Berezin Transform in Polynomial Bergman Spaces

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^()_(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^()_(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 ∞