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Aparajita Dasgupta

Publications and source records attributed to Aparajita Dasgupta.

At least 37 records · Page 2Linked to original sources
arXiv2024

Higher order hypoelliptic damped wave equations on graded Lie groups with data from negative order Sobolev spaces

Let $\mathbb G$ be a graded Lie group with homogeneous dimension $Q$. In this paper, we study the Cauchy problem for a semilinear hypoelliptic damped wave equation involving a positive Rockland operator $\mathcal{R}$ of homogeneous degree $ν\geq 2$ on $\mathbb G$ with power type nonlinearity $|u|^p$ and initial data taken from negative order homogeneous Sobolev space $\dot H^{-γ}(\mathbb G), γ>0$. In the framework of Sobolev spaces of negative order, we prove that $p_{\text{Crit}}(Q, γ, ν) :=1+\frac{2ν}{Q+2γ}$ is the new critical exponent for $γ\in (0, \frac{Q}{2})$. More precisely, we show the global-in-time existence of small data Sobolev solutions of lower regularity for $p>p_{\text{Crit}}(Q, γ, ν) $ in the energy evolution space $ \mathcal{C}\left([0, T], H^{s}(\mathbb{G})\right), s\in (0, 1]$. Under certain conditions on the initial data, we also prove a finite-time blow-up of weak solutions for $1<p<p_{\text{Crit}}(Q, γ, ν)$. Furthermore, to precisely characterize the blow-up time, we derive sharp upper bound and lower bound estimates for the lifespan in the subcritical cases. We emphasize that our results are also new, even in the setting of higher-order differential operators on $\mathbb{R}^n$, and more generally, on stratified Lie groups.

math.AP↗
arXiv2024

Semilinear damped wave equations on the Heisenberg group with initial data from Sobolev spaces of negative order

In this paper, we focus on studying the Cauchy problem for semilinear damped wave equations involving the sub-Laplacian $\mathcal{L}$ on the Heisenberg group $\mathbb{H}^n$ with power type nonlinearity $|u|^p$ and initial data taken from Sobolev spaces of negative order homogeneous Sobolev space $\dot H^{-γ}_{\mathcal{L}}(\mathbb{H}^n), γ>0$, on $\mathbb{H}^n$. In particular, in the framework of Sobolev spaces of negative order, we prove that the critical exponent is the exponent $p_{\text{crit}}(Q, γ)=1+\frac{4}{Q+2γ},$ for some $γ\in (0, \frac{Q}{2})$, where $Q:=2n+2$ is the homogeneous dimension of $\mathbb{H}^n$. More precisely, we establish a global-in-time existence of small data Sobolev solutions of lower regularity for $p>p_{\text{crit}}(Q, γ)$ in the energy evolution space; a finite time blow-up of weak solutions for $1<p<p_{\text{crit}}(Q, γ)$ under certain conditions on the initial data by using the test function method. Furthermore, to precisely characterize the blow-up time, we derive sharp upper bound and lower bound estimates for the lifespan in the subcritical case.

math.AP↗
arXiv2023

Uncertainty principles for the short-time Fourier transform on the lattice

In this paper, we study a few versions of the uncertainty principle for the short-time Fourier transform on the lattice $\mathbb Z^n \times \mathbb T^n$. In particular, we establish the uncertainty principle for orthonormal sequences, Donoho--Stark's uncertainty principle, Benedicks-type uncertainty principle, Heisenberg-type uncertainty principle and local uncertainty inequality for this transform on $\mathbb Z^n \times \mathbb T^n$. Also, we obtain the Heisenberg-type uncertainty inequality using the $k$-entropy of the short-time Fourier transform on $\mathbb Z^n \times \mathbb T^n$.

