Besov and Triebel-Lizorkin spaces on metric spaces: Embeddings and pointwise multipliers

In this paper, we obtain the Franke-Jawerth embedding property of Hajłasz-Besov and Hajłasz-Triebel-Lizorkin spaces on a measure metric space (X,d,μ) which is Ahlfors regular with dimension “Q”. As applications, we show that, when (X,d,μ) is doubling and satisfies an Ahlfors lower bound condition with Q, then the Hajłasz-Besov space Np,qs(X) with p∈(Q,∞], s∈(Qp,1] and q∈(0,∞] and the Hajłasz-Triebel-Lizorkin space Mp,qs(X) with p∈(Q,∞), s∈(Qp,1] and q∈(QQ+s,∞] are algebras under pointwise multiplication and, moreover, when X is Ahlfors Q-regular, we characterize the class of all pointwise multipliers on the Hajłasz-Triebel-Lizorkin space Mp,qs(X) for p∈(Q,∞), s∈(Qp,1] and q∈(QQ+s,∞] by its related uniform space.