Berezin transform in polynomial bergman spaces - Ameur, Yacin; Makarov, Nikolai; Hedenmalm, Håkan (Communications on pure and applied mathematics, 2010)
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 $\font\open=msbm10 at 10pt\def\C{\hbox{\open C}}f\in L^∞(\C)$. 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 ${\cal S}_τ$, 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 $\font\open=msbm10 at 10pt\def\C{\hbox{\open C}}z_0\in \C\setminus{\cal S}_τ$, the Berezin measure cannot converge to the point mass at z0. In the model case Q(z) = |z|2, when ${\cal S}_τ$ is a closed disk, we find that the Berezin measure instead converges to harmonic measure at z0 relative to $\font\open=msbm10 at 10pt\def\C{\hbox{\open C}}\C\setminus{\cal S}_τ$. 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.