The Stepanov differentiability theorem in metric measure spaces
The framework of the paper is given by a doubling metric measure space \(X\) with a strong measurable differentiable structure; examples of such spaces are given by doubling metric measure spaces supporting a Poincaré inequality, as proved by Cheeger, or doubling metric measure spaces with a chunky measure, satisfying a Lip-lip condition, as proved by Keith. In this setting, the authors prove the Stepanov differentiability theorem, that is any function \(f:X\to R\) is a.e. differentiable in the set \[ S(f)=\left\{x\in X:\limsup _{y\to x}\frac{| f(y)-f(x)| }{d(x,y)}<+\infty \right\}. \] The authors apply this result to prove an analogue of the Calderón's differentiability theorem, that is the a.e. differentiability of some \(p\)-Sobolev classes of functions. The Sobolev classes that are investigated are essentially of three types; couple of functions satisfying a \((1,p)\)-Poincaré inequality introduced by \textit{J. Hajłasz} and \textit{P. Koskela} [Mem. Am. Math. Soc. 688 (2000; Zbl 0954.46022)], the Haiłasz Sobolev space \(M^{1,p}\) introduced by \textit{P. Hajłasz} [Potential Anal. 5, No. 4, 403--415 (1996; Zbl 0859.46022)] and the Newtonian-Sobolev space \(N^{1,p}\) introduced by \textit{N. Shanmugalingam} [Rev. Mat. Iberoam. 16, No. 2, 243--279 (2000; Zbl 0974.46038)].
scientific article; zbMATH DE number 2118170