math.FA↗
arXiv2023

Localization Operators On Discrete Modulation Spaces

In this paper, we study a class of pseudo-differential operators known as time-frequency localization operators on $\mathbb Z^n$, which depend on a symbol $ς$ and two windows functions $g_1$ and $g_2$. We define the short-time Fourier transform on $ \mathbb Z^n \times \mathbb T^n $ and modulation spaces on $\mathbb Z^n$, and present some basic properties. Then, we use modulation spaces on $\mathbb Z^n \times \mathbb T^n$ as appropriate classes for symbols, and study the boundedness and compactness of the localization operators on modulation spaces on $\mathbb Z^n$. Then, we show that these operators are in the Schatten--von Neumann class. Also, we obtain the relation between the Landau--Pollak--Slepian type operator and the localization operator on $\mathbb Z^n$. Finally, under suitable conditions on the symbols, we prove that the localization operators are paracommutators, paraproducts and Fourier multipliers.

math.FA↗
arXiv2023

Non-harmonic $M$-elliptic pseudo differential operators on manifolds

In this article, we introduce and study $M$-elliptic pseudo-differential operators in the framework of non-harmonic analysis of boundary value problems on a manifold $Ω$ with boundary $\partial Ω$, introduced by Ruzhansky and Tokmagambetov ( Int. Math. Res. Not. IMRN, (12), 3548-3615, 2016) in terms of a model operator $\mathfrak{L}$. More precisely, we consider a weighted $\mathfrak{L}$-symbol class $M_{ρ, 0, Λ}^{m}, m\in \mathbb{R},$ associated to a suitable weight function $Λ$ on a countable set $\mathcal{I} $ and study elements of the symbolic calculus for pseudo-differential operators associated with $\mathfrak{L}$-symbol class $M_{ρ, 0, Λ}^{m},$ by deriving formulae for the composition, adjoint, and transpose. Using the notion of $M$-ellipticity for symbols belonging to $\mathfrak{L}$-symbol class $M_{ρ, 0, Λ}^{m}$, we construct the parametrix of $M$-elliptic pseudo-differential operators. Further, we investigate the minimal and maximal extensions for $M$-elliptic pseudo-differential operators and show that they coincide when the symbol $σ\in M_{ρ, 0, Λ}^{m}, $ is $M$-elliptic. We provide a necessary and sufficient condition to ensure that the pseudo-differential operators $T_σ$ with symbol in the $\mathfrak{L}$-symbol class $M_{ρ, 0,Λ}^{0} $ is a compact operator in $L^{2}(Ω)$ or a Riesz operator in $L^{p}(Ω).$ Finally, we prove Gärding's inequality for pseudo-differential operators associated with symbol from $M_{ρ, 0,Λ}^{0} $ in the setting of non-harmonic analysis.

math.FA↗
arXiv2023

Discrete time-dependent wave equation for the Schrödinger operator with unbounded potential

In this article, we investigate the semiclassical version of the wave equation for the discrete Schrödinger operator, $\mathcal{H}_{\hbar,V}:=-\hbar^{-2}\mathcal{L}_{\hbar}+V$ on the lattice $\hbar\mathbb{Z}^{n},$ where $\mathcal{L}_{\hbar}$ is the discrete Laplacian, and $V$ is a non-negative multiplication operator. We prove that $\mathcal{H}_{\hbar,V}$ has a purely discrete spectrum when the potential $V$ satisfies the condition $|V(k)|\to \infty$ as $|k|\to\infty$. We also show that the Cauchy problem with regular coefficients is well-posed in the associated Sobolev type spaces and very weakly well-posed for distributional coefficients. Finally, we recover the classical solution as well as the very weak solution in certain Sobolev type spaces as the limit of the semiclassical parameter $\hbar\to 0$.

math.AP↗
arXiv2023

Discrete Heat Equation with irregular thermal conductivity and tempered distributional data

