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
SN - 0022-1236
VL - 174
SP - 274
EP - 300
JO - Journal of Functional Analysis
JF - Journal of Functional Analysis
IS - 2
ER -