On estimates for oscillatory integrals with phase depending on parameters

We consider estimates for the Fourier transforms of measures supported on analytic hypersurfaces involving a damping factor. As a damper, we naturally take a power of the Gaussian curvature of the surface. It is known that if the exponent in this power is a sufficiently large positive number, then the Fourier transform of the corresponding measure has an optimal decay. C.D. Sogge and E.M. Stein formulated a problem on a minimal power of the Gaussian curvature ensuring an optimal decay for the Fourier transform. In the paper we resolve the problem by C.D. Sogge and E.M. Stein on an optimal decay for the Fourier transform with a damping factor for a particular class of families of analytic surfaces in the three-dimensional Euclidean space. We note that the power we provide is sharp not only for the families of analytic hypersurfaces but also for a fixed analytic hypersurface. The proof of main result is based on the methods of the theory of analytic functions, more precisely, on the statements like a preparation Weierstrass theorem. As D.M. Oberlin showed, similar statements fail for infinitely differentiable hypersurfaces.