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Ramesh Manna

Publications and source records attributed to Ramesh Manna.

At least 19 recordsLinked to original sources
arXiv2026

Hermite expansions of functions from the weighted Hardy class

In this paper, we analyze a function space consisting of functions for which both the function and its Fourier transform exhibit Gaussian decay together with exponential growth governed by suitable weight functions. First, we examine logarithmic-type weights, in which case these function spaces are equivalent to Pilipovi\'c spaces. In this setting, we establish a decay estimate for the Hermite coefficients of functions. Furthermore, by combining these estimates with the asymptotic behavior of Hermite functions, we prove a decay rate for solutions to the harmonic oscillator Schr\"odinger equation. Second, we consider a class of weights and prove the exponential decay of the Hermite projection operators on these spaces by analyzing Laguerre expansions and the short-time Fourier transform. Additionally, we revisit the subcritical Hardy uncertainty principle and obtain a partial improvement toward a conjecture posed by Vemuri.

math.CA↗
arXiv2025

Heat equations associated to harmonic oscillator with exponential nonlinearity

We investigate the Cauchy problem for a heat equation involving a fractional harmonic oscillator and an exponential nonlinearity. We establish local well-posedness within the appropriate Orlicz spaces. Through the examination of small initial data in suitable Orlicz spaces, we obtain the existence of global weak-mild solutions. Additionally, precise decay estimates are presented for large time, indicating that the decay rate is influenced by the nonlinearity's behavior near the origin. Moreover, we highlight that the existence of local nonnegative classical solutions is no longer guaranteed when certain nonnegative initial data is considered within the appropriate Orlicz space.

math.AP↗
arXiv2023

$L^p-L^q$ estimates for maximal operators associated to families of finite type curves

We study the boundedness problem for maximal operators $\mathbb{M}$ associated to averages along families of finite type curves in the plane, defined by $$\mathbb{M}f(x) \, := \, \sup_{1 \leq t \leq 2} \left|\int_{\mathbb{C}} f(x-ty) \, ρ(y) \, dσ(y)\right|,$$ where $dσ$ denotes the normalised Lebesgue measure over the curves $\mathbb{C}$. Let $\triangle$ be the closed triangle with vertices $P=(\frac{2}{5}, \frac{1}{5}), ~ Q=(\frac{1}{2}, \frac{1}{2}), ~ R=(0, 0).$ In this paper, we prove that for $(\frac{1}{p}, \frac{1}{q}) \in (\triangle \setminus \{P, Q\}) \cap \left\{(\frac{1}{p}, \frac{1}{q}) :q > m \right\}$, there is a constant $B$ such that $\|\mathbb{M}f\|_{L^q(\mathbb{R}^2)} \leq \, B \, \|f\|_{L^p(\mathbb{R}^2)}$. Furthermore, if $m <5,$ then we have $\|\mathbb{M}f\|_{L^{5, \infty}(\mathbb{R}^2)} \leq B \|f\|_{L^{\frac{5}{2} ,1} (\mathbb{R}^2)}.$ We shall also consider a variable coefficient version of maximal theorem and we obtain the $L^p-L^q$ boundedness result for $ (\frac{1}{p}, \frac{1}{q}) \in \triangle^{\circ} \cap \left\{(\frac{1}{p}, \frac{1}{q}) :q > m \right\},$ where $\triangle^{\circ}$ is the interior of the triangle with vertices $(0,0), ~(\frac{1}{2}, \frac{1}{2}), ~(\frac{2}{5}, \frac{1}{5}).$ An application is given to obtain $L^p-L^q$ estimates for solution to higher order, strictly hyperbolic pseudo-differential operators.

math.CA↗
arXiv2023

Multi-dimensional Bohr radii of Banach space valued holomorphic functions

In this article, we study the multi-dimensional Bohr radii of holomorphic functions defined on the Banach sequence spaces with values in the Banach spaces. For the case of finite dimensional Banach spaces, we exhibit the exact asymptotic growth of the Bohr radius. To achieve our goal in the finite case, we use $\ell_{p'}$-summability of certain coefficients of a given polynomial in terms of its uniform norm on $\ell_p^n$. The infinite case is handled using the techniques developed in recent years from the work of Defant, Maestre and Schwarting. We crucially use several properties of the symmetric $M$-linear mapping associated with a homogeneous polynomial of degree $M$ in our analysis. Furthermore, we study the bounds of the arithmetic Bohr radius of Banach space-valued holomorphic functions defined on the Banach sequence spaces, which generalises the work of Defant, Maestre, and C. Prengel in this direction.

math.FA↗
arXiv2023

Fractional Fourier transforms, harmonic oscillator propagators and Strichartz estimates on Pilipovic and modulation spaces

We show that harmonic oscillator propagators and fractional Fourier transforms are essentially the same. We deduce continuity properties and fix time estimates for such operators on modulation spaces, and apply the results to prove Strichartz estimates for the harmonic oscillator propagator when acting on modulation spaces. Especially we extend some results by Balhara, Cordero, Nicola, Rodino and Thangavelu. We also show that general forms of fractional harmonic oscillator propagators are continuous on suitable Pilipovic spaces.

