Pritam Ganguly

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• Phd
• Postdoc at Paderborn University

In this article, we establish dimension-free Fefferman-Stein inequalities for the Hardy-Littlewood maximal function associated with averages over Kor\'anyi 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 spher...

This article presents the $L^p$-Heisenberg-Pauli-Weyl uncertainty inequality for the group Fourier transform on a broad class of two-step nilpotent Lie groups, specifically the two-step MW groups. This inequality quantitatively demonstrates that on two-step MW groups, a nonzero function and its group Fourier transform cannot both be sharply localiz...

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...

We prove an analogue of Chernoff’s theorem for the Laplacian Δℍ on the Heisenberg group ℍn. As an application, we prove Ingham type theorems for the group Fourier transform on ℍn and also for the spectral projections associated to the sublaplacian.

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 fam...

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 prev...

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 fam...

In 1975, P.R. Chernoff used iterates of the Laplacian on Rn to prove an L2 version of the Denjoy-Carleman theorem which provides a sufficient condition for a smooth function on Rn 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 type using iterates of the...

We prove an uncertainty principle for certain eigenfunction expansions on L2(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 Cn and Hermite operator on Rn.

We prove an analogue of Chernoff's theorem for the Laplacian $ \Delta_{\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.

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 generalized spectral projections associated to the Laplace-Beltrami operator are given by \(P_{\lambda }f =f*\Phi _{\lambda }\), where \(\Phi _{\lambda }\) are the elementary spherical funct...

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 noncompa...

In this paper, we study an extension problem for the Ornstein-Uhlenbeck operator $L=-\Delta+2x\cdot\nabla +n$ 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...

In this paper we investigate the $L^p$ boundedness of the lacunary maximal function $ M_{\mathbb{H}^n}^{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 unweigh...

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.

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 $P_{\lambda}f =f\ast \Phi_{\lambda}$, where $\Phi_{\lambda}$ are the elementary spherical functions on $X$. I...

We prove an analogue of Chernoff's theorem for the sublaplacian on the Heisenberg group and use it prove a version of Ingham's theorem for the Fourier transform on the same 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 w...