Mapping estimates for the $k$-plane transform in Sobolev, 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.