Equivalent definition of some weighted Hardy spaces

Abstract

We present an equivalent definition of functions analytic in the half-plane ℂ+ = {z: Re z > 0} for which $$ {\mathop {\sup }\limits_{{\left| \varphi \right|} < \frac{\pi } {2}} }{\left\{ {{\int\limits_0^{ + \infty } {{\left| {f{\left( {re^{{i\varphi }} } \right)}} \right|}^{\rho } e^{{ - \rho \sigma r{\left| {\sin \varphi } \right|}}} dr} }} \right\}} < + \infty . $$ .