On projection from $A^p_ω$ to the Hardy spaces $H^p$
Armen Jerbashian, Joel E. Restrepo
TL;DR
The paper surveys the projection problem from general weighted holomorphic spaces $A^p_\omega$ to Hardy spaces $H^p$ across the unit disk, the complex plane, and the upper half-plane, synthesizing historical results and presenting new upper-half-plane projections. Central to the approach are Cauchy-type kernels $C_\omega$ (and $C_\omega^{\infty}$) and Djrbashian-type fractional operators $L_\omega$ (and $L_\omega^{\infty}$), which yield explicit representation formulas $f(z)=\frac{1}{2\pi}\int C_{\omega_1}(ze^{-i\vartheta})\varphi(e^{i\vartheta}) d\vartheta$ with $\varphi=L_{\omega_1}f$ in $H^p$, and the corresponding bounded projections with norm bounds. For the disc, the theory provides $L_{\omega_1}$-bounded projections and, in the case $p=2$, an isometry between $A^2_\omega({\mathbb D})$ and $H^2({\mathbb D})$; for the complex plane, analogous representations involve $C_\omega^{\infty}$; and the paper delivers the first comprehensive projection result in the upper half-plane, including explicit norm bounds depending on a growth parameter $\Delta_0$ and the kernel $C_{\omega_1}$. These results unify and extend classical Bergman/Wirtinger-type projections to a broad class of weighted spaces across domains, with concrete examples illustrating the constructions.
Abstract
This paper describes the known results on the projection from the most general holomorphic spaces $A^p_ω$, which depend on a functional parameter $ω$ and are over the unit disc, upper half-plane and the finite complex plane, to the classical Hardy spaces $H^p.$ The paper can be considered as a survey in the mentioned topic. A new result on the projection of the half-plane $A^p_ω$ to the half-plane Hardy space $H^p$ is obtained.