For symmetric pure jump Dirichlet forms, two-sided heat kernel estimates, jumping kernel bounds, and Sobolev/Faber-Krahn/Poincaré inequalities are mutually stable under the two-scale assumptions.
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Heat kernel estimates for general symmetric pure jump Dirichlet forms
For symmetric pure jump Dirichlet forms, two-sided heat kernel estimates, jumping kernel bounds, and Sobolev/Faber-Krahn/Poincaré inequalities are mutually stable under the two-scale assumptions.