Extremal problems related to simple singularities

In this talk we consider estimates for the Fourier transform of measures concentrated on smooth surfaces given by the graph of a smooth function with simple Arnold singularities such that both principal curvatures of the surface vanish at some point. Depending on the geometric properties of the hypersurfaces, we will consider two problems related to extremum. We are looking for maximal multiplicity of the critical points of the corresponding phase function and greatest lower bound (infimum) for integrability exponent for the Fourier transform of the corresponding measures. Therefore, we refer to them as extreme problems in geometry related to simple Arnold singularities.Then the obtained estimates for the Fourier transform of surfaces-carried measures will be applied to boundedness problems for some convolution operators with oscillatory kernels.