The author introduces the spaces and , which have previously been considered by Chen and Lau on the line, on . Let ,
be the characteristic function of , and define (1) (2) One has and for every p. (The reviewer has shown [Proc. Lond. Math. Soc., III. Ser. 29, 127-141 (1974; Zbl 0295.46051)] that (1) is an equivalent norm on the Beurling-Herz space while (2) is an equivalent norm on which gives another explanation for the duality theorem below.) The author also defines
It should be noted that Chen and Lau had already shown in one dimension that the spaces are distinct.
The author proves duality , boundedness of Calderon-Zygmund operators from and from central (1,p) atoms (which again are atoms but always having support in a ball of radius greater than or equal to one centered at the origin) into . He defines the Hardy spaces for by requiring that some functional (the non-tangential maximal function, the vertical maximal function, the tangential maximal function, the grand maximal function) be in . He shows that the resulting space is independent of the function chosen, has an atomic decomposition with central atoms, satisfies and gives a Fefferman-Stein type decomposition of as , where .