Bergman space

In complex analysis, functional analysis and operator theory, a Bergman space is a function space of holomorphic functions in a domain D of the complex plane that are sufficiently well-behaved at the boundary that they are absolutely integrable. Specifically, for 0 < p < ∞, the Bergman space Ap(D) is the space of all holomorphic functions f {displaystyle f} in D for which the p-norm is finite: sup z ∈ K | f ( z ) | ≤ C K ‖ f ‖ L p ( D ) . {displaystyle sup _{zin K}|f(z)|leq C_{K}|f|_{L^{p}(D)}.}     (1) In complex analysis, functional analysis and operator theory, a Bergman space is a function space of holomorphic functions in a domain D of the complex plane that are sufficiently well-behaved at the boundary that they are absolutely integrable. Specifically, for 0 < p < ∞, the Bergman space Ap(D) is the space of all holomorphic functions f {displaystyle f} in D for which the p-norm is finite: The quantity ‖ f ‖ A p ( D ) {displaystyle |f|_{A^{p}(D)}} is called the norm of the function f; it is a true norm if p ≥ 1 {displaystyle pgeq 1} . Thus Ap(D) is the subspace of holomorphic functions that are in the space Lp(D). The Bergman spaces are Banach spaces, which is a consequence of the estimate, valid on compact subsets K of D: Thus convergence of a sequence of holomorphic functions in Lp(D) implies also compact convergence, and so the limit function is also holomorphic. If p = 2, then Ap(D) is a reproducing kernel Hilbert space, whose kernel is given by the Bergman kernel. If the domain D is bounded, then the norm is often given by where A {displaystyle A} is a normalised Lebesgue measure of the complex plane, i.e. dA = dz/Area(D). Alternatively dA = dz/π is used, regardless of the area of D.The Bergman space is usually defined on the open unit disk D {displaystyle mathbb {D} } of the complex plane, in which case A p ( D ) := A p {displaystyle A^{p}(mathbb {D} ):=A^{p}} . In the Hilbert space case, given f ( z ) = ∑ n = 0 ∞ a n z n ∈ A 2 {displaystyle f(z)=sum _{n=0}^{infty }a_{n}z^{n}in A^{2}} , we have that is, A2 is isometrically isomorphic to the weighted ℓp(1/(n+1)) space. In particular the polynomials are dense in A2. Similarly, if D = ℂ+, the right (or the upper) complex half-plane, then where F ( z ) = ∫ 0 ∞ f ( t ) e − t z d t {displaystyle F(z)=int _{0}^{infty }f(t)e^{-tz},dt} , that is, A2(ℂ+) is isometrically isomorphic to the weighted Lp1/t (0,∞) space (via the Laplace transform). The weighted Bergman space Ap(D) is defined in an analogous way, i.e.