Harnack inequality for parabolic quasi minimizers on metric spaces

Abstract

We are concerned with local parabolic quasi-minimizers $u$ on metric measure spaces. The measure space is assumed to fulfill a doubling and an annular-decay property and to support a weak (1, $p$)-Poincaré inequality, while $u$ is associated to a Carathéodory integrand $f$ obeying $p$-growth assumptions for $p\ge 2$. We are able to show a parabolic Harnack inequality under these assumptions. The quadratic case $p=2$ has already been considered in [25], whereas the superquadratic case, at least to our knowledge, has not even been treated in the euclidean setting. The proof following the ideas of DiBenedetto, Gianazza and Vespri in [9].