Weighted norm estimates for the Semyanistyi fractional integrals and Radon transforms

TL;DR

This paper establishes sharp weighted norm inequalities and explicitly computes operator norms for Semyanistyi fractional integrals and Radon transforms on weighted L^p spaces, extending to k-plane transforms.

Contribution

It provides the first sharp weighted inequalities and explicit operator norm calculations for Semyanistyi fractional integrals and Radon transforms, including k-plane transforms.

Findings

Sharp inequalities for fractional integrals and Radon transforms.

Explicit evaluation of operator norms on weighted L^p spaces.

Extension of results to k-plane transforms for 1 ≤ k < n.

Abstract

Semyanistyi's fractional integrals have come to analysis from integral geometry. They take functions on $R^n$ to functions on hyperplanes, commute with rotations, and have a nice behavior with respect to dilations. We obtain sharp inequalities for these integrals and the corresponding Radon transforms acting on $L^p$ spaces with a radial power weight. The operator norms are explicitly evaluated. Similar results are obtained for fractional integrals associated to $k$-plane transforms for any $1\le k<n$.