| In this dissertation, we give some fundamental properties of analytic Besov space Bp(D), and provide an important application of our main result. Let D be an admissible domain which including bounded strictly pseudoconvex domain, convex domain of finite type in Cn , and pseudoconvex domain of finite type in C2 . Let P : L2( D) → A2(D) be the Bergman projection with Bergman kernel K( z,w). Let dlambda(z) = K(z,z)dv(z) be the biholomorphic invariant measure on D. We first prove that Bp(D) = P( Lp(D,dlambda)), which gives a fundamental constructural characterization for Besov space Bp( D). Second, we build up the duality theorems for the Besov space with L2(D)-pairing on D which are: B1(D) = Bn+1,0 (D)*, B1( D)* = Bn+1 (D) and Bp( D) = Aq(D, K1-qdv )*, for 1/p + 1/q = 1, and 1 < p < infinity.;To understand growth and compare with other easier and well-understood function spaces, we third prove the embedding theorem: Bp( D) ⊂ BMOA(D) (space of holomorphic functions whose boundary values have bounded mean oscillation). We also give the best growth estimates for functions f( z) ∈ Bp(D) when z near boundary ∂D. Finally, we apply our main result to give a characterization for hf belongs to the Schatten class Sp, which is: hf ∈ Sp if and only if f ∈ Bp(D) for 2 ≤ p ≤ infinity. |
