Harnack's Inequality for Parabolic De Giorgi Classes in Metric Spaces
  In this paper we study problems related to parabolic partial differential equations in metric measure spaces equipped with a doubling measure and supporting a Poincare' inequality. We give a definition of parabolic De Giorgi classes and compare this notion with that of parabolic quasiminimizers. The main result, after proving the local boundedness, is a scale and location invariant Harnack inequality for functions belonging to parabolic De Giorgi classes. In particular, the results hold true for parabolic quasiminimizers. 
J.Kinnunen, N.Marola, M.Miranda jr., F.Paronetto
math.AP