Local Vs Nonlocal De Giorgi Classes: A brief guide in the homogeneous case

We give a brief and concise guide for the analysis of the local behavior of the elements of local and nonlocal homogeneous De Giorgi classes: local boundedness, local Hölder continuity and Harnack-type inequalities. In the local case, we promote a simplified itinerary in the classic theory, propaedeutic for the successive part; while in the nonlocal case, we gather recent new developments into an unitary and concise framework. Employing a suitable definition of De Giorgi classes, we show a new proof of the Harnack inequality, way easier than in the local case, that bypasses any sort of Krylov-Safonov argument or cube decomposition.