On the equivalence of some conditions for weighted Hardy spaces

  • V. M. Dilnyi Львiв. нац. ун-т; Дрогобиц. держ. пед. ун-т


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.
How to Cite
DilnyiV. M. “On the Equivalence of Some Conditions for Weighted Hardy Spaces”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, no. 9, Sept. 2006, pp. 1257–1263, https://umj.imath.kiev.ua/index.php/umj/article/view/3526.
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