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In mathematics, in the field of harmonic analysis, an oscillatory integral operator is an integral operator of the form : where the function ''S(x,y)'' is called the phase of the operator and the function ''a(x,y)'' is called the symbol of the operator. λ is a parameter. One often considers ''S(x,y)'' to be real-valued and smooth, and ''a(x,y)'' smooth and compactly supported. Usually one is interested in the behavior of ''T''λ for large values of λ. Oscillatory integral operators often appear in many fields of mathematics (analysis, partial differential equations, integral geometry, number theory) and in physics. Properties of oscillatory integral operators have been studied by E. Stein〔Elias Stein, ''Harmonic Analysis: Real-variable Methods, Orthogonality and Oscillatory Integrals''. Princeton University Press, 1993. ISBN 0-691-03216-5〕 and his school. ==Hörmander's theorem== The following bound on the ''L''2 → ''L''2 action of oscillatory integral operators (or ''L''2 → ''L''2 operator norm) was obtained by Lars Hörmander in his paper on Fourier integral operators:〔L. Hörmander ''Fourier integral operators'', Acta Math. 127 (1971), 79–183. doi 10.1007/BF02392052, http://www.springerlink.com/content/t202410l4v37r13m/fulltext.pdf〕 Assume that ''x,y'' ∈ R''n'', ''n'' ≥ 1. Let ''S(x,y)'' be real-valued and smooth, and let ''a(x,y)'' be smooth and compactly supported. If everywhere on the support of ''a(x,y)'', then there is a constant ''C'' such that ''T''λ, which is initially defined on smooth functions, extends to a continuous operator from ''L''2(R''n'') to ''L''2(R''n''), with the norm bounded by , for any λ ≥ 1: : 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「oscillatory integral operator」の詳細全文を読む スポンサード リンク
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