tailieunhanh - Báo cáo toán học: "Maximal operators on hyperboloids "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Khai thác tối đa trên hyperboloids. | J. OPERATOR THEORY 3 1980 41 56 Copyright by INCREST 1980 MAXIMAL OPERATORS ON HYPERBOLOIDS A. EL KOHEN INTRODUCTION In a paper in 1976 8 E. M. Stein introduced and studied the following maximal function m x sup f x - ty dff y t 0 J where f is any Borel measurable function on R z the unit sphere in R and dơ the area measure on s. His positive result is the following for n 3 andp - n 1 in is a bounded operator on LP R . One of the applications of this result is a Fatou s theorem for the following Cauchy problem 0 u 0 0 Ht 0 f ớ2 ỡ2 Ớ2 where ----------- . - - ớt2 dxị dx For the local version of Stein s result say on compact Riemannian manifold one can obtain similar theorems using the theory of Fourier integral operators. This was carried out in collaboration with R. Coifman Y. Meyer A. Nagel E. Stein and s. Wainger . When studying the Cauchy problem it is natural to consider other initial hypersurfaces than R . In particular the hyperboloids 2 v-2 -2 _v2 . I -A A 2 A 2 are of special interest. In this paper we obtain global Lp estimates for various maximal functions. Here we cannot use the local version since the behavior at infinity is crucial and differs in essential ways with the Euclidean case. We thus will follow Stein s program using the intrinsic harmonic analysis of these symmetric spaces. 42 A. EL KOHEN Finally I would like to thank Professor R. Coifman for his guidance and advice in this work. 1 In this paragraph we want to give some aspects of harmonic analysis on the hyperboloids. Let G be the group of hyperbolic rotations in R L G leaves invariant the quadratic form t2 A l . x and preserves the orientation of the space. Now let K be the isotropy subgroup corresponding to the vector 1 1 0 0 . . 0 in R 1 clearly we have K 1 1 G G u ỉ 1 with R 6 sơ n l 1 0 R J Also we have G KAK the Cartan decomposition of G where A u e G u a j shs chi 0 3 0 On G we have the following integration formula. For an adequate normalization of the Haar measures and all .