On the equivalence of some conditions for weighted Hardy spaces - Dilnyi, V. M. (Ukrains’kyi Matematychnyi Zhurnal, 2006)
Let $G ∈ H_{σ}^p (ℂ+)$, where $H_{σ}^p (ℂ+)$ is the class of functions analytic in the half plane ℂ+ = {z: Re z > 0} and such that$$\mathop {\sup }\limits_{\left| φ \right| < \tfrac{π }{2}} \left\{ {\int\limits_0^{ + ∞ } {\left| {G(re^{iφ } )} \right|^p e^{ - pσ r\left| {sinφ } \right|} dr} } \right\} < + ∞ .$$In the case where a singular boundary function $G$ is identically constant and $G(z) ≠ 0$ for all $z ∈ ℂ_{+}$, we establish conditions equivalent to the condition $G(z)\exp \left\{ {\frac{{2σ }}{π }zlnz - cz} \right\} otin H^p (\mathbb{C}_+ )$, where $H^p (ℂ_{+})$ is the Hardy space, in terms of the behavior of $G$ on the real semiaxis and on the imaginary axis.