Theory of holomorphic Besov spaces in domains of complex euclidean spaces.

摘要

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.