Bessel potentials in Ahlfors regular metric spaces (Q308995)

Let \(N>0\). An Ahlfors regular \(N\)-set \((X, d, m)\) is a set \(X\) furnished with a metric \(d\) and a measure \(m\) such that \[ m \big( B(x,2r) \big) \sim r^N \] for each \(x\in X\) and \(0 <r<\)diam\(X\). Here \(B(x,2r)\) is a ball of radius \(2r\) centered at \(x\in X\). This covers Euclidean \(N\)-spaces, but also numerous distinguished fractals. There are several proposals in the literature how to define (fractional) Sobolev spaces and Besov spaces on Ahlfors regular \(N\)-sets. The paper contributes to this topic introducing related spaces in terms of Bessel potentials (guided by the well-known procedure in Euclidean \(N\)-space). This includes the notation of fractional derivatives and related embedding assertions.