Gareth Speight

Differentiability, porosity and doubling in metric measure spaces

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.

The p-weak gradient depends on p

Given a>0, we construct a weighted Lebesgue measure on R^n for which the family of non constant curves has p-modulus zero for p\leq 1+a but the weight is a Muckenhoupt A_p weight for p>1+a. In particular, the p-weak gradient is trivial for small p but non trivial for large p. This answers an open question posed by several authors. We also give a full description of the p-weak gradient for any locally finite Borel measure on the real line.

Surfaces meeting porous sets in positive measure

Let X be a Banach space and 2 < n < dimX. We show there exists a directionally porous set P in X for which the set of C1 surfaces of dimension n meeting P in positive measure is not meager. If X is separable, this leads to a decomposition of X into the union of a σ-directionally porous set and a set which is null on residually many C1 surfaces of dimension n. This is of interest in the study of Γn-null and Γ-null sets and their applications to differentiability of Lipschitz functions.

A measure zero universal differentiability set in the Heisenberg group

We show that the Heisenberg group

Lusin Approximation and Horizontal Curves in Carnot Groups

\mathbb {H}^n

Tensorization of Cheeger energies, the space H1,1 and the area formula for graphs

Hn contains a measure zero set N such that every Lipschitz function

Lusin approximation for horizontal curves in step 2 Carnot groups

f:\mathbb {H}^n \rightarrow \mathbb {R}