Cheeger-differentiability of maps belonging to certain Sobolev spaces (Q2925393)

Let \(X\) be a metric measure space supporting a \((1,p)\)-Poincaré inequality for a measure \(\mu\) with doubling constant \(\beta\) and ``dimension'' \(N:=\log_2\beta\). Let \(E\) be a Banach space with ``good finite-dimensional approximations'' (GFDA). The first main result of the paper says that every function in the Hajłasz-Sobolev space \(M^{1,q}(X,E)\), where \(q>\max(N,p)\), is \(\mu\)-a.e. differentiable in the sense of \textit{J. Cheeger} [Geom. Funct. Anal. 9, No. 3, 428--517 (1999; Zbl 0942.58018)].NEWLINENEWLINEThis relies on quite heavy preliminaries from the theories of both metric and Banach spaces. Much of the argument consists of relatively ``soft'' chaining of several hard theorems from Cheeger [loc. cit.] and others, but in the core of the proof one also meets the condition \(q>\max(N,p)\) in an explicit computation to ensure a required decay in a ``seven-epsilon estimate''.NEWLINENEWLINEIn the end it is sketched how to replace the GFDA assumption by the more general Radon-Nikodým property (RNP), if \(\mu\) is complete. This is based on the fact that separable spaces with RNP have the ``asymptotic norming property'' (ANP), which assumes the role of GFDA in the proof.