Abstract. We continue the study of the Cesàro mean $\sigma\lambda (u,v)$ of a multiplier $\lambda(x,y)$ for $L({\msbm R}^2)$. Among others, we prove that if $\lambda(x,y)$ is even in each variable, then (i) $\sigma\lambda $ is also a multiplier for $L({\msbm R}^2)$; (ii) $\sigma\lambda $ is the Fourier transform of a function $f\in L({\msbm R}^2)$ if and only if the associated finite Borel measure $\mu $ is continuous on the axes $y=0$ and $x=0$, respectively; (iii) we give necessary and sufficient conditions in order that $\sigma\lambda $ be the Fourier transform of a function $f\in{\cal H}({\msbm R}\times{\msbm R}) \subset{\cal H}({\msbm R}^2)$. We also present analogous results in the case of even multipliers for $L({\msbm T}^2)$ involving double Fourier series.

AMS Subject Classification (1991): 42B30

Keyword(s): double Fourier transform, double Hilbert transforms, Hardy space on product domain, Hardy inequality, Cesàro mean, L({\msbm R}^2), {\cal H}({\msbm R}\times{\msbm R}), multiplier forand, double Fourier series, conjugate functions, arithmetic mean, L({\msbm T}^2), {\cal H}({\msbm T}\times{\msbm T}), multiplier forand

Received August 29, 1994. (Registered under 5612/2009.)