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

Ferguson, Timothy

Rocky Mt. J. Math. 49, No. 1, 47-71 (2019).

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 \(\lim_{r\to\infty}r^{3}M_{p}^{p}(r,f)\omega(r^{2})=0\).

Reviewer: Julià Cufí (Bellaterra)

MSC:

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

Keywords:

extremal problem; regularity; Fock space; Bergman space

References: