A Hardy space related to the square root of the Poisson kernel
Volume 199 / 2010
Abstract
A real-valued Hardy space $H^{1}_{\sqrt{}}(\mathbb{T}) \subseteq L^1(\mathbb{T})$ related to the square root of the Poisson kernel in the unit disc is defined. The space is shown to be strictly larger than its classical counterpart $H^1(\mathbb{T})$. A decreasing function is in $H^{1}_{\surd}(\mathbb{T})$ if and only if the function is in the Orlicz space $L\log\log L(\mathbb{T})$. In contrast to the case of $H^{1}(\mathbb{T})$, there is no such characterization for general positive functions: every Orlicz space strictly larger than $L\log L(\mathbb{T})$ contains positive functions which do not belong to $H^{1}_{\surd}(\mathbb{T})$, and no Orlicz space of type $\Delta_2$ which is strictly smaller than $L^1(\mathbb{T})$ contains every positive function in $H^{1}_{\surd}(\mathbb{T})$. Finally, we have a characterization of certain eigenfunctions of the hyperbolic Laplace operator in terms of $H^{1}_{\surd}(\mathbb{T})$.
