We give suffcient conditions for Sobolev and Lipschitz functions in terms of their lower scaled-oscillation. The sharpness of these conditions is shown by examples. Our examples also show that a… Expand

Mathematical Proceedings of the Cambridge Philosophical Society

2016

Abstract We show that every continuous simple curve with σ-finite length has a tangent at positively many points. We also apply this result to functions with finite lower scaled oscillation; and… Expand

We consider certain properties of maps of class C 2 from R d to R d 1 that are strictly related to Sard's theorem, and show that some of them can be extended to Lipschitz maps, while others still… Expand

I report on some recent extensions of the Lusin N-property and the Sard theorem for Sobolev maps, which have been obtained in a joint work with M. Csornyei, E. D'Aniello, and B. Kirchheim. Our… Expand

We characterize the autonomous, divergence-free vector elds b on the plane such that the Cauchy problem for the continuity equation @tu + div (bu) = 0 admits a unique bounded solution (in the weak… Expand

Acknowledgements Basic notation Introduction 1. General measure theory 2. Covering and differentiation 3. Invariant measures 4. Hausdorff measures and dimension 5. Other measures and dimensions 6.… Expand