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Bergman function, Genchev transform and \(L^ 2\)-angles for multidimensional tubes. (English) Zbl 0884.46020

The author investigates properties of the Hilbert space \(L^2H(D)\) of square integrable holomorphic functions on \(D\), where \(D=\mathbb{R}^n+ iB\) is a tube over a domain \(B\subset\mathbb{R}^n\). For \(f\in L^2H(D)\), \(y\in B\), set \(f_y(x)= f(x+iy), x\in\mathbb{R}^n\), and let \(\widehat f_y\) denote the Fourier transform of \(f_y\). For \(f\in L^2H(D)\) and arbitrary \(y\in B\), the function \(G(f)(t)= e^{2\pi\langle y,t\rangle}\widehat f_y(t)\), \(t\in\mathbb{R}^n\), is called the Genchev transform of \(f\). The definition of \(G(f)\) is independent of the choice of \(y\in B\). In Theorem 3.4 and 4.1 the author proves that the Genchev transform is a unitary mapping of \(L^2H(D)\) onto \(L^2(\mathbb{R}^n, w_B)\) with weight \(w_B(t)= \int_Be^{-4\pi\langle y,t\rangle} dy\), \(t\in \mathbb{R}^n\). From this it follows that \(L^2H(D)\neq\{0\}\) if and only if \(\{t\in \mathbb{R}^n: w_B(t)<\infty\}\) has positive Lebesgue measure. Section 5 investigates properties of the weight \(w_B\), and in Section 6 an integral formula for the Bergman kernel \(K_D(z,w)\) is derived. Using this result the author computes the Bergman kernel for the tube domain over a bounded interval, the tube over a cone in \(\mathbb{R}^2\), and the tube over an open ball in \(\mathbb{R}^n\) \((n\geq 2)\). Section 7 investigates the Bergman space \(L^2H(D)\) for the case where \(B\) is a convex domain in \(\mathbb{R}^n\). In this section (Theorem 7.4) it is proved that if \(B\) is convex, then \(L^2H(D)\neq \{0\}\) if and only if \(B\) does not contain any entire straight line. The author gives two examples showing that in the case of nonconvex \(B\), the above geometric condition is neither necessary nor sufficient. This is in stark contrast to the case of the Hardy space \(H^2(D)\), where it is known that \(H^2(D)\neq \{0\}\) if and only if \(c(D)\), the convex hull of \(D\), does not contain an entire straight line. Section 8 of the paper contains some fundamental results concerning the holomorphic continuation of a function \(f\in L^2H(D)\) to a holomorphic function \(f_c\) on \(c(D)\). The function \(f_c\) however need not belong to \(L^2H(c(D))\). Theorem 8.7 provides necessary and sufficient conditions for the existence of \(L^2\)-holomorphic continuation. Section 9 contains the basic formula for the determination of \(L^2\)-angles between tube domains in \(\mathbb{C}^n\), and in section 10 the author presents some concrete examples of \(L^2\)-angles between multidimensional tubes.
Reviewer: M.Stoll (Columbia)

MSC:

46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
32A07 Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube) (MSC2010)
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