In this paper, we consider a semi-classical version of the nonhomogeneous heat equation with singular time-dependent coefficients on the lattice $\hbar \mathbb{Z}^n$. We establish the well-posedeness of such Cauchy equations in the classical sense when regular coefficients are considered, and analyse how the notion of very weak solution adapts in such equations when distributional coefficients are regarded. We prove the well-posedness of both the classical and the very weak solution in the weighted spaces $\ell^{2}_{s}(\hbar \mathbb{Z}^n)$, $s \in \mathbb{R}$, which is enough to prove the well-posedness in the space of tempered distributions $\mathcal{S}'(\hbar \mathbb{Z}^n)$. Notably, when $s=0$, we show that for $\hbar \rightarrow 0$, the classical (resp. very weak) solution of the heat equation in the Euclidean setting $\mathbb{R}^n$ is recaptured by the classical (resp. very weak) solution of it in the semi-classical setting $\hbar \mathbb{Z}^n$.

math.AP↗
arXiv2023

Two wavelet multipliers and Landau-Pollak-Slepian operators on locally compact abelian groups associated to right-$H$-translation-invariant functions

By using a coset of closed subgroup, we define a Fourier like transform for locally compact abelian (LCA) topological groups. We studied two wavelet multipliers and Landau-Pollak-Slepian operators on locally compact abelian topological groups associated to the transform and show that the transforms are $L^p$bounded linear operators, and are in Schatten p-class for $1\leq p\leq \infty$. Finally, we determine their trace class and also obtain a connection with the generalized Landau-Pollak-Slepian operators.

math.FA↗
arXiv2023

Multilinear Fourier Integral operators on modulation spaces

In this article, we study properties of multilinear Fourier integral operators on weighted modulation spaces. In particular, using the theory of Gabor frames, we study boundedness of multilinear Fourier integral operators on products of weighted modulation spaces. Further, we investigate the periodic multilinear Fourier integral operator. Finally, we study continuity of bilinear pseudo-differential operators on modulation spaces for certain symbol classes, namely $\textbf{SG}$-class.

math.FA↗
arXiv2022

Localization Operator and Weyl Transform on Reduced Heisenberg Group with Multi-dimensional Center

In this article, we study two different types of operators, the localization operator and Weyl transform, on the reduced Heisenberg group with multidimensional center $\mathcal{G}$. The group $\mathcal{G}$ is a quotient group of non-isotropic Heisenberg group with multidimensional center $\mathcal{H}^m$ by its center subgroup. Firstly, we define the localization operator using a wavelet transform on $\mathcal{G}$ and obtain the product formula for the localization operators. Next, we define the Weyl transform associated to the Wigner transform on $\mathcal{G}$ with the operator-valued symbol. Finally, we have shown that the Weyl transform is not only a bounded operator but also a compact operator when the operator-valued symbol is in $L^p,1\leq p\leq 2,$ and it is an unbounded operator when $p>2$.

math.FA↗
arXiv2022

Nonlinear fractional damped wave equation on compact Lie groups

In this paper, we deal with the initial value fractional damped wave equation on $G$, a compact Lie group, with power-type nonlinearity. The aim of this manuscript is twofold. First, using the Fourier analysis on compact Lie groups, we prove a local in-time existence result in the energy space for the fractional damped wave equation on $G$. Moreover, a finite time blow-up result is established under certain conditions on the initial data. In the next part of the paper, we consider fractional wave equation with lower order terms, that is, damping and mass with the same power type nonlinearity on compact Lie groups, and prove the global in-time existence of small data solutions in the energy evolution space.

math.AP↗
arXiv2022

Weighted periodic and discrete Pseudo-Differential Operators

In this paper, we study elements of symbolic calculus for pseudo-differential operators associated with the weighted symbol class $M_{ρ, Λ}^m(\mathbb{ T}\times \mathbb{Z})$ (associated to a suitable weight function $Λ$ on $\mathbb{Z}$) by deriving formulae for the asymptotic sums, composition, adjoint, transpose. We also construct the parametrix of $M$-elliptic pseudo-differential operators on $\mathbb{ T}$. Further, we prove a version of Gohberg's lemma for pseudo-differetial operators with weighted symbol class $M_{ρ, Λ}^0(\mathbb{ T}\times \mathbb{Z})$ and as an application, we provide a sufficient and necessary condition to ensure that the corresponding pseudo-differential operator is compact on $L^2(\mathbb{T})$. Finally, we provide Gårding's and Sharp Gårding's inequality for $M$-elliptic operators on $\mathbb{Z}$ and $\mathbb{T}$, respectively, and present an application in the context of strong solution of the pseudo-differential equation $T_σ u=f$ in $L^{2}\left(\mathbb{T}\right)$.

