Resistance conditions, Poincaré inequalities, the Lip-lip condition and Hardy's inequalities

This note investigates weaker conditions than a Poincaré inequality in analysis on metric measure spaces. We discuss two resistance conditions which are stated in terms of capacities. We show that these conditions can be characterized by versions of Sobolev–Poincaré inequalities. As a consequence, we obtain so-called Lip-lip condition related to pointwise Lipschitz constants. Moreover, we show that the pointwise Hardy inequalities and uniform fatness conditions are equivalent under an appropriate resistance condition.