ABSTRACT
This chapter describes some results concerning oscillatory integrals and, in particular, their application to Radon-like transforms. It briefly discusses three classes of oscillatory integrals. The first class consists of maximal oscillatory integrals. The second is made up of the oscillatory integrals arising in restrictions theorems and Bochner–Riesz summability. The third class contains the oscillatory integrals related to the Radon-like transforms, which is most closely related to Fourier integral operators. The chapter shows that the study of Radon-like transforms has an impact on the estimates for averaging operators involving integration over lower dimensional manifolds and connections with hyperbolic equations, relations of Radon transforms with Fourier integral operators, the initial study of singular Radon transforms as “Hilbert transforms along curves,” and the relevance to several complex variables.
