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Hardy spaces and Beurling algebras. (English) Zbl 0681.42014
The author introduces the spaces $A\sp p$ and $B\sp p$, which have previously been considered by Chen and Lau on the line, on ${\bbfR}\sp n$. Let $B(x,R)=\{y\in {\bbfR}\sp n\vert \vert x-y\vert <R\}$, $$C\sb k=\overline{B(0,2\sp k)}\setminus \overline{B(0,2\sp{k-1})}=\{x\in R\sp n\vert 2\sp{k-1}<\vert x\vert \le 2\sp k\},$$ $\chi\sb k$ be the characteristic function of $C\sb k$, and define (1) $\Vert f\Vert\sb{A\sp p}=\sum\sp{\infty}\sb{k=0}2\sp{kn/p}{}'\Vert f\chi\sb k\Vert\sb p<+\infty,$ (2) $\Vert f\Vert\sb{B\sp p}=\sup\sb{k\ge 0}2\sp{-kn/p}\Vert f\chi\sb k\Vert\sb p.$ One has $L\sp{\infty}\subseteq B\sp p$ and $A\sp p\subseteq L\sp 1$ 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 $K\sb{p,1}\sp{n/p'}$ while (2) is an equivalent norm on $K\sb{p,\infty}\sp{-n/p}$ which gives another explanation for the duality theorem below.) The author also defines $$CMO\sp p=\{f\in L\sp p\sb{loc}({\bbfR}\sp n)\vert \sup\sb{R\ge 1}(\frac{1}{\vert B(0,R)\vert}\int\sb{B(0,R)}\vert f(x)-C\sb R\vert\sp pdx)\sp{1/p}<\infty \}.$$ It should be noted that Chen and Lau had already shown in one dimension that the spaces $CMO\sp p$ are distinct. The author proves duality $(A\sp p)\sp*=B\sp{p'}$, boundedness of Calderon-Zygmund operators from $B\sp p\to CMO\sp p$ 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 $A\sp p$. He defines the Hardy spaces for $A\sp p$ by requiring that some functional (the non-tangential maximal function, the vertical maximal function, the tangential maximal function, the grand maximal function) be in $A\sp p$. He shows that the resulting space $HA\sp p$ is independent of the function chosen, has an atomic decomposition with central atoms, satisfies $(HA\sp p)\sp*=CMO\sp{p'}$ and gives a Fefferman-Stein type decomposition of $CMO\sp p$ as $g=g\sb 0+\sum\sp{n}\sb{j=1}R\sb jg\sb j$, where $g\sb j\in B\sp p$.
Reviewer: R.Johnson

##### MSC:
 42B30 $H^p$-spaces (Fourier analysis) 42B20 Singular and oscillatory integrals, several variables
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