Bergman space

Summary

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 in D for which the p-norm is finite:

The quantity is called the norm of the function f; it is a true norm if . 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:

 

 

 

 

(1)

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.

Special cases and generalisations

If the domain D is bounded, then the norm is often given by

where 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 of the complex plane, in which case . In the Hilbert space case, given , we have

that is, A2 is isometrically isomorphic to the weighted p(1/(n+1)) space.[1] In particular the polynomials are dense in A2. Similarly, if D = +, the right (or the upper) complex half-plane, then

where , that is, A2(+) is isometrically isomorphic to the weighted Lp1/t (0,∞) space (via the Laplace transform).[2][3]

The weighted Bergman space Ap(D) is defined in an analogous way,[1] i.e.

provided that w : D → [0, ∞) is chosen in such way, that is a Banach space (or a Hilbert space, if p = 2). In case where , by a weighted Bergman space [4] we mean the space of all analytic functions f such that

and similarly on the right half-plane (i.e. ) we have[5]

and this space is isometrically isomorphic, via the Laplace transform, to the space ,[6][7] where

(here Γ denotes the Gamma function).

Further generalisations are sometimes considered, for example denotes a weighted Bergman space (often called a Zen space[3]) with respect to a translation-invariant positive regular Borel measure on the closed right complex half-plane , that is

Reproducing kernels

The reproducing kernel of A2 at point is given by[1]

and similarly for we have[5]

.

In general, if maps a domain conformally onto a domain , then[1]

In weighted case we have[4]

and[5]

References

  1. ^ a b c d Duren, Peter L.; Schuster, Alexander (2004), Bergman spaces, Mathematical Series and Monographs, American Mathematical Society, ISBN 978-0-8218-0810-8
  2. ^ Duren, Peter L. (1969), Extension of a theorem of Carleson (PDF), 75, Bulletin of the American Mathematical Society, pp. 143–146
  3. ^ a b Jacob, Brigit; Partington, Jonathan R.; Pott, Sandra (2013-02-01). "On Laplace-Carleson embedding theorems". Journal of Functional Analysis. 264 (3): 783–814. arXiv:1201.1021. doi:10.1016/j.jfa.2012.11.016. S2CID 7770226.
  4. ^ a b Cowen, Carl; MacCluer, Barbara (1995-04-27), Composition Operators on Spaces of Analytic Functions, Studies in Advanced Mathematics, CRC Press, p. 27, ISBN 9780849384929
  5. ^ a b c Elliott, Sam J.; Wynn, Andrew (2011), Composition Operators on the Weighted Bergman Spaces of the Half-Plane, 54, Proceedings of the Edinburgh Mathematical Society, pp. 374–379
  6. ^ Duren, Peter L.; Gallardo-Gutiérez, Eva A.; Montes-Rodríguez, Alfonso (2007-06-03), A Paley-Wiener theorem for Bergman spaces with application to invariant subspaces, 39, Bulletin of the London Mathematical Society, pp. 459–466, archived from the original on 2015-12-24
  7. ^ Gallrado-Gutiérez, Eva A.; Partington, Jonathan R.; Segura, Dolores (2009), Cyclic vectors and invariant subspaces for Bergman and Dirichlet shifts (PDF), 62, Journal of Operator Theory, pp. 199–214

Further reading

  • Bergman, Stefan (1970), The kernel function and conformal mapping, Mathematical Surveys, 5 (2nd ed.), American Mathematical Society
  • Hedenmalm, H.; Korenblum, B.; Zhu, K. (2000), Theory of Bergman Spaces, Springer, ISBN 978-0-387-98791-0
  • Richter, Stefan (2001) [1994], "Bergman spaces", Encyclopedia of Mathematics, EMS Press.

See also