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

In this paper, we obtain the Franke-Jawerth embedding property of Hajlasz-Besov and Hajlasz-Triebel-Lizorkin spaces on a measure metric space (chi, d, mu) which is Ahlfors regular with dimension "Q". As applications, we show that, when (chi, d, mu) is doubling and satisfies an Ahlfors lower bound condition with Q, then the Hajlasz-Besov space N-p,q(s) (chi) with p is an element of (Q, infinity], s is an element of (Q/p, 1] and q is an element of (0, infinity] and the Hajlasz-Triebel-Lizorkin space MA, (chi) with p is an element of (Q, infinity), s is an element of,(Q/p, 1] and q is an element of (Q/Q+s, infinity] are algebras under pointwise multiplication and, moreover, when chi is Ahlfors Q-regular, we characterize the class of all pointwise multipliers on the Hajlasz-Triebel-Lizorkin space M-p,q(s) (chi) for p is an element of (Q, infinity), s is an element of(Q/p, 1] and q is an element of (Q/Q+s, infinity] by its related uniform space. (C) 2017 Elsevier Inc. All rights reserved.