Bergman space

In complex analysis, a branch of mathematics, 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, \( A^p(D) \) is the space of holomorphic functions in D such that the p-norm

\( \|f\|_p = \left(\int_D |f(x+iy)|^p\,dx\,dy\right)^{1/p} < \infty. \)

Thus \( A^p(D) \) is the subspace of homolorphic 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:

\( \sup_{z\in K} |f(z)| \le C_K\|f\|_{L^p(D)}. \) (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.
References

Richter, Stefan (2001), "Bergman spaces", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1556080104.

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