Strichartz transforms with Riesz potentials and Semyanistyi integrals

TL;DR

This paper investigates the Strichartz transform involving Riesz potentials and Semyanistyi integrals, establishing existence conditions, relation formulas, and explicit inversion formulas, thereby generalizing previous specific cases.

Contribution

It provides new existence conditions, relation formulas, and inversion formulas for the Strichartz transform, extending previous results to a more general setting.

Findings

Established sharp existence conditions for the transform on L^p spaces.

Derived relation formulas connecting Strichartz transform and Semyanistyi integrals.

Provided explicit inversion formulas for the Strichartz transform.

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.