Composition Operators and Isometries on Holomorphic Function Spaces over Domains in C

Let D be a bounded domain in C with C boundary. Many holomorphic function spaces over D have been introduced in last half century, such as Hardy, Bergman, Besov and Sobolev spaces. Properties (boundary behaviour ect.) of functions in those function spaces have been received a great deal of studies. Readers can see parts of them from the following books [26] [29], [66], [71], [75], [76] and references therein. For any 0 < p ≤ ∞, we let L(∂D) be Lebesgue p−space with respect to Lebesgue surface measure on ∂D, let H(D) be Hardy space of holomorphic functions onD with their boundary value functions belonging to L(∂D). Let L(D) be the Lebesgue p-space with respect to the Lebesgue volume measure on D, and let A(D) be its holomorphic subspace. It is well-known that H(D), L(∂D), L(D) and A(D) are Hilbert spaces; H(D) is a closed subspace of L(∂D) while A(D) is a closed subspace of L(D). There are two important orthogonal projection operators: S : L(∂D)→ H(D) and P : L(D)→ A(D). S is called Szegö projection and P is called Bergman projection. Both of them can be written as integral operators: