Elements of functional calculus and L-2 regularity for some classes of Fourier integral operators

摘要

We employ the method of slices to develop a rudimentary calculus describing the nature of operators T*T (respectively, TT*), vi here 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 it result allows its to reduce the L-2 regularity problem for operators at it 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 L-2-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 carved. lilt, 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 bar right arrow (t, t(k)). Appropriate notions of singularity, strong singularity, and type are also developed.