Integrability of the fourier transforms of measures concentrated on hypersurfaces

Abstract

This paper considers estimates for the Fourier transforms of signed measures concentrated on families of hypersurfaces. A theorem about the integrability of Randol-type maximal functions related to a certain class of nonconvex hypersurfaces is presented. The results are applied to study the integrability of the Fourier transforms of signed measures concentrated on certain hypersurfaces. In a special case, the exact integrability exponent of the Fourier transforms of measures is specified. The results improve a recent theorem of L. Erdős and M. Salmhofer.