On the equivalence of some conditions for weighted Hardy spaces
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 $$\mathop {\sup }\limits_{\left| \varphi \right| < \tfrac{\pi }{2}} \left\{ {\int\limits_0^{ + \infty } {\left| {G(re^{i\varphi } )} \right|^p e^{ - p\si...
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| Date: | 2006 |
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| Main Authors: | , |
| Format: | Article |
| Language: | Ukrainian English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2006
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/3526 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
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Ukrains’kyi Matematychnyi Zhurnal| Summary: | Let $G ∈ H_{σ}^p (ℂ+)$, where $H_{σ}^p (ℂ+)$ is the class of functions analytic in the half plane ℂ+ = {z: Re z > 0} and such that
$$\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. |
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