We show that the Heisenberg group contains a measure zero set N such that every real-valued Lipsc... more We show that the Heisenberg group contains a measure zero set N such that every real-valued Lipschitz function is Pansu differentiable at a point of N. The proof adapts the construction of small 'universal differentiability sets' in the Euclidean setting: we find a point of N and a horizontal direction where the directional derivative in horizontal directions is almost locally maximal, then deduce Pansu differentiability at such a point.
We show that given an absolutely continuous horizontal curve g in the Heisenberg group, there is ... more We show that given an absolutely continuous horizontal curve g in the Heisenberg group, there is a C1 horizontal curve G such that G=g and G'=g' outside a set of small measure. Conversely, we construct an absolutely continuous horizontal curve in the Engel group with no C1 horizontal approximation.
First we study in detail the tensorization properties of weak gradients in metric measure spaces ... more First we study in detail the tensorization properties of weak gradients in metric measure spaces (X,d,m). Then, we compare potentially different notions of Sobolev space H^{1,1}(X,d,m) and of weak gradient with exponent 1. Eventually we apply these results to compare the area functional with the perimeter of the subgraph of f, in the same spirit as the classical theory.
We investigate weighted Sobolev spaces on metric measure spaces (X,d,m). Denoting by rho the weig... more We investigate weighted Sobolev spaces on metric measure spaces (X,d,m). Denoting by rho the weight function, we compare the space W^{1,p}(X,d,rho m) (which always concides with the closure H^{1,p}(X,d,rho m) of Lipschitz functions) with the weighted Sobolev spaces W^{1,p}_{rho}(X,d,m) and H^{1,p}_{rho}(X,d,m) defined as in the Euclidean theory of weighted Sobolev spaces. Under mild assumptions on the metric measure structure and on the weight we show that W^{1,p}(X,d,rho m)=H^{1,p}_{rho}(X,d,m). We also adapt results by Muckenhoupt and a recent paper of Zhikov to the metric measure setting, considering appropriate conditions on rho that ensure the equality W^{1,p}_{rho}(X,d,m)=H^{1,p}_{rho}(X,d,m).
Given a>0, we construct a weighted Lebesgue measure on R^n for which the family of non constant c... more 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.
We show that if n>1 then there exists a Lebesgue null set in R^n containing a point of differenti... more We show that if n>1 then there exists a Lebesgue null set in R^n containing a point of differentiability of each Lipschitz function f:R^n→R^(n-1); in combination with the work of others, this completes the investigation of when the classical Rademacher theorem admits a converse. Avoidance of σ-porous sets, arising as irregular points of Lipschitz functions, plays a key role in the proof.
We investigate differences between upper and lower porosity. In finite dimensional Banach spaces ... more We investigate differences between upper and lower porosity. In finite dimensional Banach spaces every upper porous set is directionally upper porous. We show the situation is very different for lower porous sets; there exists a lower porous set in R^2 which is not even a countable union of directionally lower porous sets.
Let X be a Banach space and 2 < n < dimX. We show there exists a directionally porous set P in X ... more 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 C^1 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 C^1 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.
We show if a metric measure space admits a differentiable structure then porous sets have measure... more 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.
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Papers by Gareth Speight
construct an absolutely continuous horizontal curve in the Engel group with no C1 horizontal approximation.