Related papers: Measurable Differentiable Structures on Doubling M…

We adapt the classical theory of local well-posedness of evolution problems to cases in which the nonlinearity can be accurately quantified by two different norms. For ordinary differential equations, we consider $\dot{x} = f(x,x)$ for a…

In a complete metric space that is equipped with a doubling measure and supports a Poincar\'e inequality, we study strict subsets, i.e. sets whose variational capacity with respect to a larger reference set is finite, in the case $p=1$.…

As a continuation of \cite{NSY:local}, we mainly discuss the global structure of two-dimensional locally compact geodesically complete metric spaces with curvature bounded above. We first obtain the result on the Lipschitz homotopy…

We study Neumann functions for divergence form, second order elliptic systems with bounded measurable coefficients in a bounded Lipschitz domain or a Lipschitz graph domain. We establish existence, uniqueness, and various estimates for the…

We study a characterization of BV and Sobolev functions via nonlocal functionals in metric spaces equipped with a doubling measure and supporting a Poincar\'e inequality. Compared with previous works, we consider more general functionals.…

It is known that a Lipschitz continuous map from the Euclidean domain to a metric space is metrically differentiable almost everywhere. When the metric space is a Banach space dual to separable, the metric differential has its linear…

By differentiating a concavity principle arising from the Pr\'ekopa-Leindler inequality, we obtain a statement simultaneously strengthening the weighted boundary Poincar\'e inequality and the Brascamp-Lieb variance inequality. The resulting…

Let $X$ be a noncomplete metric space satisfying the usual (local) assumptions of a doubling property and a Poincar\'e inequality. We study extensions of Newtonian Sobolev functions to the completion $\widehat{X}$ of $X$ and use them to…

Given a Lipschitz map $f$ from a cube into a metric space, we find several equivalent conditions for $f$ to have a Lipschitz factorization through a metric tree. As an application we prove a recent conjecture of David and Schul. The…

We prove two-sided inequalities for the $L^p$-norm of a pushforward or pullback (with respect to an orientation-preserving diffeomorphism) on oriented volume and Riemannian manifolds. For a function or density on a volume manifold, these…

Given a pointed metric space $M$, we study when there exist $n$-dimensional linear subspaces of $\operatorname{Lip}_0(M)$ consisting of strongly norm-attaining Lipschitz functionals, for $n\in\mathbb{N}$. We show that this is always the…

We study the relationships between Gateaux, weak Hadamard and Frechet differentiability and their bornologies for Lipschitz and for convex functions. In particular, Frechet and weak Hadamard differentiabily coincide for all Lipschitz…

The main purpose of this paper is to investigate the behaviour of fractional integral operators associated to a measure on a metric space satisfying just a mild growth condition, namely that the measure of each ball is controlled by a fixed…

The theory of differentiation of measures originates from works of Besicovitch in the 1940's. His pioneering works, as well as subsequent developments of the theory, rely as fundamental tools on suitable covering properties. The first aim…

We prove optimal ${L}^2$ bounds for a pair of Hilbert space valued differentially subordinate martingales under a change of law. The change of law is given by a process called a weight and sharpness in this context refers to the optimal…

In this paper, we prove that if a metric measure space satisfies the volume doubling condition and the Gagliardo-Nirenberg inequality with the same exponent $n$ $(n\geq 2)$, then it has exactly the $n$-dimensional volume growth. Besides,…

In this note we study the Dirichlet problem associated with a version of prime end boundary of a bounded domain in a complete metric measure space equipped with a doubling measure supporting a Poincare inequality. We show the resolutivity…

We prove that the Lipschitz-free space over a countable proper metric space is isometric to a dual space and has the metric approximation property. We also show that the Lipschitz-free space over a proper ultrametric space is isometric to…

We introduce the property of countable separation for a locally convex Hausdorff space $X$ and relate it to the existence of a metrizable coarser topology. Building on this, we demonstrate how the separability of $X$ is equivalent to the…

In this paper, we prove that if a metric measure space satisfies the volume doubling condition and the Caffarelli-Kohn-Nirenberg inequality with same exponent n(n>1), then it has exactly n-dimensional volume growth. As application, we…