Rudra P. Sarkar

FOURIER AND RADON TRANSFORM ON HARMONIC NA GROUPS

In this article we study the Fourier and the horocyclic Radon transform on harmonic N A groups (also known as Damek-Ricci spaces). We consider the geometric Fourier transform for functions on L p -spaces and prove an analogue of the L 2 -restriction theorem. We also prove some mixed norm estimates for the Fourier transform generalizing the Hausdorff-Young and Hardy-Littlewood-Paley inequalities. Unlike Euclidean spaces the domains of the Fourier transforms are various strips in the complex plane. All the theorems are considered on these entire domains of the Fourier transforms. Finally we deal with the existence of the Radon transform on L P -spaces and obtain its continuity property.

The Helgason-Fourier Transform for Symmetric Spaces II

We formulate analogues of the Hausdor-Young and Hardy- Littlewood-Paley inequalities, the Wiener Tauberian theorem, and some un- certainty theorems on Riemannian symmetric spaces of noncompact type using the Helgason-Fourier transform.

Cowling-Price theorem and characterization of heat kernel on symmetric spaces

We extend the uncertainty principle, the Cowling-Price theorem, on noncompact Riemannian symmetric spacesX. We establish a characterization of the heat kernel of the Laplace-Beltrami operator onX from integral estimates of the Cowling-Price type.

Beurling's Theorem and Characterization of Heat Kernel for Riemannian Symmetric Spaces of Noncompact Type

We prove Beurlings theorem for rank 1 Riemmanian symmetric spaces and relate it to the characterization of the heat kernel of the symmetric space.

A complete analogue of Hardy's theorem on SL2.R/ and characterization of the heat kernel

A theorem of Hardy characterizes the Gauss kernel (heat kernel of the Laplacian) on ℝ from estimates on the function and its Fourier transform. In this article we establisha full group version of the theorem for SL2(ℝ) which can accommodate functions with arbitraryK-types. We also consider the ‘heat equation’ of the Casimir operator, which plays the role of the Laplacian for the group. We show that despite the structural difference of the Casimir with the Laplacian on ℝn or the Laplace—Beltrami operator on the Riemannian symmetric spaces, it is possible to have a heat kernel. This heat kernel for the full group can also be characterized by Hardy-like estimates.

A theorem of Beurling and Hörmander on Damek–Ricci spaces

Abstract We prove Beurlings theorem and Lp –Lq Morgans theorem for Damek–Ricci spaces. These two theorems exhaust a family of theorems which illustrate a well-known paradigm that a function and its Fourier transform cannot be simultaneously localized.

Beurling’s theorem for Riemannian symmetric spaces II

We prove two versions of Beurlings theorem for Riemannian symmetric spaces of arbitrary rank. One of them uses the group Fourier transform and the other uses the Helgason Fourier transform. This is the master theorem in the quantitative uncertainty principle.

The Helgason Fourier Transform for semisimple Lie groups I: the case of SL 2 (R)

We consider a Helgason-type Fourier transform on SL 2 (ℝ) and prove various results on L 1 -harmonic analysis on the full group analogous to those on symmetric spaces.