Differentiability, porosity and doubling in metric measure spaces

  title={Differentiability, porosity and doubling in metric measure spaces},
  author={David Bate and Gareth Speight},
  journal={arXiv: Metric Geometry},
We show that 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 that if we require only an approximate differentiable structure, the measure need no longer be pointwise doubling. 


We use porosity to study differentiability of Lipschitz maps on Carnot groups. Our first result states that directional derivatives of a Lipschitz function act linearly outside a σ-porous set. The

On Whitney-type Characterization of Approximate Differentiability on Metric Measure Spaces

Abstract We study approximately differentiable functions on metric measure spaces admitting a Cheeger differentiable structure. The main result is a Whitney-type characterization of approximately

An example of a differentiability space which is PI-unrectifiable

We construct a (Lipschitz) differentiability space which has at generic points a disconnected tangent and thus does not contain positive measure subsets isometric to positive measure subsets of

Infinitesimal Structure of Differentiability Spaces, and Metric Differentiation

Abstract We prove metric differentiation for differentiability spaces in the sense of Cheeger [10, 14, 27]. As corollarieswe give a new proof of one of the main results of [14], a proof that the

Porosity, Differentiability and Pansu’s Theorem

We use porosity to study differentiability of Lipschitz maps on Carnot groups. Our first result states that directional derivatives of a Lipschitz function act linearly outside a $$\sigma $$σ-porous

N-Lusin property in metric measure spaces: A new sufficient condition

In this work, we are concerned with the study of the N-Lusin property in metric measure spaces. A map satisfies that property if sets of measure zero are mapped to sets of measure zero. We prove a

Measurable differentiable structures on doubling metric spaces

  • Jasun Gong
  • Mathematics
    Transactions of the American Mathematical Society
  • 2019
On metric spaces equipped with doubling measures, we prove that a differentiability theorem holds for Lipschitz functions if and only if the space supports nontrivial (metric) derivations in the

Structure of measures in Lipschitz differentiability spaces

We prove the equivalence of two seemingly very di erent ways of generalising Rademacher's theorem to metric measure spaces. The rst was introduced by Cheeger and is based upon di erentiation with

The Lip-lip condition on metric measure spaces

On complete metric spaces that support doubling measures, we show that the validity of a Rademacher theorem for Lipschitz functions can be characterised by Keith's "Lip-lip" condition. Roughly



A differentiable structure for metric measure spaces

Differentiability of Lipschitz Functions on Metric Measure Spaces

Abstract. ((Without Abstract)).

Porosity, σ-porosity and measures

We show that given a σ-finite Borel regular measure μ in a metric space X, every σ-porous subset of X of finite measure can be approximated by strongly porous sets. It follows that every σ-porous set

Measurable differentiable structures and the Poincaré inequality

The main result of this paper is an improvement for the differentiable structure presented in Cheeger [2, Theorem 4.38] under the same assumptions of [2] that the given metric measure space admits a

Ahlfors Q-regular spaces with arbitrary Q > 1 admitting weak Poincaré inequality

Abstract. Given Q > 1, we construct an Ahlfors Q-regular space that admits a weak (1,1)-Poincaré inequality.

Geometric Measure Theory

Introduction Chapter 1 Grassmann algebra 1.1 Tensor products 1.2 Graded algebras 1.3 Teh exterior algebra of a vectorspace 1.4 Alternating forms and duality 1.5 Interior multiplications 1.6 Simple

Porosity, σ−porosity and measures, Nonlinearity, Volume

  • Number
  • 2003

On approximate identities in abstract, measure spaces

Geometric Measure Theory, Classics

  • 1996