arXiv subjects

Mithun Bhowmik

Publications and source records attributed to Mithun Bhowmik.

13 recordsLinked to original sources

arXiv2026

Uniqueness results for quasi-analytic functions on compact Lie groups and homogeneous spaces

In this article, we establish a quantitative uniqueness theorem for quasi-analytic functions defined on compact, connected Lie groups $G$ and on homogeneous spaces $G/H$, where $H$ is any closed subgroup of $G$. Our result extends classical Logvinenko-Sereda-type theorems to the setting of quasi-analytic functions on compact Lie groups and their homogeneous spaces. We introduce the quasi-analytic class of functions using iterates of the Casimir operator on $G$. This construction is justified by establishing that every function in this class possesses the strong unique continuation property. In particular, our result extends a result of P. Chernoff (Bull. Amer. Math. Soc., 1975) to the framework of compact Lie groups and their homogeneous spaces.

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arXiv2026

Quantitative uniqueness properties for functions on compact quasi-analytic manifolds

In this article, we establish quantitative uniqueness results for a class of functions defined on a quasi-analytic compact manifold $X$ without boundary. This function class is characterized by the iterates of a positive elliptic linear differential operator on $X$ and, notably, encompasses all functions with a finite spectrum. By employing a relatively dense observable set, we extend classical Logvinenko-Sereda-type results to this quasi-analytic framework. Furthermore, we demonstrate that if these functions satisfy a doubling property, observability holds from any measurable set of positive measure. Our results generalise the propagation of smallness from finite sums of eigenfunctions to infinite sums with an appropriate energy-parameter decay, thereby extending recent findings by Kukavica-Li (Proc. Lond. Math. Soc., 2025) to the quasi-analytic setting.

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arXiv2023

Spectral projections and resolvent estimates on Damek-Ricci spaces and their applications

We prove $L^p-L^{p^\prime}$ boundedness of spectral projections and the resolvent of the Laplace-Beltrami operator on Damek-Ricci spaces with the explicit norms in terms of the spectral parameter. To prove these results we established pointwise sharp bounds on the spherical functions and their derivatives. As an application, we study the eigenvalue bounds of Schrödinger operators with complex valued potential.

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arXiv2021

Sharp Adams type inequalities for the fractional Laplace-Beltrami operator on noncompact symmetric spaces

We establish sharp Adams type inequalities on Sobolev spaces $W^{α, n/α}(X)$ of any fractional order $α< n$ on Riemannian symmetric space $X$ of noncompact type with dimension $n$ and of arbitrary rank. We also establish sharp Hardy-Adams inequalities on the Sobolev spaces $W^{n/2, 2}(X)$. For the real hyperbolic spaces, such results were recently obtained by J. Li et al. (Trans. AMS, 2020). We use Fourier analysis on the symmetric spaces to obtain these results.

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arXiv2021

A theorem of Chernoff on quasi-analytic functions for Riemannian symmetric spaces

An $L^2$ version of the classical Denjoy-Carleman theorem regarding quasi-analytic functions was proved by P. Chernoff on $\mathbb R^n$ using iterates of the Laplacian. We give a simple proof of this theorem which generalizes the result on $\mathbb R^n$ for any $p\in [1, 2]$. We then extend this result to Riemannian symmetric spaces of compact and noncompact type for $K$-biinvariant functions.

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arXiv2021

An extension problem and Hardy's inequality for the fractional Laplace-Beltrami operator on Riemannian symmetric spaces of noncompact type

In this paper we study an extension problem for the Laplace-Beltrami operator on Riemannian symmetric spaces of noncompact type and use the solution to prove Hardy-type inequalities for fractional powers of the Laplace-Beltrami operator. Next, we study the mapping properties of the extension operator. In the last part we prove Poincaré-Sobolev inequalities on these spaces.

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arXiv2019

A theorem of Levinson for Riemannian symmetric spaces of noncompact type

A classical result of N. Levinson characterizes the existence of a nonzero integrable function vanishing on a nonempty open subset of the real line in terms of the pointwise decay of its Fourier transform. We prove an analogue of this result for Riemannian symmetric spaces of noncompact type.

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arXiv2019

A local Levinson theorem for compact symmetric spaces

A classical result due to Levinson characterizes the existence of non-zero functions defined on a circle vanishing on an open subset of the circle in terms of the pointwise decay of their Fourier coefficients [13]. We prove certain analogue of this result on compact symmetric spaces.

math.CA↗

arXiv2019

Analogs of certain quasi-analiticity results on Riemannian symmetric spaces of noncompact type

An $L^2$ version of the celebrated Denjoy-Carleman theorem regarding quasi-analytic functions was proved by Chernoff \cite{CR} on $\mathbb R^d$ using iterates of the Laplacian. In $1934$ Ingham \cite{I} used the classical Denjoy-Carleman theorem to relate the decay of Fourier transform and quasi-analyticity of integrable functions on $\mathbb R$. In this paper we extend both these theorems to Riemannian symmetric spaces of noncompact type and show that the theorem of Ingham follows from that of Chernoff.

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arXiv2018

Improved Ingham-type result on $\mathbb R^d$ and on connected, simply connected nilpotent Lie Groups

In \cite{BRS} we have characterized the existance of a non zero function vanishing on an open set in terms of the decay of it's Fourier transform on the $d$-dimensional Euclidean space, the $d$-dimensional torus and on connected, simply connected two step nilpotent Lie groups. In this paper we improved these results on $\mathbb R^d$ and prove analogus results on connected, simply connected nilpotent Lie groups.

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arXiv2016

Uncertainty Principles of Ingham and Paley-Wiener on Semisimple Lie Groups

Classical results due to Ingham and Paley-Wiener characterize the existence of nonzero functions supported on certain subsets of the real line in terms of the pointwise decay of the Fourier transforms. Viewing these results as uncertainty principles for Fourier transforms, we prove certain analogues of these results on connected, noncompact, semisimple Lie groups with finite center. We also use these results to show unique continuation property of solutions to the initial value problem for time-dependent Schrödinger equations on Riemmanian symmetric spaces of noncompact type.

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arXiv2016

An Uncertainty Principle of Paley and Wiener on Euclidean Motion Group

A classical result due to Paley and Wiener characterizes the existence of a non-zero function in $L^2(\mathbb{R})$, supported on a half line, in terms of the decay of its Fourier transform. In this paper we prove an analogue of this result for compactly supported continuous functions on the Euclidean motion group $M(n)$. We also relate this result to a uniqueness property of solutions to the initial value problem for time-dependent Schrödinger equation on $M(n)$.

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arXiv2016

Around Uncertainty Principles of Ingham-type on $\R^n$, $\T^n$ and Two Step Nilpotent Lie Groups

Classical results due to Ingham and Paley-Wiener characterize the existence of nonzero functions supported on certain subsets of the real line in terms of the pointwise decay of the Fourier transforms. We view these results as uncertainty principles for Fourier transforms. We prove certain analogues of these uncertainty principles on the $n$-dimensional Euclidean space, the $n$-dimensional torus and connected, simply connected two step nilpotent Lie groups. We also use these results to show a unique continuation property of solutions to the initial value problem for time-dependent Schrödinger equations on the Euclidean space and a class of connected, simply connected two step nilpotent Lie groups.

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