Estimates on level set integral operators in dimension two

Abstract

Both oscillatory integral operators and level set operators appear naturally in the study of propertiesofdegenerateFourierintegraloperators(suchasgeneralizedRadontransforms). Theproperties of oscillatory integral operators have a longer history and are better understood. On the other hand, level set operators, while sharing many common characteristics with oscillatory integral operators, are easier to handle. We study L 2 -estimates onlevel set operators indimension two and compare them with what isknown about oscillatory integral operators. The cases include operators with non-degenerate phase functions and the level set version of Melrose-Taylor transform (as an example of a degenerate phase function). The estimates are formulated in terms of the Newton polyhedra and type conditions.