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1 Bulletin of the Polish Academy of Sciences. Mathematics

1 2001

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Horocyclic Radon transform on Damek-Ricci space

Abouelaz A., El Fourchi O.

Bulletin of the Polish Academy of Sciences. Mathematics

2001

Vol. 49, no 2

107-140

We study the horocyclic Radon transform, defined in [5], of Damek-Ricci space. This transform R is obtained by integration over an orbitals family of NA space. We establish a Plancherel's formula for this transform. In particular, we characterize the range of horocyclic Radon transform of certain subspace of [L^2] ( NA, dx). The operators R and R*, where R* is the dual Radon transform of R, are inverted by a differential operator with constant coefficients (if dim NA is odd), or an integro-differential operator (if dim NA is even). The inversion formulas obtained are similar to the inversion formulas in symmetric space of noncompact type (see [7], [18], [19]). Also, we prove a radial Paley-Wiener-Schwartz's theorem for Damek-Ricci space.