Abstract. Following R. Fefferman, the Hardy space ${\cal H}^1({\msbm R}\times{\msbm R})$ of functions $f\in L^1({\msbm R}^2)$ is defined by the requirement that its Hilbert transforms $H_1f, H_2f$, and $H_1H_2f$ also belong to $L^1({\msbm R}^2)$. The proof of the statement claimed in the title relies on the closed graph theorem and on the fact that if a function $f\in L^1({\msbm R}^2)$ is such that its Fourier transform ${\hat f}(u,v)=0$ unless $u\ge0, v\ge0$, then $f\in{\cal H}^1 ({\msbm R}\times{\msbm R})$. The following reversed statement is also proved: If $f\in{\cal H}^1 ({\msbm R}\times{\msbm R})$, then $f$ can be represented in the form $f=f_1+f_2+f_3+f_4$, where each $f_j\in{\cal H}^1({\msbm R}\times{\msbm R})$ and ${\hat f}_1(u,v)=0$ unless $u\ge0, v\ge0$; ${\hat f}_2(u,v)=0$ unless $u\le0, v\ge0$; ${\hat f}_3(u,v)=0$ unless $u\ge0, v\le0$; and ${\hat f}_4(u,v)=0$ unless $u\le0, v\le0$.

AMS Subject Classification (1991): 47D05

Keyword(s): Hilbert transforms, ${\cal H}^1({\msbm R}\times{\msbm R})$, Hardy space, Fourier transform, Cesàro operator, closed graph theorem

Received September 6, 1996. (Registered under 6115/2009.)