Spaces of generalized smoothness on h-sets and related Dirichlet forms

Abstract

Abstract. The paper is devoted to spaces of generalized smoothness on so-called h-sets. First we find quarkonial representations of isotropic spaces of generalized smoothness on R n and on an h-set. Then we investigate representations of such spaces via differences, which are very helpful when we want to find an explicit representation of the domain of a Dirichlet form on h-sets. We prove that both representations are equivalent, and also find the domain of some time-changed Dirichlet form on an h-set.1. Introduction. The paper is devoted to Besov-type spaces on h-sets. Such sets were defined and studied in Edmunds and Triebel [5], [6]; see also Bricchi [3] and [4]. Our goal is to study more general spaces on such sets, and show how these results can be applied when we look for the domain of Dirichlet forms on an h-set, associated with restrictions of certain symmetric Lévy processes to this set.

Homogeneity, non-smooth atoms and Besov spaces of generalised smoothness on quasi-metric spaces

“…In that paper the spaces B It is an immediate consequence of Definition 5.11 and the existence, under some restrictions, of characterisations of B σ 1/ε p,q (Γ) in terms quarkonial decompositions (cf. [3] and [17]) that something analogous could be obtained for the spaces defined above. So, to prove that the definition of Besov spaces on abstract h-spaces using ε-charts is independent of the charts, under some restrictions, it has been enough to prove that this construction works (in the sense of being independent of the charts) in the particular case of h-sets.…”

“…This can be obtained following directly the proofs of Theorem 4.4.3, p. 49, in [11] and all the intermediate results and relying also on the characterisations of B σ,N p,q (R n ) with quarkonial decompositions located in these more general approximate lattices obtained by Knopova and Zähle in [17].…”

“…[17,Theorem 18]). Most of the conditions on f were applied to prove the existence and continuity of the restriction operator.…”

“…This kind of condition was considered in [14] and [17]. In these papers Besov spaces on h-sets were defined following the approach of Jonsson and Wallin [15] for d-sets.…”

“…There he also introduced and studied Besov spaces of generalised smoothness on these fractal sets. We also refer to [17] for characterisations of these function spaces, which will be used later in this paper. In the particular case where h(r) = r d for some d > 0, we say that Γ is a d-set.…”