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.)
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