Hardy Spaces on the Plane and Double Fourier Transforms

Abstract

We provide a direct computational proof of the known inclusion ${\cal H}({\bf R} \times {\bf R}) \subseteq {\cal H}({\bf R}^2),$ where ${\cal H}({\bf R} \times {\bf R})$ is the product Hardy space defined for example by R. Fefferman and ${\cal H}({\bf R}^2)$ is the classical Hardy space used, for example, by E.M. Stein. We introduce a third space ${\cal J}({\bf R} \times {\bf R})$ of Hardy type and analyze the interrelations among these spaces. We give simple sufficient conditions for a given function of two variables to be the double Fourier transform of a function in $L({\bf R}^2)$ and ${\cal H}({\bf R} \times {\bf R}),$ respectively. In particular, we obtain a broad class of multipliers on $L({\bf R}^2)$ and ${\cal H}({\bf R}^2),$ respectively. We also present analogous sufficient conditions in the case of double trigonometric series and, as a by-product, obtain new multipliers on $L({\bf T}^2)$ and ${\cal H}({\bf T}^2),$ respectively.