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

In the context of a metric measure space $(X,d,\mu )$, we explore the potential-theoretic implications of having a finite-dimensional Besov space. We prove that if the dimension of the Besov space $B_{p,p}^{\theta }(X)$ is $k>1$, then $X$ can be decomposed into $k$ number of irreducible components (Theorem 1.1). Note that $\theta$ 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 ${\theta }_p(X)$ and ${\theta }_p^{\ast }(X)$ 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 Sierpiński gaskets, and of the Sierpiński carpet.