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 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 = ℂ+, the right (or the upper) complex half-plane, then

where , that is, A2(ℂ+) 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. 1 2 3 4 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. 1 2 Jacob, Brigit; Partington, Jonathan R.; Pott, Sandra (2013-02-01), On Laplace-Carleson embedding theorems, 264 (3), Journal of Functional Analysis, pp. 783–814
  4. 1 2 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. 1 2 3 Elliott, Sam J.; Wynn, Andrew (2011), Composition Operators on the Weighted Bergman Spaces of the Half-Plane, 54 (2), 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 (PDF), 39 (3), Bulletin of the London Mathematical Society, pp. 459–466
  7. Gallrado-Gutiérez, Eva A.; Partington, Jonathan R.; Segura, Dolores (2009), Cyclic vectors and invariant subspaces for Bergman and Dirichlet shifts (PDF), 62 (1), Journal of Operator Theory, pp. 199–214

Further reading

See also


This article is issued from Wikipedia - version of the 7/1/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.