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Hardy spaces and Beurling algebras. (English) Zbl 0681.42014

The author introduces the spaces A p and B p , which have previously been considered by Chen and Lau on the line, on n . Let B(x,R)={y n ||x-y|<R},

C k =B(0,2 k ) ¯B(0,2 k-1 ) ¯={xR n |2 k-1 <|x|2 k },

χ k be the characteristic function of C k , and define (1) f A p = k=0 2 kn/p ' fχ k p <+, (2) f B p =sup k0 2 -kn/p fχ k p · One has L B p and A p L 1 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 K p,1 n/p ' while (2) is an equivalent norm on K p, -n/p which gives another explanation for the duality theorem below.) The author also defines

CMO p ={fL loc p ( n )|sup R1 (1 |B(0,R)| B(0,R) |f(x)-C R | p dx) 1/p <}·

It should be noted that Chen and Lau had already shown in one dimension that the spaces CMO p are distinct.

The author proves duality (A p ) * =B p ' , boundedness of Calderon-Zygmund operators from B p CMO p 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 A p . He defines the Hardy spaces for A p by requiring that some functional (the non-tangential maximal function, the vertical maximal function, the tangential maximal function, the grand maximal function) be in A p . He shows that the resulting space HA p is independent of the function chosen, has an atomic decomposition with central atoms, satisfies (HA p ) * =CMO p ' and gives a Fefferman-Stein type decomposition of CMO p as g=g 0 + j=1 n R j g j , where g j B p .

Reviewer: R.Johnson

MSC:
42B30H p -spaces (Fourier analysis)
42B20Singular and oscillatory integrals, several variables