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Let (E, F, µ) be a probability space, (E, D(E)) a (not necessarily symmetric) Dirichlet form on L 2 (µ), and P t the associated sub-Markov semigroup. The equivalence of the following eight properties is studied: (i) the L 2 -uniform integrability of the unit ball in the Sobolev space; (ii) the super-Poincaré inequality (1.2); (iii) the F -Sobolev inequality (1.3); (iv) the L 2 -uniform integrability of P t ; (v) the L 2uniform integrability of the associated resolvents; (vi) the compactness ofdoi:10.1515/form.2002.013 fatcat:4qzf6h63vngdliujlpqsb4qzlu