Proposal: DMS-9877101 Principal Investigator: Allan T. Greenleaf
Abstract: Estimates for Fourier integral operators associated with canonical relations whose projections exhibit stable classes of singularities, higher-order cusps in particular, will be investigated. For two-sided cusps, new decompositions of the operators near the singular varieties will be developed. These estimates and decompositions will have applications to boundedness properties of generalized Radon transforms, particularly averages over families of curves and restricted X-ray transforms, on Sobolev and Lebesgue spaces. In addition, classes of degenerate Fourier integral operators arising in integral geometry and inverse seismic problems will be studied and composition calculi obtained for them. Real analogues of Goncharov's complexes of hyperplane sections of algebraic varieties of minimal degree and the forward operator for linearized source problems will provide important prototypes of such operators.
The operators analyzed in this project are special classes of Fourier integral operators, which have been used for the last thirty years to construct approximate solutions to linear partial differential equations. Such equations govern basic physical processes, such as vibration of membranes and propagation of electromagnetic fields. More recently, it has been found that Fourier integral operators arise in other settings, such as tomography (medical imaging) and geophysical exploration. More detailed understanding of these operators under a variety of geometric assumptions will improve our understanding of both the qualitative and quantitative behavior of the physical systems being modelled.
