On the equivalence of some conditions for weighted Hardy spaces

Abstract

Let G ∈ H ^p_σ (ℂ₊), where H ^p_σ (ℂ₊) is the class of functions analytic in the half plane ℂ₊ = {z: Re z > 0} and such that

$$\mathop {\sup }\limits_{\left| \varphi \right| < \tfrac{\pi }{2}} \left\{ {\int\limits_0^{ + \infty } {\left| {G(re^{i\varphi } )} \right|^p e^{ - p\sigma r\left| {sin\varphi } \right|} dr} } \right\} < + \infty .$$

. 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 $G(z)\exp \left\{ {\frac{{2\sigma }}{\pi }zlnz - cz} \right\} \notin H^p (\mathbb{C}_ + )$, where H ^p(ℂ₊) is the Hardy space, in terms of the behavior of G on the real semiaxis and on the imaginary axis.