On a quasi-additivity formula for the capacity on Ahlfors-regular spaces

In this work we prove formulas of quasi-additivity for the capacity associated to kernels of radial type in the setting of the boundary of a tree structure and in the setting of compact Ahlfors-regular spaces. We also define a notion of harmonic extension, to one additional variable, of a function defined over a compact Ahlfors-regular space, and we prove a result of tangential convergence of the harmonic extension to the values at the boundary.