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S. Thangavelu

Publications and source records attributed to S. Thangavelu.

At least 19 recordsLinked to original sources
arXiv2021

On the maximal function associated to the spherical means on the Heisenberg group

In this paper we deal with lacunary and full versions of the spherical maximal function on the Heisenberg group $\mathbb{H}^n$, for $n\ge 2$. By suitable adaptation of an approach developed by M. Lacey in the Euclidean case, we obtain sparse bounds for these maximal functions, which lead to new unweighted and weighted estimates. In particular, we deduce the $L^p$ boundedness, for $1<p<\infty$, of the lacunary maximal function associated to the spherical means on the Heisenberg group. In order to prove the sparse bounds, we establish $ L^p-L^q $ estimates for local (single scale) variants of the spherical means.

math.CA↗
arXiv2020

Holomorphic extensions of eigenfunctions on $NA$ groups

Let $ X = G/K $ be a rank one Riemannian symmetric space of noncompact type. In view of the Iwasawa decomposition $ G = NAK $ of the underlying semisimple Lie group, we can also view $ X $ as the solvable extension $ S = NA $ of the Iwasawa group $ N$ which is known to be a $H$-type group. In this work we study the holomorphic extendability of eigenfunctions of the Laplace-Beltrami operator $ Δ_S$ on $ S $ to certain domains in the complexification of the nilpotent group $ N$. We can also do the same for any $H$-type group $ N $ not necessarily an Iwasawa group. The results are accomplished by making use of the connection with solutions of the extension problem for the Laplacian or the sublaplacian on the corresponding $ N$.

math.CA↗
arXiv2016

Hardy-type inequalities for fractional powers of the Dunkl--Hermite operator

We prove Hardy-type inequalities for a fractional Dunkl--Hermite operator which incidentally give Hardy inequalities for the fractional harmonic oscillator as well. The idea is to use $h$-harmonic expansions to reduce the problem in the Dunkl--Hermite context to the Laguerre setting. Then, we push forward a technique based on a non-local ground representation, initially developed by R. L. Frank, E. H. Lieb and R. Seiringer in the Euclidean setting, to get a Hardy inequality for the fractional-type Laguerre operator. The above-mentioned method is shown to be adaptable to an abstract setting, whenever there is a "good" spectral theorem and an integral representation for the fractional operators involved.

math.CA↗
arXiv2016

Hardy's inequality for fractional powers of the sublaplacian on the Heisenberg group

We prove Hardy inequalities for the conformally invariant fractional powers of the sublaplacian on the Heisenberg group $\mathbb{H}^n$. We prove two versions of such inequalities depending on whether the weights involved are non-homogeneous or homogeneous. In the first case, the constant arising in the Hardy inequality turns out to be optimal. In order to get our results, we will use ground state representations. The key ingredients to obtain the latter are some explicit integral representations for the fractional powers of the sublaplacian and a generalized result by M. Cowling and U. Haagerup. The approach to prove the integral representations is via the language of semigroups. As a consequence of the Hardy inequalities we also obtain versions of Heisenberg uncertainty inequality for the fractional sublaplacian.

math.CA↗
arXiv2015

Mixed norm estimates for the Cesàro means associated with Dunkl--Hermite expansions

Our main goal in this article is to study mixed norm estimates for the Cesàro means associated with Dunkl--Hermite expansions on $\mathbb{R}^d$. These expansions arise when one consider the Dunkl--Hermite operator (or Dunkl harmonic oscillator) $H_κ:=-Δ_κ+|x|^2$, where $Δ_κ$ stands for the Dunkl--Laplacian. It is shown that the desired mixed norm estimates are equivalent to vector-valued inequalities for a sequence of Cesàro means for Laguerre expansions with shifted parameter. In order to obtain the latter, we develop an argument to extend these operators for complex values of the parameters involved and apply a version of three lines lemma.

