The Daugavet property for Sobolev spaces over the plane

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.

💡 Research Summary

The paper investigates two distinct normed structures on the Sobolev space (W^{1,1}(\mathbb{R}^{2})) and establishes contrasting geometric properties for each.

First, the authors consider the homogeneous Sobolev norm (|f|{\widetilde W^{1,1}}:=|\nabla f|{L^{1}(\mathbb{R}^{2})}), which ignores the (L^{1})-norm of the function itself. They prove that (\widetilde W^{1,1}(\mathbb{R}^{2})) enjoys the Daugavet property, i.e. every rank‑one operator (T) satisfies (|{\rm Id}+T|=1+|T|). The proof proceeds by showing that for any unit vector (f) in the unit ball, its gradient can be approximated arbitrarily well by convex combinations of gradients of functions whose supports have arbitrarily small Lebesgue measure. The key ingredients are:

• Sard’s theorem and the regular‑value pre‑image theorem, which guarantee that for a smooth compactly supported function most level sets are finite unions of Jordan curves.
• Lemma 2.4, which asserts that the complement of a finite family of disjoint Jordan curves in the plane is path‑connected.
• A Vitali covering argument to select a countable family of regular level intervals whose total measure covers almost all values of the function.
• The coarea formula, which translates integrals of (|\nabla f|) into integrals over the 1‑dimensional Hausdorff measure of level sets.

By constructing, for each selected interval, a piecewise‑defined function that coincides with the original function on the interior of the corresponding level region and is constant elsewhere, the authors obtain functions (f_i) with (\nabla f_i=\nabla f) on the region and zero outside. The coarea formula shows that the contribution of the omitted level sets to the (L^{1})-norm of the gradient can be made arbitrarily small. Consequently, any unit vector can be approximated by convex combinations of vectors whose gradients are supported on sets of arbitrarily small measure, which fulfills the geometric condition equivalent to the Daugavet property (Lemma 1.1 (ii)). Hence (\widetilde W^{1,1}(\mathbb{R}^{2})) is a Daugavet space.

In the second part, the authors turn to the usual Sobolev norm (|f|{W^{1,1}}:=|f|{L^{1}}+|\nabla f|_{L^{1}}). They demonstrate that (W^{1,1}(\mathbb{R}^{2})) fails the slice diameter‑two property, which is a necessary consequence of the Daugavet property. They construct a norm‑one functional \