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Singular Berezin transforms. (English) Zbl 1138.32002

Let \(\Omega\) be a domain in \(\mathbb{C}^n\), not necessarily bounded, and \(L^2_{\text{hol}}(\Omega)\) be the Bergman space formed by the holomorphic functions in \(L^2(\Omega)\). It is well known that \(L^2_{\text{hol}}(\Omega)\) has a reproducing kernel, meaning that for each \(y\in\Omega\), there exists a function \(K_y\in L^2_{\text{hol}}(\Omega)\) such that
\[ f(y)=\langle f, K_y\rangle= \int f(x) K(y, x)\,dx\quad \forall f\in L^2_{\text{hol}}(\Omega), \] where \(K(x, y):=\langle K_y, K_x\rangle\), and \(dx\) is the Lebesgue measure. The Berezin transform on \(\Omega\) is the integral operator
\[ Bf(y):= K(y, y)^{-1}\int f(x)|K(x,y)|^2\;dx. \]
Although the integral is well defined for any \(f\in L^\infty\), the Berezin transform could have singularities at the points of \(N:= \{z\in\Omega: K(z, z)= 0\}\). It will be assumed for the rest of this review that \(N\neq\Omega\), since otherwise the Bergman space reduces to \(\{0\}\). That means that \(N\) is a proper real analytic variety in \(\Omega\), and therefore has trivial Lebesgue measure. The paper deals with the possible singularities of the Berezin transform for unweighted and weighted Bergman spaces with reproducing kernel. First it is shown that if \(\Omega\subset\mathbb{C}\) then \(N= \emptyset\) in the unweighted case, and that \(B(f)\) extends to a real-analytic function on \(\Omega\) in the weighted case.
A domain \(\Omega\subset\mathbb{C}^n\) is called a complete Reinhardt domain if whenever \((z_1,\dots, z_n)\in \Omega\) and \(|x_j|\leq |z_j|\) for \(1\leq j\leq n\), then \((x_1,\dots, x_n)\in\Omega\). For the unweighted case, the author exhibits a pseudoconvex complete Reinhardt domain \(\Omega\subset\mathbb{C}^3\) and \(f\in L^\infty(\Omega)\) such that \(Bf\) has a discontinuity, and shows that this phenomenon does not happen for complete Reinhardt domains in \(\mathbb{C}^2\). However, it is shown that for the bidisc in \(\mathbb{C}^2\) and the weight \(w(z)=(|z_1|^2+ |z_2|^2)^{-1}\), the associated Berezin transform of \(f(z)= |z_1|^2\) has a discontinuity at the origin.
The last section is related to some examples of J. J. O. O. Wiegerinck [Math. Z. 187, 559–562 (1984; Zbl 0534.32001)] who showed that there are Reinhardt domains in \(\mathbb{C}^n\) for which the unweighted Bergman space is nonzero but finite-dimensional. The author shows that this cannot happen for a pseudoconvex Reinhardt domain.

MSC:

32A36 Bergman spaces of functions in several complex variables
32A07 Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube) (MSC2010)
32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)

Citations:

Zbl 0534.32001
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