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Models of degenerate Fourier integral operators and Radon transforms. (English) Zbl 0833.43004

Let S(x,y) be a homogeneous polynomial of degree n, and consider the oscillatory integral T λ defined by

T λ (f)(x)= - e iλS(x,y) Ψ(x,y)f(y)dy

where f (resp. Ψ) denotes a smooth compactly supported function on (resp. 2 ). The operator T λ is closely related to a class of Radon transforms which we describe as follows: The phase S(x,y) gives rise to a family (t,x)γ (t,x) of curves in the plane via

γ (t,x) ={(s,y);s=t+S(x,y)}

with curves γ (t,x) and suitable densities dμ (t,x) defined on them; the associated Radon transform R is given by R(f)(t,x)= γ (t,x) fdμ (t,x) · In this paper the authors give necessary and sufficient conditions for optimal L 2 estimates of the operator T λ . Next, by viewing R as a pseudo-differential operator whose symbol is the operator-valued function λT λ they deduce the L p -L q boundedness and Sobolev regularity of the transform R.


MSC:
43A32Other transforms and operators of Fourier type
35S30Fourier integral operators