Elements of functional calculus andL 2 regularity for some classes of fourier integral operators

Abstract

We employ the method of slices to develop a rudimentary calculus describing the nature of operators T*T (respectively, TT*), where T are Fourier integral operators with one-sided right (respectively, left) singularities; this idea has its roots in the work of Greenleaf and Seeger. Such a result allows us to reduce the L2 regularity problem for operators in n dimensions with one-sided singularities to the L2 regularity problem for operators with two-sided singularities in n − 1 dimensions. As a consequence we deduce almost sharp L2-Sobolev estimates for operators in three-dimensions; an interesting special case is provided by certain restricted X-ray transforms associated to line complexes which are not well curved. We also provide a proof of almost-sharpness by looking at a restricted X-ray transform associated to the line complex generated by the curve t → (t, tk). Appropriate notions of singularity, strong singularity, and type are also developed.