Finite dimensionality of Besov spaces and potential-theoretic decomposition of metric spaces

In the context of a metric measure space (X,d,μ), we explore the potential-theoretic implications of having a finite-dimensional Besov space. We prove that if the dimension of the Besov space B^θp,p(X) is $k>1$, then X can be decomposed into k number of irreducible components (Theorem 1.1). Note that θ may be bigger than $1$, as our framework includes fractals. We also provide sufficient conditions under which the dimension of the Besov space is $1$. We introduce critical exponents θₚ(X) and θₚ^(X) for the Besov spaces. As examples illustrating Theorem 1.1, we compute these critical exponents for spaces X formed by glueing copies of n-dimensional cubes, the Sierpi\'{n}ski gaskets, and of the Sierpi\'{n}ski carpet.