Abstract. We prove that if $\lambda(x,y)$ is an odd multiplier for $L({\msbm R}^2)$, then its Ces?ro mean defined by $$\sigma\lambda(u,v) : = (uv)^{-1} \int^u_0 \int^v_0 \lambda(x,y) dx dy$$ is the double Fourier transform of some function in ${\cal H}({\msbm R}\times{\msbm R})$, the Hardy space on the plane defined by R. Fefferman. As a corollary, we obtain that if $\lambda(x,y)$ is an arbitrary multiplier for $L({\msbm R}^2)$, then we have necessarily $$\int^\infty_0 \int^\infty_0 \Bigl|{1\over uv} \int^u_0 \int^v_0 \{\lambda(x,y) - \lambda(x, -y) -\lambda(-x,y) + \lambda(-x,-y)\} dx dy\Bigr| {du\over u} {dv\over v} < \infty.$$ We also present analogous results in the case of odd multipliers for $L({\cal T}^2)$ involving double Fourier series.
AMS Subject Classification
(1991): 42B30
Keyword(s):
double Fourier transform,
double Hilbert transforms,
Hardy space on product domain,
Hardy inequality,
Ces?ro 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 June 20, 1995 and in revised form April 30, 1996. (Registered under 6127/2009.)
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