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 _| φ | < π 2 { _0^ + ∞ | G(re^iφ ) |^p e^ - pσ r| sinφ | dr } < + ∞ . 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){ 2σ π zlnz - cz } otin H^p (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.