The Yamabe problem on Dirichlet spaces

Abstract: We continue our previous work studying critical exponent semilinear elliptic (and subelliptic) problems which generalize the classical Yamabe problem. In [3] the focus was on metric-measure spaces with an almost smooth' structure, with stratified spaces furnishing the key examples. The criterion for solvability there is phrased in terms of a strict inequality of the global Yamabe invariant with alocal Yamabe invariant', which captures information about the local singular structure. All of this is generalized here to the setting of Dirichlet spaces which admit a Sobolev inequality and satisfy a few other mild hypotheses. Applications include a new approach to the nonspherical part of the CR Yamabe problem.