Duals of holomorphic Besov spaces on the polydisks and diagonal mappings

Let Un be the unit polydisk in Cn and S be the space of functions of regular variation. Let 1 ≤ p < ∞, ω = (ω1, ..., ωn), ωj ∈ S(1 ≤ j ≤ n) and f ∈ H(Un). The function f is said to be in holomorphic Besov space Bp(ω) if $$ \left\| f \right\|_{B_p (\omega )}^p = \int_{U^n } {\left| {Df(z)} \right|^p \prod\limits_{j = 1}^n {\frac{{\omega _j (1 - |z_j |)}} {{(1 - |z_j |^{2 - p} )}}} dm_{2n} (z) < + \infty } $$ where dm2n(z) is the 2n-dimensional Lebesgue measure on Un and D stands for the fractional differentation of f. This work gives a complete description of (Bp(ω))*, where X* means the dual space of X.. Also the problem of diagonal mapping is completely solved.