Bounded projections, duality and representations on large mixed norm spaces
Recall that, for \(0<p\leq\infty\) and an analytic function \(f\) on the open unit disk \(\mathbb{D}\), the integral mean \(M_p(r,f)\) is defined as a function of \(0\leq r<1\) in the following way: \[ M_p(r,f) = \Big(\int_{0}^{2\pi}|f(re^{i\theta})|^{p}\frac{d\theta}{2\pi}\Big)^{1/p} \] if \(0<p<\infty\) and \[ M_{p}(r,f)=\sup_{0<\theta<2\pi}|f(re^{i\theta})| \] if \(p=\infty\). Let \(\omega\) be a radial weight function, that is, a positive function belonging to \(L^1(0,1)\). For \(0<p\leq\infty\) and \(0<q<\infty\), the weighted mixed norm Bergman space \(A^{p,q}(\omega)\) consists of analytic functions \(f\) on \(\mathbb{D}\) for which \[ \|f\|_{p,q,\omega} = \Big(\int_{0}^{1}M_{p}^{q}(r,f)\omega(r)\, dr\Big)^{1/q} < \infty. \] Furthermore, for \(0<p\leq\infty\), the space \(A^{p,\infty}(\omega)\) consists of analytic functions \(f\) for which \[ \|f\|_{p,\infty,\omega} = \sup_{0\leq r <1}M_{p}(r,f)\omega(r) < \infty. \] Appropriate choices of \(p, q\), and \(\omega\) give the classical Hardy and Bergman spaces. In particular, the case \(p=q=2\) gives rise to the weighted Bergman space \(A^2(\omega)\), which is a reproducing kernel Hilbert space. Using the reproducing kernel of \(A^2(\omega)\), one defines the Bergman projection \(P_{\omega}\). \textit{H. Arroussi} and \textit{J. Pau} [J. Geom. Anal. 25, No. 4, 2284--2312 (2015; Zbl 1341.30045] proved the boundedness of \(P_{\omega}\) from \(L^p(\omega^{p/2})\) to \(A^p(\omega^{p/2})\) for \(1\leq p<\infty\) and identified the duals of \(A^2(\omega^{p/2})\). The paper under review generalizes these results to weighted mixed norm Bergman spaces under certain assumptions on the weight \(\omega\). More specifically, it is shown that the Bergman projection \(P_{\omega}\) is bounded from \(L^{p,q}(\omega^{q/2})\) into \(A^{p,q}(\omega^{q/2})\) for \(1\leq p,q<\infty\). It is also proved that \(P_{\omega}\) is bounded from \(L^{p,\infty}(\omega^{1/2})\) into \(A^{p,\infty}(\omega^{1/2})\) for \(1\leq p<\infty\). As an application, the dual of \(A^{p,q}(\omega^{q/2})\) is identified. Using duality, the author obtains norm representations and reproducing kernel series representations for elements in these spaces.
scientific article; zbMATH DE number 6370928