math.CA↗
arXiv2014

On the chaotic behavior of the Dunkl heat semigroup on weighted $ L^p $ spaces

In this paper we study the chaotic behaviour of the heat semigroup generated by the Dunkl-Laplacian on weighted $ L^p$ spaces. In the case of the heat semigroup associated to the standard Laplacian we obtain a complete picture on the spaces $ L^p(\R^n, (φ_{iρ}(x))^2 dx) $ where $ φ_{iρ} $ is the Euclidean spherical function. The behaviour is very similar to the case of the Laplace-Beltrami operator on non-compact Riemannian symmetric spaces studied by Pramanik and Sarkar.

math.FA↗
arXiv2014

Mixed norm estimates for the Riesz transforms associated to Dunkl harmonic oscillators

In this paper we study weighted mixed norm estimates for Riesz transforms associated to Dunkl harmonic oscillators. The idea is to show that the required inequalities are equivalent to certain vector valued inequalities for operator defined in terms of Laguerre expansions. In certain cases the main result can be deduced from the corresponding result for Hermite Riesz transforms.

math.FA↗
arXiv2013

Revisiting Riesz transforms for Hermite and Special Hermite Operators

In this paper we prove weighted mixed norm estimates for Riesz transforms associated to Hermite and special Hermite operators. The estimates are shown to be equivalent to vectorvalued esimates for a sequence of operators defined in terms of Laguerre functions of different type.

math.CA↗
arXiv2013

Mixed norm estimates for Hermite multipliers

In this article mixed norm estimates are obtained for some integral operators, from which those for the Hermite semigroup and the Bochner Riesz means associated with the Hermite expansions are deduced. Also, mixed norm estimates for the Littlewood Paley g functions and g* functions for the Hermite expansions are obtained, which lead to those for Hermite multipliers

math.CA↗
arXiv2011

A characterisation of the Fourier transform on the Heisenberg group

The aim of this paper is to show that any continuous *-homomorphism of L1(Cn)(with twisted convolution as multipli- cation) into B(L2(Rn)) is essentially a Weyl transform. From this we deduce a similar characterisation for the group Fourier transform on the Heisenberg group, in terms of convolution.

math.FA↗
arXiv2011

Riesz transforms and multipliers for the Grushin operator

We show that Riesz transforms associated to the Grushin operator G = -Δ- |x|^2\partial_t^2 are bounded on L^p(R^n+1). We also establish an analogue of Hörmander-Mihlin multiplier theorem and study Bochner-Riesz means associated to the Grushin operator. The main tools used are Littlewood-Paley theory and an operator valued Fourier multiplier theorem due to L. Weis.

math.FA↗
arXiv2011

Revisiting Riesz transforms on Heisenberg groups

We characterise higher order Riesz transforms on the Heisenberg group and also show that they satisfy dimension-free bounds under some assumptions on the multipliers. Using transfer- ence theorems, we deduce boundedness theorems for Riesz trans- forms on the reduced Heisenberg group and hence also for the Riesz transforms associated to multiple Hermite and Laguerre ex- pansions.

math.FA↗
arXiv2010

Heat kernel transform on nilmanifolds associated to H-type groups

We study the heat kernel transform on a nilmanifold M associated to a H-type group. Using a reduction technique we reduce the problem to the case of Heisenberg groups. The image of $ L^2(M) $ under the heat kernel transform is shown to be a direct sum of weighted Bergman spaces.

math.FA↗
arXiv2009

Toeplitz operators with special symbols on Segal-Bargmann spaces

We study the boundedness of Toeplitz operators on Segal-Bargmann spaces in various contexts. Using Gutzmer's formula as the main tool we identify symbols for which the Toeplitz operators correspond to Fourier multipliers on the underlying groups. The spaces considered include Fock spaces, Hermite and twisted Bergman spaces and Segal-Bargmann spaces associated to Riemannian symmetric spaces of compact type.

math.FA↗
arXiv2007

Holomorphic Sobolev spaces, Hermite ans special Hermite semigroups and a Paley-Wiener theorem for the windowed Fourier transform

The images of Hermite and Laguerre Sobolev spaces under the Hermite and special Hermite semigroups (respectively) are characterised. These are used to characterise the Schwartz class of rapidly decreasing functions. The image of the space of all tempered distributions is also considered and a Paley-Wiener theorem for the windowed Fourier transform is proved.

math.FA↗