Oscillating multipliers on rank one symmetric spaces

In this paper we consider the family m_{\u3b1,\u3b2} of radial multipliers on rank one symmetric spaces G/K of the non compact type, defined by the formula m_{\u3b1,\u3b2}(\u3bb)=(\u3bb2+\u3c12)^(-\u3b2/2) exp(i((\u3bb2+\u3c12)^(\u3b1/2)), Re\u3b2 650, \u3b1>0. Let T_{\u3b1,\u3b2} denote the associated convolution operator. Our main result is the following: if \u3b1 > 1, T_{\u3b1,\u3b2} is bounded on L^p(G/K) if and only if p=2; if \u3b1 < 1, T_{\u3b1,\u3b2} is bounded on L^p(G/K) if |1/p-1/2| < Re\u3b2/(\u3b1 dim(G/K)); if \u3b1 = 1, T_{\u3b1,\u3b2} is bounded on L^p(G/K) if |1/p-1/2| < Re\u3b2/(dim(G/K) 121). The values \u3b1=1 and \u3b1=1 are of particular interest, since che corresponding operators are closely related to the wave and the Schr\uf6dinger equations on G/K, the elliptic part of them being given by the Laplace-Beltrami operator. The case \u3b1>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 \u3b1 641 resemble closely the Euclidean ones because \u201cthe part at infinity\u201d of T_{\u3b1,\u3b2} behaves nicely, while the \u201clocal part\u201d is assentially Euclidean