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Generalized capacity, Harnack inequality and heat kernels of Dirichlet forms on metric measure spaces. (English) Zbl 1331.35152
In this interesting paper, the authors are concerned with heat kernel estimates in the setting of Dirichlet forms on metric measure spaces. More precisely, they give necessary and sufficient conditions for sub-Gaussian estimates of the heat kernel of a strongly local regular Dirichlet form. The conditions for two-sided estimates are given in terms of the generalized capacity inequality and the Poincaré inequality. The conditions for the upper bound are given in terms of the generalized capacity inequality and the Faber-Krahn inequality. The main difficulty lies in obtaining the elliptic Harnack inequality under these assumptions.

MSC:
35K08 Heat kernel
28A80 Fractals
31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
35J08 Green’s functions for elliptic equations
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
47D07 Markov semigroups and applications to diffusion processes
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