Covering Theorems, Inequalities on Metric Spaces and applications to PDE's

We establish a covering lemma of Besicovitch type for metric balls in the setting of Hölder quasimetric spaces of homogenous type and use it to prove a covering theorem for measurable sets. For families of measurable functions, we introduce the notions of power decay, critical density and double ball property and with the aid of the covering theorem we show how these notions are related. Next we present an axiomatic procedure to establish Harnack inequality that permits to handle both divergence and non divergence linear equations.