The Daugavet property for Sobolev spaces over the plane

TL;DR

This work identifies a sharp geometric dichotomy for Sobolev spaces on the plane: the homogeneous Sobolev space $tildeW^{1,1}(\mathbb{R}^2)$, with norm given by the $L^1$-norm of the gradient, enjoys the Daugavet property, while the standard Sobolev space $W^{1,1}(\mathbb{R}^2)$ with the full norm fails the slice diameter two property. The authors prove the former by decomposing gradients into finite convex combinations supported on sets of arbitrarily small measure using level-set geometry, Sard’s theorem, the coarea formula, and Vitali covering arguments. They prove the latter by constructing a rank-one functional and a family of slices that forces the supports to shrink, preventing diameter two growth in any slice. The results illuminate how the choice of norm in Sobolev spaces affects global geometric properties and embeddings, with implications for operator equations on these spaces.

Abstract

We show that $W^{1,1}(\mathbb{R}^2)$ has the Daugavet property when endowed with the norm induced by the $L^1$-norm of the gradient, but fails to have the slice diameter two property when equipped with the usual Sobolev norm.

Theorems & Definitions (11)

• Lemma 1.1
• Definition 1.2
• Definition 2.1
• Theorem 2.2: Sard's Theorem
• Theorem 2.3: Preimage Theorem
• Lemma 2.4
• Theorem 2.5: Coarea formula
• Theorem 3.1
• proof
• Theorem 3.2