Ruzhansky M. Regularity theory of Fourier integral operators with complex phases and singularities of affine fibrations

Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, 2001. — 144 p.The main subject of the present monograph is the regularity theory of Fourier integral operators with real and complex phases and related questions of the singularity theory of affine fibrations. It is known that the geometry and singularities of the canonical relation of an operator are reflected in the boundedness properties of operators in Lp-spaces.
Therefore, a part of the book is devoted to the general singularity theory of affine fibrations. Conditions for the boundedness of Fourier integral operators are related to the singularities of affine fibrations in (subsets of) Cn, defined by the Hessians of phase functions.
One of the new aspects of the current work is that we deal with operators with complex phase functions. The theory of such operators is well developed but their regularity has not been much studied. In a way, the use of complex phases provides a more natural approach to Fourier integral operators. In the monograph the so-called smooth factorization condition is extended to to the complex phases and a number of regularity properties are derived in the complex setting.
The use of the complex phase allows to treat several new examples, such as nonhyperbolic Cauchy problems for pseudo-differential equations and the oblique derivative problem. The background information on Fourier integral operators with real and complex phases as well as the singularity theory of affine fibrations and relevant methods of complex analytic geometry is provided in the book.