On the equivalence of some conditions for weighted Hardy spaces

Abstract

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