Sobolev and dirichlet spaces over maps between metric spaces

We construct the (1, p)-Sobolev spaces and energy functional over L^p-maps between metric spaces for p ≥ 1 under the condition so-called strong measure contraction property of Bishop-Gromov type (SMCPBG in short). Under this property, we also prove the existence of energy measures, and the weak Poincaré inequality, which extends some parts of the results of Schoen and Sturm. Alexandrov spaces are included in this formulation and we show that the constructed Sobolev spaces are compatible with (1, p)-Sobolev spaces over L^p-functions on Alexandrov spaces.