Berezin transform in polynomial bergman spaces
Fix a smooth weight function Q in the plane, subject to a growth condition from below. Let Km,n denote the reproducing kernel for the Hilbert space of analytic polynomials of degree at most n − 1 of finite L2‐norm with respect to the measure e−mQ dA. Here dA is normalized area measure, and m is a po...
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| Vydáno v: | Communications on pure and applied mathematics Ročník 63; číslo 12; s. 1533 - 1584 |
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| Hlavní autoři: | , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Hoboken
Wiley Subscription Services, Inc., A Wiley Company
01.12.2010
Wiley John Wiley and Sons, Limited |
| Témata: | |
| ISSN: | 0010-3640, 1097-0312, 1097-0312 |
| On-line přístup: | Získat plný text |
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| Shrnutí: | Fix a smooth weight function Q in the plane, subject to a growth condition from below. Let Km,n denote the reproducing kernel for the Hilbert space of analytic polynomials of degree at most n − 1 of finite L2‐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
${\rm d}B_{m,n}^{\langle z_0\rangle}(z)=K_{m,n}(z_0,z_0)^{-1}|K_{m,n}(z,z_0)|^2{\rm e}^{-mQ(z)}\,{\rm d}A(z)$
for the point z0 is a probability measure that defines the (polynomial) Berezin transform
$\font\open=msbm10 at 10pt\def\C{\hbox{\open C}}{\rm B}_{m,n}f(z_0)=\int\limits_{\C} f\,{\rm d}B_{m,n}^{\langle z_0\rangle}$
for continuous
$\font\open=msbm10 at 10pt\def\C{\hbox{\open C}}f\in L^\infty(\C)$. 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 z0 converges weak‐star to the unit point mass at the point z0 provided that Δ Q(z0) > 0 and that z0 is contained in the interior of a compact set ${\cal S}_\tau$, 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 $\font\open=msbm10 at 10pt\def\C{\hbox{\open C}}z_0\in \C\setminus{\cal S}_\tau$, the Berezin measure cannot converge to the point mass at z0. In the model case Q(z) = |z|2, when ${\cal S}_\tau$
is a closed disk, we find that the Berezin measure instead converges to harmonic measure at z0 relative to $\font\open=msbm10 at 10pt\def\C{\hbox{\open C}}\C\setminus{\cal S}_\tau$.
Our results have applications to the study of the eigenvalues of random normal matrices. The auxiliary results include weighted L2‐estimates for the equation
${\bar \partial} u=f$
when f is a suitable test function and the solution u is restricted by a polynomial growth bound at ∞. © 2009 Wiley Periodicals, Inc. |
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| Bibliografie: | ArticleID:CPA20329 istex:578BB420667D3DC6FECAF92EEFF2B7C9B89ED075 ark:/67375/WNG-HHCF6120-B SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 14 |
| ISSN: | 0010-3640 1097-0312 1097-0312 |
| DOI: | 10.1002/cpa.20329 |