On theorems of Chernoff and Ingham on the Heisenberg group
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.
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arXiv subjects
Sundaram Thangavelu
Publications and source records attributed to Sundaram Thangavelu.
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.
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Homogeneous Fourier and Weyl multipliers on Sobolev spaces related to the Heisenberg group
Inspired by the work of A. Bonami and S. Poornima that a non-constant function which is homogeneous of degree $0$ cannot be a Fourier multiplier on homogeneous Sobolev spaces, we establish analogous results for Fourier multipliers on the Heisenberg group $ \mathbb{H}^n $ and Weyl multipliers on $ \mathbb{C}^n $ acting on Sobolev Spaces.
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Segal-Bargmann transform and holomorphic Sobolev spaces of fractional order
In this note we investigate the image of Sobolev spaces of fractional order on a compact Lie group $ K $ under the Segal-Bargmann transform. We show that the image can be characterised in terms of certain weighted Bergman spaces of holomorphic functions on the complexification extending a theorem of Hall and Lewkeeratiyutkul. We also treat the heat kernel transform associated to the Hermite operator.
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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 $M^{\infty,1}$.
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On the lacunary spherical maximal function on the Heisenberg 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 weighted estimates. The key ingredients in the proof are the $L^p$ improving property of the operator $A_rf$ and a continuity property of the difference $A_rf-τ_y A_rf$, where $τ_yf(x)=f(xy^{-1})$ is the right translation operator.
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A note on fractional powers of the Hermite operator
We give a very short proof of a result proved by Cappiello-Rodino-Toft on the Weyl symbol of the inverse of the Harmonic oscillator. We also extend their results to fractional powers.
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Hermite Multipliers on Modulation Spaces
We study multipliers associated to the Hermite operator $H=-Δ+ |x|^2$ on modulation spaces $M^{p,q}(\mathbb R^d)$. We prove that the operator $m(H)$ is bounded on $M^{p,q}(\mathbb R^d)$ under standard conditions on $m,$ for suitable choice of $p$ and $q$. As an application, we point out that the solutions to the free wave and Schrödinger equations associated to $H$ with initial data in a modulation space will remain in the same modulation space for all times. We also point out that Riesz transforms associated to $H$ are bounded on some modulation spaces.
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Weighted norm inequalities for Weyl multipliers and Fourier multipliers on the Heisenberg group
In this paper we prove weighted norm inequalities for Weyl multipliers satisfying Mauceri's condition. As applications of this we obtain some estimates for $L^p$ multipliers on the Heisenberg group and also show in the context of a theorem of Weis on operator valued Fourier multipliers that the R-boundedness of the derivative of the multiplier is not necessary for the boundedness of the multiplier transform.
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On Hermite pseudo-multipliers
In this article we deal with a variation of a theorem of Mauceri concerning the $ L^p $ boundedness of operators $ M $ which are known to be bounded on $ L^2.$ We obtain sufficient conditions on the kernel of the operaor $ M $ so that it satisfies weighted $ L^p $ estimates. As an application we prove $ L^p $ boundedness of Hermite pseudo-multipliers.
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Hardy's theorem for compact Lie groups
We show that Hardy's uncertainty principle can be reformulated in such a way that it has an analogue even for compact Lie groups and symmetric spaces of compact type.
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Variations on a theorem of Beurling
We consider functions satisfying the subcritical Beurling's condition, viz., $$\int_{\R^n}\int_{\R^n} |f(x)| |\hat{f}(y)| e^{a |x \cdot y|} \, dx \, dy < \infty$$ for some $ 0 < a < 1.$ We show that such functions are entire vectors for the Schrödinger representations of the Heisenberg group. If an eigenfunction $f$ of the Fourier transform satisfies the above condition we show that the Hermite coefficients of $f$ have certain exponential decay which depends on $a$.
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Uniqueness of solutions to the Schrodinger equation on the Heisenberg group
This paper deals with the Schr{ö}dinger equation $i\partial_s u({\bf z},t;s)-\cal L u({\bf z}, t;s)=0,$ where $\cal L$ is the sub-Laplacian on the Heisenberg group. Assume that the initial data $f$ satisfies $| f({\bf z},t)| \leq C q_a({\bf z},t),$ where $q_s$ is the heat kernel associated to $\cal L.$ If in addition $ |u({\bf z},t;s_0)|\leq C q_b({\bf z},t),$ for some $s_0\in \R^*,$ then we prove that $u({\bf z},t;s)=0$ for all $s\in \R $ whenever $ab<s_0^2.$ This result also holds true on $H$-type groups.
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On the structure of analytic vectors for the schrodinger representation
This article deals with the structure of analytic and entire vectors for the Schrödinger representations of the Heisenberg group. Using refined versions of Hardy's theorem and their connection with Hermite expansions we obtain very precise representation theorems for analytic and entire vectors.
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Convolution operators defined by singular measures on the motion group
This paper contains an $L^{p}$ improving result for convolution operators defined by singular measures associated to hypersurfaces on the motion group. This needs only mild geometric properties of the surfaces, and it extends earlier results on Radon type transforms on $\mathbb{R}^{n}$. The proof relies on the harmonic analysis on the motion group.
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On the Hermite expansions of functions from Hardy class
Considering functions $ f $ on $ \R^n $ for which both $ f $ and $ \hat{f} $ are bounded by the Gaussian $ e^{-{1/2}a|x|^2}, 0 < a < 1 $ we show that their Fourier-Hermite coefficients have exponential decay. Optimal decay is obtained for $ O(n)-$finite functions thus extending the one dimensional result of Vemuri.
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A Paley-Wiener theorem for the inverse Fourier transform on some homogeneous spaces
We formulate and prove a version of Paley-Wiener theorem for the inverse Fourier transform on non-compact Riemannian symmetric spaces and Heisenberg groups. The main ingredient in the proof is the Gutzmer's formula.
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Convolution operator and maximal function for Dunkl transform
For a family of weight functions, $h_κ$, invariant under a finite reflection group on $\RR^d$, analysis related to the Dunkl transform is carried out for the weighted $L^p$ spaces. Making use of the generalized translation operator and the weighted convolution, we study the summability of the inverse Dunkl transform, including as examples the Poisson integrals and the Bochner-Riesz means. We also define a maximal function and use it to prove the almost everywhere convergence.
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