Strichartz transforms with Riesz potentials and Semyanistyi integrals

Abstract

In this paper, we study the general orthogonal Radon transform $R_{p,q}^k$ first studied by R.S Strichartz in \cite{Stri}. An sharp existence condition of $R_{p,q}^k f$ on $L^p$-spaces will be given. Then we devote to the relation formulas connecting Strichartz transform $R_{p,q}^k$ and Semyanistyi integrals. We prove the corresponding Fuglede type formulas, through which a number of explicit inversion formulas for $R_{p,q}^k f$ will be given. Different from the inclusion Radon transform and "Gonzalez" type orthogonal transform, Strichartz transform is more complicated. Our conclusions generalize the corresponding results of the two particular cases above.