Weighted Composition Operator Between Different Weighted B_log Space And Q_k (p, Q) Space

There are three parts in this paper. In the first part, we introduce some definations and give a number of lemmas about theβ_log space, (?)_K(p,q) spaces and H_α^∞spaces.The second part focus on the boundedness and compactness of the composition operator C_? from theβ_log space to (?)_K(p, q). The boundedness and compactness of the weighted composition operator C_? from the (?)_K(p, q) space toβ_log are investigated.At last, we study the boundedness and compactness of the weighted composition operator and weighted composition operator from theβ_log(β_log0) space to the H_α^∞space. The main results are as follows:i) Some necessary and sufficient conditions are given for which uC_? is a bounded or a compact operator from theβ_log space to the (?)_K(p, q) space.ii) Some necessary and sufficient conditions are given for which uC_? is a bounded or a compact operator from the (?)_K(p, q) space to theβ_log space.iii) some necessary and sufficient conditions are given for which uC_? is a bounded or a compact operator betweenβ_log(β_log0) space and H_α^∞(H_α0^∞) space.