Besov spaces via hyperbolic fillings

We establish a new characterization of the homogeneous Besov spaces ${\dot{\mathcal{B}}}_{p,q}^s(Z)$ with smoothness $s\in (0,1)$ in the setting of doubling metric measure spaces $(Z,d,\mu )$. The characterization is given in terms of a hyperbolic filling of the metric space $(Z,d)$, a construction which has previously appeared in the context of other function spaces in [3,1,2]. We use the characterization to obtain results concerning the density of Lipschitz functions in the spaces ${\dot{\mathcal{B}}}_{p,q}^s(Z)$ and a general complex interpolation formula in the smoothness range $0<s<1$.