Oscillatory integral operator
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 Tλ 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 and his school.
Assume that x,y ∈ Rn, 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 Tλ, which is initially defined on smooth functions, extends to a continuous operator from L2(Rn) to L2(Rn), with the norm bounded by , for any λ ≥ 1:
- Elias Stein, Harmonic Analysis: Real-variable Methods, Orthogonality and Oscillatory Integrals. Princeton University Press, 1993. ISBN 0-691-03216-5
- L. Hörmander Fourier integral operators, Acta Math. 127 (1971), 79–183. doi 10.1007/BF02392052, http://www.springerlink.com/content/t202410l4v37r13m/fulltext.pdf