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 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

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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

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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

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  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), vol. 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", Proceedings of the Edinburgh Mathematical Society, 54 (2): 374–379, arXiv:0910.0408, doi:10.1017/S0013091509001412, S2CID 18811195
  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, vol. 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), vol. 62, Journal of Operator Theory, pp. 199–214

Further reading

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See also

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