Let ΔG be a sublaplacian on a Carnot group, and let μ be a local measure on the open set Ω ⊂ G. If u ∈ L1l oc(Ω) is such that −ΔGu = μ, u ≥ 0 on Ω, then μc ≥ 0, where μc is the concentrated component of μ with respect to the G-capacity. This extends to the Carnot group setting a result contained in [9].
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 H¨older continuity and Harnacktype 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.
