Generalized Sobolev classes on metric measure spaces (Q2388203)

Let \((X,d,\mu)\) be a homogeneous space consisting of a set \(X\), a quasi-metric \(d\) and a regular doubling Borel measure \(\mu\). The paper deals with generalised Sobolev spaces consisting of all \(f \in L_p (X)\), \(1<p< \infty\), such that \[ | f(x) - f(y)| \leq \eta \left( d(x,y) \right) \cdot \left( g(x) + g(y) \right) \] for some \(g \in L_p (X)\) and \(\mu\)-almost all \(x\in X\), \(y \in X\). Here \(\eta\) is a positive increasing function on \([0, \infty)\) with some properties, covering especially \(\eta(t) = t^\alpha\), \( \alpha >0\).