Mapping estimates for the $k$-plane transform in Sobolev, Besov, and Triebel--Lizorkin Spaces

TL;DR

This paper investigates the mapping properties of the $k$-plane transform across various function spaces, extending classical results and establishing new boundedness and stability estimates.

Contribution

It extends Sobolev stability estimates and isometry identities from classical transforms to the general $k$-plane transform, and proves boundedness in Besov and Triebel--Lizorkin spaces.

Findings

Established Sobolev stability estimates for compactly supported functions.

Extended isometry identities to the $k$-plane transform.

Proved boundedness of the $k$-plane transform in Besov and Triebel--Lizorkin spaces.

Abstract

We study mapping properties of the $k$-plane transform in Sobolev, Besov, and Triebel--Lizorkin spaces. For $1\le k\le d-1$, the $k$-plane transform integrates a function over $k$-dimensional affine planes in $\mathbb{R}^d$, yielding a function on the affine Grassmannian $\mathcal{G}_{k,d}$. First, we establish Sobolev stability estimates for compactly supported functions, extending classical results of Natterer for the X-ray ($k=1$) and Radon ($k=d-1$) transforms to the general $k$-plane transform. Second, we extend isometry identities for the Radon and X-ray transforms, due to Reshetnyak, Sharafutdinov, and Kindermann--Hubmer, to the $k$-plane transform. Finally, we prove boundedness of the $k$-plane transform in Besov and Triebel--Lizorkin spaces.