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Regularity of extremal functions in weighted Bergman and Fock type spaces. (English) Zbl 1429.30050

This article deals with the regularity of solutions to extremal problems in certain weighted Bergman spaces in discs, as well as in Fock spaces.
For \(0 < R\le \infty\), let \(\mathbb{D}_{R}\) be the open disc of radius \(R\) centered at the origin and let \(\nu(z)=\omega (|z|^{2})\) where \(\omega\) is a positive, decreasing and non-constant function on \([0,R^{2}]\) that is analytic in a complex neighborhood of \([0,R^{2}]\). The space \(A_{R}^{p}(\nu)=A^{p}_{R} (\omega(|z|^{2}))\) consists of all analytic functions on \(\mathbb{D}_{R}\) such that \[ \|f\|_{A_{R}^{p} (\omega (|z|^{2}))}=\left( \int_{\mathbb{D}_{R}} |f(z)|^{p} \omega (|z|^{2})\,dA(z)\right)^{1/p}<+\infty, \] where \(1 < p <\infty\).
If \(k\in A_{R}^{p'}(\nu)\), where \(\frac{1}{p}+\frac{1}{p'}=1\), \(f\to \int_{\mathbb{D}_{R}}f(z)\overline{k(z)} \nu(z)\,dA(z)\) defines a linear functional \(\Phi_{k}\) on \(A^{p}_{R}(\nu)\) with norm denoted by \(\|k\|^{*}\).
One says that \(f\) is the extremal function for the integral kernel \(k\) if \(\|f\|_{A^{p}_{R}(\nu)}=1\) and \[ \operatorname{Re}\Phi_{k}(f)=\sup_{\|g\|_{A^{p}_{R}(\nu)=1}}\left( \operatorname{Re}\int_{\mathbb{D}_{R}}g(z)\overline{k(z)} \nu(z)\,dA(z)\right). \] There always exists a unique solution to this extremal problem.
Writing \[ \begin{aligned} M_{p}(r,f)&=\left(\int_{0}^{2\pi} |f(r e^{i\theta})|^{p}\,d\theta\right)^{1/p},\quad r<R,\\ M_{p}(R,f)&=\lim_{r\to R^{-}}M_{p}(r,f), \\ D_{p}(r,f)&= D_{p}(r,f;\omega)=\left(-\int_{\mathbb{D}_{R}}|z|^{2} |f(z)|^{p}\omega' (|z|^{2})\,dA\right)^{1/p}\end{aligned} \] and \(\widehat p=\max (p-1,1)\), one can state the result obtained for the Bergman space:
Theorem 1. Let \(1<p<\infty\), and let \(0<R<\infty\). Let the function \(\omega\) be analytic in a neighborhood of \([0,R^{2})\), and let \(\omega\) be positive, non-increasing and non-constant on \([0,R^{2})\). Suppose that \(f\) is the extremal function in \(A^{p}_{R}(\omega(|z|^{2}))\) for the integral kernel \(k\). Then \[ \frac{R^{2}}{2} \omega (R^{2})M_{p}^{p}(R,f)+D_{p}^{p}(R,f)\le \frac{2^{1/q}\widehat{p}}{\|k\|^{*}} \left[ \left( \frac{R^{2}}{2}\omega (R^{2})\right)^{1/q}M_{q}(R,k)+D_{q}(R,k)\right]^{q}. \]
In order to provide a result about the regularity of extremal functions in the Fock-type spaces \(A_{\infty}^{p}(\nu)\), one needs to put some restrictions on the weight \(\nu(z)=\omega(|z|^{2})\). The result is the following one:
Theorem 2. Let \(1<p<\infty\). Let the function \(\omega\) be analytic in a neighborhood of \([0,\infty)\), and let \(\omega\) be positive, non-increasing and non-constant on \([0,\infty)\). Also, suppose that \(\lim_{r\to\infty}r^{n}\omega(r^{2})= \lim_{r\to\infty}r^{n}\omega'(r^{2})=0\) for all integers \(n\), and that the polynomials are dense in \(A^{p}_{\infty}(\omega(|z|^{2}))\) and in \(A^{p}_{\infty}(\omega(|z|^{2}))- |z|^{2}\omega'(|z|^{2}))\). Suppose that \(f\) is the extremal function in \(A_{R}^{p}(\omega (|z|^{2}))\) for the integral kernel \(k\). Then: \[ D_{p}(\infty,f)\le \left[ \frac{\widehat{p}}{\|k\|^{*}} D_{q}(\infty,k)\right]^{1/(p-1)}. \]
Theorem 2 does not bound the quantity \(\lim\limits_{r\to\infty}r^{2}\omega (r^{2})M_{p}^{p}(r,f)\), although a similar term is bounded in Theorem 1. It can be shown that \(r^{3}\omega (r^{2}) M_{p}^{p}(r,f)\to 0\) as \(r\to\infty\), as a consequence of \(D_{p}(f,\infty)<\infty\) for certain functions \(\omega\). More precisely, if one writes \[ S(x,\lambda)=\int_{x}^{\infty}\frac{\lambda(x)}{\lambda(t)} \left(\frac{t}{x}\right)^{x\lambda'(x)/\lambda(x)}\,dt, \] for \(\lambda(x)\) some positive, increasing, smooth function defined for \(x\ge R\), one gets:
{Theorem 3}. Suppose that \(\lim\inf_{x\to\infty}S(x,-1/\omega'(r^{2}))>0\) and that there is some positive constant \(C\) such that \(-\omega'(r)\ge C\omega(r)\) for all sufficiently large \(r\). If \(D_{p}(\infty,f;\omega)<\infty\), then \(\lim_{r\to\infty}r^{3}M_{p}^{p}(r,f)\omega(r^{2})=0\).

MSC:

30H20 Bergman spaces and Fock spaces
30C75 Extremal problems for conformal and quasiconformal mappings, other methods
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