Non-local Dirichlet forms generated by pseudodifferential operators on compact abelian groups

Let G_i, 1 ≤ i ≤ n, be compact abelian groups and let Γ_i , 1 ≤ i ≤ n, be countable dual groups. We consider G = G₁ ⊕ G₂ ⊕ … ⊕ G_n and Γ = Γ₁ ⊕ Γ₂ ⊕ … ⊕ Γ_n . For 1 ≤ j ≤ n, let a_j be a negative definite function on Γ_j and a (γ) = . For φ ∈ S (G), the set of all generalized trigonometrical polynomials on G, we define , where (γ) = a_j (γ_j) (γ), 1 ≤ j ≤ n. Then is a Dirichlet form with the domain on L² (G). (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)