A Hardy space related to the square root of the Poisson kernel

Journal article, 2010

A real-valued Hardy space H-1 (T) subset of L-1 (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(T). A decreasing function is in H-i(T) if and only if the function is in the Orlicz space L log log L(T). In contrast to the case of H-1(T), there is no such characterization for general positive functions: every Orlicz space strictly larger than L log L(T) contains positive functions which do not belong to H-1(T), and no Orlicz space of type Delta(2) which is strictly smaller than L-1(T) contains every positive function in H-1(T): Finally, we have a characterization of certain eigenfunctions of the hyperbolic Laplace operator in terms of H-1(T).