math.FA↗
arXiv2022

Pseudo-Differential Operators, Wigner Transform, and Weyl Transform on the Affine Poincaré Group

In this paper, we study harmonic analysis on the affine Poincaré group $\mathcal{P}_{aff}$, which is a non-unimodular group, and obtain pseudo-differential operators with operator valued symbols. More precisely, we study the boundedness properties of pseudo-differential operators on $\mathcal{P}_{aff}$. We also provide a necessary and sufficient condition on the operator-valued symbols such that the corresponding pseudo-differential operators are in the class of Hilbert--Schmidt operators. Consequently, we obtain a characterization of the trace class pseudo-differential operators on the Poincaré affine group $\mathcal{P}_{aff}$, and provide a trace formula for these trace class operators. Finally, we study the Wigner transform, and Weyl transform associated with the operator valued symbol on the Poincaré affine group $\mathcal{P}_{aff}$.

math.FA↗
arXiv2022

Nonlinear fractional wave equation on compact Lie groups

Let $G$ be a compact Lie group. In this article, we consider the initial value fractional wave equation with power-type nonlinearity on $G$. Mainly, we investigate some $L^{2}-L^{2}$ estimates of the solutions to the homogeneous fractional wave equation on $G$ with the help of the group Fourier transform on $G$. Further, using the Fourier analysis on compact Lie groups, we prove a local in-time existence result in the energy space. Moreover, under certain conditions on the initial data, a finite time blow-up result is established. We also derive a sharp lifespan for local (in-time) solutions. Finally, we consider the space-fractional wave equation with a regular mass term depending on the position and study the well-posedness of the fractional Klein-Gordon equation on compact Lie groups.

math.AP↗
arXiv2022

Discrete Time-Dependent Wave Equations II. Semiclassical Fractional Klein-Gordon Equation

In this paper, we consider a semiclassical version of the fractional Klein-Gordon equation on the lattice, $h{\mathbb{Z}}^n.$ Contrary to the Euclidean case that was considered in [2], the discrete fractional Klein-Gordon equation is well-posed in $\ell^2(h{\mathbb{Z}}^n).$ However, we also recover the well-posedness results in the certain Sobolev spaces in the limit of the semiclassical parameter $h\rightarrow 0.$

math.AP↗
arXiv2021

Discrete time-dependent wave equations I. Semiclassical analysis

In this paper we consider a semiclassical version of the wave equations with singular Hölder time-dependent propagation speeds on the lattice $\hbar\mathbb{Z}^{n}$. We allow the propagation speed to vanish leading to the weakly hyperbolic nature of the equations. Curiously, very much contrary to the Euclidean case considered by Colombini, de Giorgi and Spagnolo [2] and by other authors, the Cauchy problem, in this case, is well-posed in $\ell^2(\hbar\mathbb{Z}^{n})$. However, we also recover the well-posedness results in the intersection of certain Gevrey and Sobolev spaces in the limit of the semiclassical parameter $\hbar\to 0$.

math.AP↗
arXiv2019

Ellipticity and Fredholmness of pseudo-differential operators on $\ell^2(\Zn)$

The minimal operator and the maximal operator of an elliptic pseudo-differential operator with symbols on $\Z^n\times \mathbb{T}^n$ are proved to coincide and the domain is given in terms of a Sobolev space. Ellipticity and Fredholmness are proved to be equivalent for pseudo-differential operators on $\Z^n$. The index of an elliptic pseudo-differential operator on $\Z^n$ is also computed.

math.SP↗