Characterizations of Hardy spaces on tube domains over polyhedral cones

Source abstract

Abstract This paper is devoted to the equivalence of various characterizations of holomorphic Hardy spaces on tube domains over polyhedral cones. We establish a new iterated Poisson integral formula that reproduces holomorphic functions on such domains. However, this formula shows that holomorphic functions have boundary values in a new type of Hardy space of real variables on their Shilov boundaries , which cannot be treated by standard multiparameter harmonic analysis. We overcome this difficulty by developing techniques suitably adapted to this setting. Using the iterated Poisson integral as our approximation to the identity and employing a lifting technique, we introduce various notions of multiparameter analysis adapted to tube domains, such as twisted rectangles, new nontangential approach regions, nontangential maximal functions, and Littlewood–Paley‐type functions. All these notions exhibit new geometric features associated with polyhedral cones and involve hidden parameters, as in the flag setting. We develop the necessary multiparameter tools to investigate these new Hardy spaces. In particular, we apply these tools to obtain equivalent characterizations of the holomorphic Hardy spaces on tube domains in terms of nontangential maximal, Lusin–Littlewood–Paley area, and Littlewood–Paley ‐functions.