• S Thangavelu

Articles written in Proceedings – Mathematical Sciences

• Multipliers for the Weyl transform and Laguerre expansions

Let Pk denote the projection of L2(RR) onto the kth eigenspace of the operator (-δ+⋎x⋎2 andSNα=(1/ANαkN=0AN−kαPk. We study the multiplier transformTNα for the Weyl transform W defined byW(TNαf)=SnαW(f). Applications to Laguerre expansions are given.

• Some uncertainty inequalities

We prove an uncertainty inequality for the Fourier transform on the Heisenberg group analogous to the classical uncertainty inequality for the Euclidean Fourier transform. Inequalities of similar form are obtained for the Hermite and Laguerre expansions.

• Riesz means for the sublaplacian on the Heisenberg group

The uniform boundedness of the Riesz means for the sublaplacian on the Heisenberg groupHn is considered. It is proved thatSRα are uniformly bounded onLp(Hn) for 1≤p≤2 provided α&gt;α(p)=(2n+1)[(1/p)−(1/2)].

• On regularity of twisted spherical means and special Hermite expansions

Regularity properties of twisted spherical means are studied in terms of certain Sobolev spaces defined using Laguerre functions. As an application we prove a localisation theorem for special Hermite expansions.

• Uncertainty principles on certain Lie groups

There are several ways of formulating the uncertainty principle for the Fourier transform on ℝn. Roughly speaking, the uncertainty principle says that if a functionf is ‘concentrated’ then its Fourier transform$$\tilde f$$ cannot be ‘concentrated’ unlessf is identically zero. Of course, in the above, we should be precise about what we mean by ‘concentration’. There are several ways of measuring ‘concentration’ and depending on the definition we get a host of uncertainty principles. As several authors have shown, some of these uncertainty principles seem to be a general feature of harmonic analysis on connected locally compact groups. In this paper, we show how various uncertainty principles take form in the case of some locally compact groups including ℝn, the Heisenberg group, the reduced Heisenberg groups and the Euclidean motion group of the plane.

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