Differentiability of Lipschitz Functions in Lebesgue Null Sets

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 mapping from R^n to 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 sigma-porous sets, arising as irregular points of Lipschitz functions, plays a key role in the proof.Comment: 33 pages. Corrected minor misprints and added more detail to the proofs of Lemma 3.2 and Lemma 8.

Porosity and Differentiability of Lipschitz Maps from Stratified Groups to Banach Homogeneous Groups

Let $f$ be a Lipschitz map from a subset $A$ of a stratified group to a Banach homogeneous group. We show that directional derivatives of $f$ act as homogeneous homomorphisms at density points of $A$ outside a $\sigma$-porous set. At density points of $A$ we establish a pointwise characterization of differentiability in terms of directional derivatives. We use these new results to obtain an alternate proof of almost everywhere differentiability of Lipschitz maps from subsets of stratified groups to Banach homogeneous groups satisfying a suitably weakened Radon-Nikodym property. As a consequence we also get an alternative proof of Pansu's Theorem.Comment: 23 page

Weighted Sobolev Spaces on Metric Measure Spaces

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 recent work by Zhikov to the metric measure setting, considering appropriate conditions on $\rho$ that ensure the equality $W^{1,p}_\rho(X,d,m)=H^{1,p}_\rho(X,d,m)$.Comment: 26 page