This paper introduces the Morrey Banach spaces on spaces of homogeneous type. We establish the Rubio de Francia extrapolation theory for Morrey Banach spaces on spaces of homogeneous type. Consequently, we obtain the boundedness of the Calderón–Zygmund operators and the Fefferman–Stein inequalities on Morrey Banach spaces on spaces of homogeneous type. We apply the results obtained for Morrey Banach space to study the Morrey spaces with variable exponents on spaces of homogeneous type.
We observe that the diameter of small (in a locally uniform sense) balls in C^1,1 sub-Riemannian manifolds equals twice the radius. We also prove that, when the regularity of the structure is further lowered to C^0, the diameter is arbitrarily close to twice the radius. Both results hold independently of the bracket-generating condition.
The Gromov-Wasserstein (GW) distances define a family of metrics, based on ideas from optimal transport, which enable comparisons between probability measures defined on distinct metric spaces. They are particularly useful in areas such as network analysis and geometry processing, as computation of a GW distance involves solving for registration between the objects which minimizes geometric distortion. Although GW distances have proven useful for various applications in the recent machine learning literature, it has been observed that they are inherently sensitive to outlier noise and cannot accommodate partial matching. This has been addressed by various constructions building on the GW framework; in this article, we focus specifically on a natural relaxation of the GW optimization problem, introduced by Chapel et al., which is aimed at addressing exactly these shortcomings. Our goal is to understand the theoretical properties of this relaxed optimization problem, from the viewpoint of metric geometry. While the relaxed problem fails to induce a metric, we derive precise characterizations of how it fails the axioms of non-degeneracy and triangle inequality. These observations lead us to define a novel family of distances, whose construction is inspired by the Prokhorov and Ky Fan distances, as well as by the recent work of Raghvendra et al. on robust versions of classical Wasserstein distance. We show that our new distances define true metrics, that they induce the same topology as the GW distances, and that they enjoy additional robustness to perturbations. These results provide a mathematically rigorous basis for using our robust partial GW distances in applications where outliers and partial matching are concerns.
We disprove the conjecture of Georgakopoulos and Papasoglu that a graph with no K-fat H minor is quasi-isometric to a graph with no H minor. On the other hand, we show that the following weakening holds: any graph with no K-fat H minor is quasi-isometric to a graph with no 3-fat H minor.
In this article, using the existence of infinite equidistant subsets of closed balls, we first characterize the injectivity of ultrametric spaces for finite ultrametric spaces. This method also gives characterizations of the Urysohn universal ultrametric spaces. As an application, we find that the operations of the Cartesian product and the hyperspaces preserve the structures of the Urysohn universal ultrametric spaces. Namely, let ( X , d ) \left(X,d) be the Urysohn universal ultrametric space. Then, we show that ( X × X , d × d ) \left(X\times X,d\times d) is isometric to ( X , d ) \left(X,d) , and show that the hyperspace consisting of all non-empty compact subsets of ( X , d ) \left(X,d) and symmetric products of ( X , d ) \left(X,d) are isometric to ( X , d ) \left(X,d) . We also establish that every complete ultrametric space injective for finite ultrametric space contains a subspace isometric to ( X , d ) \left(X,d) .