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↗
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↗
arXiv2020

Carleman estimates for a class of variable coefficient degenerate elliptic operators with applications to unique continuation

In this paper, we obtain new Carleman estimates for a class of variable coefficient degenerate elliptic operators whose constant coefficient model at one point is the so called Baouendi-Grushin operator. This generalizes the results obtained by the two of us with Garofalo in [9] where similar estimates were established for the "constant coefficient" Baouendi-Grushin operator. Consequently, we obtain: (i) a Bourgain-Kenig type quantitative uniqueness result in the variable coefficient setting; (ii) and a strong unique continuation property for a class of degenerate sublinear equations. We also derive a subelliptic version of a scaling critical Carleman estimate proven by Regbaoui in the Euclidean setting using which we deduce a new unique continuation result in the case of scaling critical Hardy type potentials.

math.AP↗
arXiv2020

Phase space analysis of the Hermite semigroup and applications to nonlinear global well-posedness

We study the Hermite operator $H=-Δ+|x|^2$ in $\mathbb{R}^d$ and its fractional powers $H^β$, $β>0$ in phase space. Namely, we represent functions $f$ via the so-called short-time Fourier, alias Fourier-Wigner or Bargmann transform $V_g f$ ($g$ being a fixed window function), and we measure their regularity and decay by means of mixed Lebesgue norms in phase space of $V_g f$, that is in terms of membership to modulation spaces $M^{p,q}$, $0< p,q\leq \infty$. We prove the complete range of fixed-time estimates for the semigroup $e^{-tH^β}$ when acting on $M^{p,q}$, for every $0< p,q\leq \infty$, exhibiting the optimal global-in-time decay as well as phase-space smoothing. As an application, we establish global well-posedness for the nonlinear heat equation for $H^β$ with power-type nonlinearity (focusing or defocusing), with small initial data in modulation spaces or in Wiener amalgam spaces. We show that such a global solution exhibits the same optimal decay $e^{-c t}$ as the solution of the corresponding linear equation, where $c=d^β$ is the bottom of the spectrum of $H^β$. This is in sharp contrast to what happens for the nonlinear focusing heat equation without potential, where blow-up in finite time always occurs for (even small) constant initial data - hence in $M^{\infty,1}$.

math.FA↗
arXiv2020

Translation and modulation invariant Hilbert spaces

We show that for any Hilbert space of distributions on $\textbf{R}^d$ which is translation and modulation invariant, is equal to $L^2(\textbf{R}^d)$, with the same norm apart from a multiplicative constant.

math.FA↗
arXiv2019

Borderline gradient estimates at the boundary in Carnot groups

In this article, we prove the continuity of the horizontal gradient near a $C^{1,\text{Dini}}$ non-characteristic portion of the boundary for solutions to $Γ^{0, \text{Dini}}$ perturbations of horizontal Laplaceans as in (1.1) below where the scalar term is in scaling critical Lorentz space $L(Q,1)$ with $Q$ being the homogeneous dimension of the group. This result can be thought of both as a sharpening of the $Γ^{1, α}$ boundary regularity result in [4] as well as a subelliptic analogue of the main result in [1] restricted to linear equations.

math.AP↗
arXiv2019

Momentum Ray Transforms, II: Range Characterization In the Schwartz space

The momentum ray transform $I^k$ integrates a rank $m$ symmetric tensor field $f$ over lines of ${\R}^n$ with the weight $t^k$: $ (I^k\!f)(x,ξ)=\int_{-\infty}^\infty t^kłf(x+tξ),ξ^m\r\,dt. $ We give the range characterization for the operator $f\mapsto(I^0\!f,I^1\!f,\dots, I^m\!f)$ on the Schwartz space of rank $m$ smooth fast decaying tensor fields. In dimensions $n\ge3$, the range is characterized by certain differential equations of order $2(m+1)$ which generalize the classical John equations. In the two-dimensional case, the range is characterized by certain integral conditions which generalize the classical Gelfand -- Helgason -- Ludwig conditions.

math.AP↗
arXiv2018

Space like strong unique continuation for sublinear parabolic equations

In this paper, we establish space like strong unique continuation property (sucp) for uniformly parabolic sublinear equations under appropriate structural assumptions. Our main result Theorem 1.1 constitutes the parabolic counterpart of the strong unique continuation result recently established by Ruland in [Ru] for analogous elliptic sublinear equations. Similar to that in [Ru], this is accomplished via a new $L^{2}-L^{2}$ type Carleman estimate for a class of sublinear parabolic operators.

math.AP↗
arXiv2018

Momentum ray transforms

The momentum ray transform $I^k$ integrates a rank $m$ symmetric tensor field $f$ over lines with the weight $t^k$: $ (I^k\!f)(x,ξ)=\int_{-\infty}^\infty t^k\langle f(x+tξ),ξ^m\rangle\,dt. $ In particular, the ray transform $I=I^0$ was studied by several authors since it had many tomographic applications. We present an algorithm for recovering $f$ from the data $(I^0\!f,I^1\!f,\dots, I^m\!f)$. In the cases of $m=1$ and $m=2$, we derive the Reshetnyak formula that expresses $\|f\|_{H^s_t({\mathbb{R}}^n)}$ through some norm of $(I^0\!f,I^1\!f,\dots, I^m\!f)$. The $H^{s}_{t}$-norm is a modification of the Sobolev norm weighted differently at high and low frequencies. Using the Reshetnyak formula, we obtain a stability estimate.

math.AP↗