More Like this

1. The space is primary for 1 < p < ∞

   https://doi.org/10.1017/fms.2022.25  

   Lechner, Richard; Motakis, Pavlos; Müller, Paul F.X.; Schlumprecht, Thomas (January 2022, Forum of Mathematics, Sigma)

   Abstract The classical Banach space $$L_1(L_p)$$ consists of measurable scalar functions f on the unit square for which $$ \begin{align*}\|f\| = \int_0^1\Big(\int_0^1 |f(x,y)|^p dy\Big)^{1/p}dx < \infty.\end{align*} $$ We show that $$L_1(L_p) (1 < p < \infty )$$ is primary, meaning that whenever $$L_1(L_p) = E\oplus F$$ , where E and F are closed subspaces of $$L_1(L_p)$$ , then either E or F is isomorphic to $$L_1(L_p)$$ . More generally, we show that $$L_1(X)$$ is primary for a large class of rearrangement-invariant Banach function spaces. 

   more » « less
2. Two-Sided Kirszbraun Theorem

   https://doi.org/10.4230/LIPIcs.SoCG.2021.13  

   Backurs, Arturs; Mahabadi, Sepideh; Makarychev, Konstantin; Makarychev, Yury (June 2021, Leibniz international proceedings in informatics)

   Buchin, Kevin; Colin de Verdiere, Eric (Ed.)

   In this paper, we prove a two-sided variant of the Kirszbraun theorem. Consider an arbitrary subset X of Euclidean space and its superset Y. Let f be a 1-Lipschitz map from X to ℝ^m. The Kirszbraun theorem states that the map f can be extended to a 1-Lipschitz map ̃ f from Y to ℝ^m. While the extension ̃ f does not increase distances between points, there is no guarantee that it does not decrease distances significantly. In fact, ̃ f may even map distinct points to the same point (that is, it can infinitely decrease some distances). However, we prove that there exists a (1 + ε)-Lipschitz outer extension f̃:Y → ℝ^{m'} that does not decrease distances more than "necessary". Namely, ‖f̃(x) - f̃(y)‖ ≥ c √{ε} min(‖x-y‖, inf_{a,b ∈ X} (‖x - a‖ + ‖f(a) - f(b)‖ + ‖b-y‖)) for some absolutely constant c > 0. This bound is asymptotically optimal, since no L-Lipschitz extension g can have ‖g(x) - g(y)‖ > L min(‖x-y‖, inf_{a,b ∈ X} (‖x - a‖ + ‖f(a) - f(b)‖ + ‖b-y‖)) even for a single pair of points x and y. In some applications, one is interested in the distances ‖f̃(x) - f̃(y)‖ between images of points x,y ∈ Y rather than in the map f̃ itself. The standard Kirszbraun theorem does not provide any method of computing these distances without computing the entire map ̃ f first. In contrast, our theorem provides a simple approximate formula for distances ‖f̃(x) - f̃(y)‖. 

   more » « less
3. Examples of non-Dini domains with large singular sets

   https://doi.org/10.1515/ans-2022-0058  

   Kenig, Carlos; Zhao, Zihui (January 2023, Advanced Nonlinear Studies)

   Abstract Let u u be a nontrivial harmonic function in a domain D ⊂ R d D\subset {{\mathbb{R}}}^{d} , which vanishes on an open set of the boundary. In a recent article, we showed that if D D is a C 1 {C}^{1} -Dini domain, then, within the open set, the singular set of u u , defined as { X ∈ D ¯ : u ( X ) = 0 = ∣ ∇ u ( X ) ∣ } \left\{X\in \overline{D}:u\left(X)=0=| \nabla u\left(X)| \right\} , has finite ( d − 2 ) \left(d-2) -dimensional Hausdorff measure. In this article, we show that the assumption of C 1 {C}^{1} -Dini domains is sharp, by constructing a large class of non-Dini (but almost Dini) domains whose singular sets have infinite ℋ d − 2 {{\mathcal{ {\mathcal H} }}}^{d-2} -measures. 

   more » « less
4. Boundary behavior of compact manifolds with scalar curvature lower bounds and static quasi-local mass of tori

   https://doi.org/10.1090/proc/17223  

   Alaee, Aghil; Hung, Pei-Ken; Khuri, Marcus (March 2025, Proceedings of the American Mathematical Society)

   A classic result of Shi and Tam states that a 2-sphere of positive Gauss and mean curvature bounding a compact 3-manifold with nonnegative scalar curvature must have total mean curvature not greater than that of the isometric embedding into Euclidean 3-space, with equality only for domains in this reference manifold. We generalize this result to 2-tori of Gauss curvature greater than $-<\#comment/>1$, which bound a compact 3-manifold having scalar curvature not less than $-<\#comment/>6$and at least one other boundary component satisfying a ‘trapping condition’. The conclusion is that the total weighted mean curvature is not greater than that of an isometric embedding into the Kottler manifold, with equality only for domains in this space. Examples are given to show that the assumption of a secondary boundary component cannot be removed. The result gives a positive mass theorem for the static Brown-York mass of tori, in analogy to the Shi-Tam positivity of the standard Brown-York mass, and represents the first such quasi-local mass positivity result for nonspherical surfaces. Furthermore, we prove a Penrose-type inequality in this setting. 

   more » « less
5. Rectifiability of planes and Alberti representations

   https://doi.org/10.2422/2036-2145.201701_015  

   David, Guy C.; Kleiner, Bruce (January 2019, Annali della Scuola normale superiore di Pisa. Classe di scienze)

   We study metric measure spaces that have quantitative topological control, as well as a weak form of differentiable structure. In particular, let X be a pointwise doubling metric measure space. Let U be a Borel subset on which the blowups of X are topological planes. We show that U can admit at most 2 independent Alberti representations. Furthermore, if U admits 2 Alberti representations, then the restriction of the measure to U is 2-rectifiable. This is a partial answer to the case n=2 of a question of the second author and Schioppa. 

   more » « less