Axiomatic theory of Sobolev spaces (Q1348585)

Let \((X,d,\mu)\) be a set \(X\), equipped with a metric \(d\) and a Borel measure \(\mu\). For any \(u\in L^{\text{loc}}_p (X)\), so-called pseudo-gradients \(D[u]\) are introduced axiomatically, and on this basis \(p\)-Dirichlet energies, which, in turn, are used to introduce Sobolev spaces \(W^1_p (X)\). The aim of this paper is a systematic study of these spaces and to show that other well-known definitions of Sobolev spaces of first order are special cases of this axiomatic approach.