Weighted composition operators between Bloch-type spaces in the polydisc

Abstract

For p > 0, let βp(Dn) denote the p-Bloch space on the unit polydisc Dn of Cn and ρ(z) = (ρ1(z), ⋯ , ρn(z)) a holomorphic self-map of Dn. We investigate the boundedness and compactness of the weighted composition uCρf(z) = u(z)f(∞(z)) between p-Bloch space βp(Dn) (little p-Bloch space β 0p (Dn)) and q-Bloch space Bq(Dn) (little q-Bloch space β0q(Dn)). The most important result in the paper is that conditions for the compactness are different for the cases p ∈ (0, 1) and p ≥ 1, unlike for the case of the weighted operators on the unit disc.