Results in estimates for $k$-plane transforms

TL;DR

This paper provides detailed proofs of classical results related to $k$-plane transforms and explores invariant measures on homogeneous manifolds, clarifying foundational aspects of integral geometry.

Contribution

It offers rigorous proofs of key theorems by Christ and Drury on $k$-plane transforms and discusses invariant measures on Lie group manifolds, enhancing understanding of these topics.

Findings

Proofs of Christ's and Drury's results on $k$-plane transforms

Existence of invariant measures on homogeneous Lie group manifolds

Clarification of foundational integral geometry results

Abstract

This is an expository paper. We give proofs of some results of M. Christ (1984) and S. W. Drury (1984) for $k$-plane transforms. Also, we give proofs for some related results including that for the existence of invariant measures on certain homogeneous manifolds of Lie groups.