ACTA issues

The Ces?ro operator on the Banach algebra of $L({\msbm R}^2)$ multipliers. I (Odd case)

Dang Vu Giang, Ferenc Móricz

Acta Sci. Math. (Szeged) 62:3-4(1997), 433-456

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.)