TY - JOUR

T1 - Fourier Integral Operators on Noncompact Symmetric Spaces of Real Rank One

AU - Ionescu, Alexandru D.

N1 - Funding Information:
1The author was supported by a Princeton University fellowship assistantship. This work is part of the author’s dissertation at Princeton University. I thank my advisor, Professor Elias M. Stein, for his guidance and support as well as for several valuable suggestions regarding the formulation of Theorem B.

PY - 2000/7/10

Y1 - 2000/7/10

N2 - Let X=G/K be a noncompact symmetric space of real rank one. The purpose of this paper is to investigate Lp boundedness properties of a certain class of radial Fourier integral operators on the space X. We will prove that if uτ is the solution at some fixed time τ of the natural wave equation on X with initial data f and g and 1τLp(X)≤Cp(τ)(f Lpbp(X)+(1+τ)gLpbp-1(X)). We will obtain both the precise behavior of the norm Cp(τ) and the sharp regularity assumptions on the functions f and g (i.e., the exponent bp) 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 Lp norm of supT≤τ≤T+1f*dστ(z), where dστ is a normalized spherical measure.

AB - Let X=G/K be a noncompact symmetric space of real rank one. The purpose of this paper is to investigate Lp boundedness properties of a certain class of radial Fourier integral operators on the space X. We will prove that if uτ is the solution at some fixed time τ of the natural wave equation on X with initial data f and g and 1τLp(X)≤Cp(τ)(f Lpbp(X)+(1+τ)gLpbp-1(X)). We will obtain both the precise behavior of the norm Cp(τ) and the sharp regularity assumptions on the functions f and g (i.e., the exponent bp) 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 Lp norm of supT≤τ≤T+1f*dστ(z), where dστ is a normalized spherical measure.

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U2 - 10.1006/jfan.2000.3572

DO - 10.1006/jfan.2000.3572

M3 - Article

AN - SCOPUS:0000752189

VL - 174

SP - 274

EP - 300

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 2

ER -