Local properties of fractional parabolic De Giorgi classes of order s,p

Abstract

We discuss homogeneous fractional parabolic De Giorgi classes of order s,p and establish the nonlocal weak Harnack estimate with a time gap of order ϱsp, whereas the elliptic case were studied in Cozzi (2017) [4]. The proof relies on a heavy expansion of positivity in the nonlocal framework by-passing the Krylov-Safonov covering argument. Besides, we shall prove the local boundedness and the semicontinuity property. Our results come along with the De Giorgi method.