Dirichlet forms and convergence of Besov norms on self-similar sets

. Let B σ 2 , ∞ , B σ 2 , 2 denote the Besov spaces defined on a compact set K ⊂ R d that is equipped with an α -regular measure µ ( K is called an α -set). The critical exponent σ ∗ is the supremum of the σ such that B σ 2 , 2 ∩ C ( K ) is dense in C ( K ) . It is known that B σ 2 , 2 is the domain of a non-local regular Dirichlet form, and for certain standard self-similar set, B σ ∗ 2 , ∞ is the domain of a local regular Dirichlet form. In this paper, we study, on the homogenous p.c.f. self-similar sets (which are α -sets), the convergence of the B σ 2 , 2 -norm to the B σ ∗ 2 , ∞ -norm as σ (cid:37) σ ∗ and the associated Dirichlet forms. The theorem extends a celebrate result of Bourgain, Brezis and Mironescu [4] on Euclidean domains, and the more recent results on some self-similar sets [10, 22, 29].