Ameur, Yacin and Hedenmalm, Håkan and Makarov, Nikolai
(2010)
*Berezin Transform in Polynomial Bergman Spaces.*
Communications on Pure and Applied Mathematics, 63
(12).
pp. 1533-1584.
ISSN 0010-3640.
https://resolver.caltech.edu/CaltechAUTHORS:20101111-144213512

PDF
- Published Version
Restricted to Repository administrators only See Usage Policy. 505Kb |

Use this Persistent URL to link to this item: https://resolver.caltech.edu/CaltechAUTHORS:20101111-144213512

## Abstract

Fix a smooth weight function Q in the plane, subject to a growth condition from below. Let K_(m,n) denote the reproducing kernel for the Hilbert space of analytic polynomials of degree at most n - 1 of finite L^2-norm with respect to the measure e^(-mQ) dA. Here dA is normalized area measure, and m is a positive real scaling parameter. The (polynomial) Berezin measure dB^(<Z0>)_(m,n)(z)= K_(m,n)(z_0,z_0)^(-1) │K_(m,n)(z,z_0)│^2e^(-mQ(z)) dA(z) for the point z_0 is a probability measure that defines the (polynomial) Berezin transform B_(m,n f)(z_0)= ʃC f dB^(<z0>)_(m,n) for continuous f є L^∞(C). We analyze the semiclassical limit of the Berezin measure (and transform) as m → +∞ while n = m τ + o(1), where τ is fixed, positive, and real. We find that the Berezin measure for z_0 converges weak-star to the unit point mass at the point z_0 provided that ΔQ(z_0) > 0 and that z_0 is contained in the interior of a compact set S_ τ, defined as the coincidence set for an obstacle problem. As a refinement, we show that the appropriate local blowup of the Berezin measure converges to the standardized Gaussian measure in the plane. For points z_0 є C\S _τ , the Berezin measure cannot converge to the point mass at z_0. In the model case Q(z)= │z│^2, when S_ τ is a closed disk, we find that the Berezin measure instead converges to harmonic measure at z_0 relative to C\S_ τ. Our results have applications to the study of the eigenvalues of random normal matrices. The auxiliary results include weighted L^2 -estimates for the equation ∂[overscore]u = f when f is a suitable test function and the solution u is restricted by a polynomial growth bound at ∞.

Item Type: | Article | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

Related URLs: |
| |||||||||

Additional Information: | © 2010 Wiley Periodicals, Inc. Received October 2009. Article first published online: 4 Aug 2010. This research was partially supported by grants from the Swedish Science Council (VR) and from the the Göran Gustafsson Foundation. | |||||||||

Funders: |
| |||||||||

Issue or Number: | 12 | |||||||||

Record Number: | CaltechAUTHORS:20101111-144213512 | |||||||||

Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20101111-144213512 | |||||||||

Official Citation: | Ameur, Y., Makarov, N. and Hedenmalm, H. (2010), Berezin transform in polynomial bergman spaces. Communications on Pure and Applied Mathematics, 63: 1533–1584. doi: 10.1002/cpa.20329 | |||||||||

Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | |||||||||

ID Code: | 20765 | |||||||||

Collection: | CaltechAUTHORS | |||||||||

Deposited By: | Ruth Sustaita | |||||||||

Deposited On: | 15 Nov 2010 16:46 | |||||||||

Last Modified: | 03 Oct 2019 02:14 |

Repository Staff Only: item control page