Randol Maximal Functions and the Integrability of the Fourier Transform of Measures

Estimates of the Fourier transform of charges (measures) concentrated on smooth hypersurfaces are considered. Following M. Sugumoto, three classes of smooth hypersurfaces are defined. Depending on the class, estimates of the Fourier transform of charges are obtained in terms of Randol maximal functions. The obtained estimates are applied to the solution of the integrability problem for the Fourier transform of measures concentrated on some nonconvex hypersurfaces. The sharpness of the obtained estimates is shown.

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