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A real-valued Hardy space $H{¹}_{\surd }\left(\right)\subseteq L¹\left(\right)$ 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 H¹(). A decreasing function is in $H{¹}_{\surd }\left(\right)$ if and only if the function is in the Orlicz space LloglogL(). In contrast to the case of H¹(), there is no such characterization for general positive functions: every Orlicz space strictly larger than L log L() contains positive functions which do not belong to $H{¹}_{\surd }\left(\right)$, and no Orlicz space...