Stability estimates show the k-plane transform on Radon measures is bi-Lipschitz equivalent to a Fourier metric and Hölder equivalent to Wasserstein distance, with a strong Sobolev equivalence for bounded-density measures.
Mapping estimates for the $k$-plane transform in Sobolev, Besov, and Triebel--Lizorkin Spaces
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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.
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math.FA 1years
2026 1verdicts
UNVERDICTED 1representative citing papers
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Stability Estimates for the $k$-plane Transform on Measures and a H\"older-Type Comparison Between Wasserstein and Max-Sliced Wasserstein Distances
Stability estimates show the k-plane transform on Radon measures is bi-Lipschitz equivalent to a Fourier metric and Hölder equivalent to Wasserstein distance, with a strong Sobolev equivalence for bounded-density measures.