Related papers: Space-Filling vs. Luzin's Condition (N)
Let $\{x\_n\}\_{n\geq 0}$ be a sequence of $[0,1]^d$, $\{\lambda\_n\} \_{n\geq 0}$ a sequence of positive real numbers converging to 0, and $\delta>1$. Let $\mu$ be a positive Borel measure on $[0,1]^d$, $\rho\in (0,1]$ and $\alpha>0$.…
The main goals of this paper are: i) To develop an abstract differential calculus on metric measure spaces by investigating the duality relations between differentials and gradients of Sobolev functions. This will be achieved without…
We show that if $X$ is a uniformly perfect complete metric space satisfying the finite doubling property, then there exists a fully supported measure with lower regularity dimension as close to the lower dimension of $X$ as we wish.…
We prove that for all $b$, the Hausdorff dimension of the set of $m \times n$ matrices $\epsilon$-badly approximable for the target $b$ is not full. The doubly metric case follows. It was known that for almost every matrix $A$, the…
In the setting of a metric space equipped with a doubling measure that supports a Poincar\'e inequality, we show that a set $E$ is of finite perimeter if and only if $\mathcal H(\partial^1 I_E)<\infty$, that is, if and only if the…
We consider Non Autonomous Conformal Iterative Function Systems (NACIFS) and their limit set. Our main concern is harmonic measure and its dimensions : Hausdorff and Packing. We prove that this two dimensions are continuous under…
The possibility of matter coupling to two metrics at once is considered. This appears natural in the most general ghost-free, bimetric theory of gravity, where it unlocks an additional symmetry with respect to the exchange of the metrics.…
We consider a generalization of the hyperplane problem to arbitrary measures in place of volume and to sections of lower dimensions. We prove this generalization for unconditional convex bodies and for duals of bodies with bounded volume…
We prove the equivalence of two seemingly very different ways of generalising Rademacher's theorem to metric measure spaces. One such generalisation is based upon the notion of forming partial derivatives along a very rich structure of…
The possibility of introducing a positive metric on the states of the massless scalar field in 1+1 dimensions by mean of Krein spaces is examined. Two different realisations in Krein spaces for the massless scalar field are compared. It is…
The magnitude of a metric space is a real-valued function whose parameter controls the scale of the metric. A metric space is said to have the one-point property if its magnitude converges to 1 as the space is scaled down to a point. Not…
We study the problem of whether the curvature-dimension condition with negative values of the generalized dimension parameter is stable under a suitable notion of convergence. To this purpose, first of all we introduce an appropriate…
We discuss five simple functions on finite multisets of metric spaces. The first four are all metrics iff the underlying space is bounded and are complete metrics iff it is also complete. Two of them, and the fifth function, all generalise…
It is known [Dai and Sun, J. Funct. Anal. 268 (2015), 2464--2477] that there exist spectral measures with arbitrary Hausdorff dimensions, and it is natural to pose the question of whether similar phenomena occur for other dimensions of…
In this work, we are interested in characterizing typical (generic) dimensional properties of invariant measures associated with the full-shift system, $T$, in a product space whose alphabet is a perfect and separable metric space (thus,…
Given a nonnegative function $\psi : \N \to \R $, let $W(\psi)$ denote the set of real numbers $x$ such that $|nx -a| < \psi(n) $ for infinitely many reduced rationals $a/n (n>0) $. A consequence of our main result is that $W(\psi)$ is of…
The main purpose of the note is to explore the invariant properties of sphericalization and flattening and their applications in quasi-metric spaces. We show that sphericalization and flattening procedures on a quasimetric spaces preserving…
We define various notions of Lagrangian solution in physical space for 3-d incompressible geostrophic system with free upper boudary under different conditions for initial data,then prove their existence via the minimization with respect to…
We study the Hausdorff dimension of a measure related to a positive weak solution of a certain partial differential equation in a simply connected domain in the plane. Our work generalizes work of Lewis and coauthors when the measure is $p$…
We investigate variants of the Erd\H{o}s similarity problem for Cantor sets. We prove that under a mild Hausdorff or packing logarithmic dimension assumption, Cantor sets are not full measure universal, significantly improving the known…