Bergman space

In complex analysis, functional analysis and operator theory, a Bergman space, named after Stefan Bergman, 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 A^p(D) is the space of all holomorphic functions ${\displaystyle f}$ in D for which the p-norm is finite:

${\displaystyle \|f\|_{A^{p}(D)}:=\left(\int _{D}|f(x+iy)|^{p}\,\mathrm {d} x\,\mathrm {d} y\right)^{1/p}<\infty .}$

The quantity ${\displaystyle \|f\|_{A^{p}(D)}}$ is called the norm of the function f; it is a true norm if ${\displaystyle p\geq 1}$. Thus A^p(D) is the subspace of holomorphic functions that are in the space L^p(D). The Bergman spaces are Banach spaces, which is a consequence of the estimate, valid on compact subsets K of D:

${\displaystyle \sup _{z\in K}|f(z)|\leq C_{K}\|f\|_{L^{p}(D)}.}$

(1)

Thus convergence of a sequence of holomorphic functions in L^p(D) implies also compact convergence, and so the limit function is also holomorphic.

If p = 2, then A^p(D) is a reproducing kernel Hilbert space, whose kernel is given by the Bergman kernel.