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.
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