Oscillating multipliers on rank one symmetric spaces

In this paper we consider the family m_{α,β} of radial multipliers on rank one symmetric spaces G/K of the non compact type, defined by the formula m_{α,β}(λ)=(λ2+ρ2)^(-β/2) exp(i((λ2+ρ2)^(α/2)), Reβ≥0, α>0. Let T_{α,β} denote the associated convolution operator. Our main result is the following: if α > 1, T_{α,β} is bounded on L^p(G/K) if and only if p=2; if α < 1, T_{α,β} is bounded on L^p(G/K) if |1/p-1/2| < Reβ/(α dim(G/K)); if α = 1, T_{α,β} is bounded on L^p(G/K) if |1/p-1/2| < Reβ/(dim(G/K)−1). The values α=1 and α=1 are of particular interest, since che corresponding operators are closely related to the wave and the Schrödinger equations on G/K, the elliptic part of them being given by the Laplace-Beltrami operator. The case α>1 differs considerably from the corresponding Euclidean result and this fact is a consequence of the properties of the spherical Fourier transform on symmetric spaces. On the other side, the results for α≤1 resemble closely the Euclidean ones because “the part at infinity” of T_{α,β} behaves nicely, while the “local part” is assentially Euclidean.