6 - Fourier Integral Operators

Summary

We start out with a rapid and somewhat sketchy introduction to Fourier integral operators, emphasizing the role of stationary phase and only presenting material that will be needed later. In Section 2 we give the standard proof of the L² boundedness of Fourier integral operators whose canonical relations are locally a canonical graph and we state and prove a special case of the composition theorem in which one of the operators is assumed to be of this form. The same proof of course shows that this theorem holds under the weaker assumption that C₁ × C₂ intersects {(x, ξ, y, η, y, η, z, ζ) : (x, ξ) ∈ T^*X\0, (y, η) ∈ T^*Y\0, (z, ζ) ∈ T^*Z\0} transversally, although it is a little harder to check here that the phase function arising in the proof of the composition theorem is non-degenerate. The next thing we do is to prove the pointwise and L^P regularity theorems for Fourier integral operators and show that these are sharp if the operators are conormal with largest possible singular supports. Although this theorem came first, its proof uses the decomposition used in the proof of the maximal theorems for Riesz means and the circular maximal theorem given in Section 2.4. In the last section we apply the estimates for Fourier integral operators to give a proof of Stein's spherical maximal theorem and its variable coefficient generalizations involving the assumption of rotational curvature. In anticipation of the last chapter, we point out how this assumption is inadequate for variable coefficient maximal theorems in the plane.