## Abstract

Let X=G/K be a noncompact symmetric space of real rank one. The purpose of this paper is to investigate L^{p} boundedness properties of a certain class of radial Fourier integral operators on the space X. We will prove that if u_{τ} is the solution at some fixed time τ of the natural wave equation on X with initial data f and g and 1<p<∞, then u_{τLp(X)}≤C_{p}(τ)(f _{Lpbp(X)}+(1+τ)g_{Lpbp-1(X)}). We will obtain both the precise behavior of the norm C_{p}(τ) and the sharp regularity assumptions on the functions f and g (i.e., the exponent b_{p}) that make this inequality possible. In the second part of the paper we deal with the analog of E. M. Stein's maximal spherical averages and prove exponential decay estimates (of a highly non-euclidean nature) on the L^{p} norm of sup_{T≤τ≤T+1}f*dσ_{τ}(z), where dσ_{τ} is a normalized spherical measure.

Original language | English (US) |
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Pages (from-to) | 274-300 |

Number of pages | 27 |

Journal | Journal of Functional Analysis |

Volume | 174 |

Issue number | 2 |

DOIs | |

State | Published - Jul 10 2000 |

## All Science Journal Classification (ASJC) codes

- Analysis