A note on Gehring's lemma.
We prove that several classical Banach space properties are equivalent to separability for the class of Lipschitz-free spaces, including Corson’s property (), Talponen’s countable separation property, or being a Gâteaux differentiability space. On the other hand, we single out more general properties where this equivalence fails. In particular, the question whether the duals of nonseparable Lipschitz-free spaces have a weak sequentially compact ball is undecidable in ZFC. Finally, we provide an...
The - regularity of the gradient of weak solutions to nonlinear elliptic systems is proved.
It is proved that for any , where is the Poulsen simplex, so that and , are smooth points, there is a rotation of carrying in .
The Poincaré inequality is extended to uniformly doubling metric-measure spaces which satisfy a version of the triangle comparison property. The proof is based on a generalization of the change of variables formula.