arXiv subjects

David Bate

Publications and source records attributed to David Bate.

26 records · Page 2Linked to original sources

arXiv2020

Purely unrectifiable metric spaces and perturbations of Lipschitz functions

We characterise purely $n$-unrectifiable subsets $S$ of a complete metric space $X$ with finite Hausdorff $n$-measure by studying arbitrarily small perturbations of elements of the set of all bounded 1-Lipschitz functions $f\colon X \to \mathbb R^m$ with respect to the supremum norm. In one such characterisation it is shown that, if $S$ has positive lower density almost everywhere, then the set of all $f$ with $\mathcal H^n(f(S))=0$ is residual. Conversely, if $E\subset X$ is $n$-rectifiable with $\mathcal H^n(E)>0$, the set of all $f$ with $\mathcal H^n(f(E))>0$ is residual. These results provide a replacement for the Besicovitch-Federer projection theorem in arbitrary metric spaces, which is known to be false outside of Euclidean spaces.

math.MG↗

arXiv2019

Quantitative absolute continuity of planar measures with two independent Alberti representations

We study measures $μ$ on the plane with two independent Alberti representations. It is known, due to Alberti, Csörnyei, and Preiss, that such measures are absolutely continuous with respect to Lebesgue measure. The purpose of this paper is to quantify the result of A-C-P. Assuming that the representations of $μ$ are bounded from above, in a natural way to be defined in the introduction, we prove that $μ\in L^{2}$. If the representations are also bounded from below, we show that $μ$ satisfies a reverse Hölder inequality with exponent $2$, and is consequently in $L^{2 + ε}$ by Gehring's lemma. A substantial part of the paper is also devoted to showing that both results stated above are optimal.

math.CA↗

arXiv2018

Differentiability and Poincaré-type inequalities in metric measure spaces

We demonstrate the necessity of a Poincaré type inequality for those metric measure spaces that satisfy Cheeger's generalization of Rademacher's theorem for all Lipschitz functions taking values in a Banach space with the Radon-Nikodym property. This is done by showing the existence of a rich structure of curve fragments that connect near by points, similar in nature to Semmes's pencil of curves for the standard Poincaré inequality. Using techniques similar to Cheeger-Kleiner, we show that our conditions are also sufficient. We also develop another characterization of "RNP Lipschitz differentiability spaces" by connecting points by curves that form a rich structure of partial derivatives that were first discussed in work by the first author.

math.MG↗

arXiv2018

On the conformal dimension of product measures

Given a compact set $E \subset \mathbb{R}^{d - 1}$, $d \geq 1$, write $K_{E} := [0,1] \times E \subset \mathbb{R}^{d}$. A theorem of C. Bishop and J. Tyson states that any set of the form $K_{E}$ is minimal for conformal dimension: if $(X,d)$ is a metric space and $f \colon K_{E} \to (X,d)$ is a quasisymmetric homeomorphism, then $$\dim_{\mathrm{H}} f(K_{E}) \geq \dim_{\mathrm{H}} K_{E}.$$ We prove that the measure-theoretic analogue of the result is not true. For any $d \geq 2$ and $0 \leq s < d - 1$, there exist compact sets $E \subset \mathbb{R}^{d - 1}$ with $0 < \mathcal{H}^{s}(E) < \infty$ such that the conformal dimension of $ν$, the restriction of the $(1 + s)$-dimensional Hausdorff measure on $K_{E}$, is zero. More precisely, for any $ε> 0$, there exists a quasisymmetric embedding $F \colon K_{E} \to \mathbb{R}^{d}$ such that $\dim_{\mathrm{H}} F_{\sharp}ν< ε$.

math.CA↗

arXiv2015

The Besicovitch-Federer projection theorem is false in every infinite dimensional Banach space

We construct a purely unrectifiable set of finite $\mathcal H^1$-measure in every infinite dimensional separable Banach space $X$ whose image under every $0\neq x^*\in X^*$ has positive Lebesgue measure. This demonstrates completely the failure of the Besicovitch-Federer projection theorem in infinite dimensional Banach spaces.

math.FA↗

arXiv2015

Characterizations of rectifiable metric measure spaces

We characterize $n$-rectifiable metric measure spaces as those spaces that admit a countable Borel decomposition so that each piece has positive and finite $n$-densities and one of the following: is an $n$-dimensional Lipschitz differentiability space; has $n$-independent Alberti representations; satisfies David's condition for an $n$-dimensional chart. The key tool is an iterative grid construction which allows us to show that the image of a ball with a high density of curves from the Alberti representations under a chart map contains a large portion of a uniformly large ball and hence satisfies David's condition. This allows us to apply previously known "biLipschitz pieces" results on the charts.

math.MG↗

arXiv2013

Structure of measures in Lipschitz differentiability spaces

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 Lipschitz curves in a way analogous to the differentiability theory of Euclidean spaces. This approach to differentiability in this generality appears here for the first time and by examining this structure further, we naturally arrive to several descriptions of Lipschitz differentiability spaces.

math.MG↗

arXiv2011

Differentiability, Porosity and Doubling in Metric Measure Spaces

We show if a metric measure space admits a differentiable structure then porous sets have measure zero and hence the measure is pointwise doubling. We then give a construction to show if we only require an approximate differentiable structure the measure need no longer be pointwise doubling.

math.MG↗