Boundedness of certain linear operators on twisted Fock spaces

On the twisted Fock spaces $ \mathcal{F}^λ(\mathbb{C}^{2n}) $ we consider two types of convolution operators $ S_φ$ and $ \widetilde{S}_φ$ associated to an element $ φ\in \mathcal{F}^λ(\mathbb{C}^{2n}) .$ We find a necessary and sufficient condition on $ φ$ so that $ S_φ$ (resp. $ \widetilde{S}_φ$ ) is bounded on $ \mathcal{F}^λ(\mathbb{C}^{2n}).$ We show that for any given non constant $ φ$ at least one of these two operators is unbounded.

math.FA↗
arXiv2023

Fourier multipliers and pseudo-differential operators on Fock-Sobolev spaces

Any bounded linear operator $ T $ on $ L^2(\mathbb{R}^n) $ gives rise to the operator $ S= B \circ T \circ B^\ast $ on the Fock space $ \mathcal{F}(\C^n) $ where $ B $ is the Bargmann transform. In this article we identify those $ S $ which correspond to Fourier multipliers and pseudo-differential operators on $ L^2(\mathbb{R}^n)$ and study their boundedness on the Fock-Sobolev spaces $ \mathcal{F}^{s,2}(\C^n).$

math.FA↗
arXiv2022

On heat equations associated with fractional harmonic oscillators

We establish some fixed-time decay estimates in Lebesgue spaces for the fractional heat propagator $e^{-tH^β}$, $t, β>0$, associated with the harmonic oscillator $H=-Δ+ |x|^2$. We then prove some local and global wellposedness results for nonlinear fractional heat equations.

math.AP↗
arXiv2022

A scalar valued Fourier transform for the Heisenberg group

We define a scalar valued Fourier transform for functions on the Heisenberg group and establish some of its basic properties like inversion formula, Plancherel theorem and Riemann-Lebesgue lemma. We also restate certain well known theorems for the group Fourier transform in terms of the new transform which we would like to call Strichartz Fourier transform.

math.FA↗
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

Beurling's theorem on the Heisenberg group

We formulate and prove an analogue of Beurling's theorem for the Fourier transform on the Heisenberg group. As a consequence we deduce Hardy and Cowling-Price theorems.

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↗