Hardy spaces on the plane and double Fourier transforms. (Q1386221)

In this paper, very accurate consideration of the interplay between \(H(\mathbf R^2)\), the space of integrable functions with integrable Riesz transforms, \[ H(\mathbf R^2\times \mathbf R^2)=\{f\in L(\mathbf R^2): H_1f,H_2f,H_1H_2f\in L(\mathbf R^2)\}, \] where \(H_jf\) is the Hilbert transform with respect to the \(j\)th variable, and \[ J(\mathbf R^2\times \mathbf R^2)=\{f\in L(\mathbf R^2): H_1f,H_2f\in L(\mathbf R^2)\}, \] is given. Theorem 2, the main result of this paper, gives simple sufficient conditions for a function of two variables to be the double Fourier transform of a function in \(L(\mathbf R^2)\) or \(H(\mathbf R^2\times \mathbf R^2)\). The author gives an extension to double Fourier series. For two open problems, conjectures are formulated. Note that a result analogous to Theorem 2 was proved in a paper by the reviewer [Anal. Math. 19, No. 2, 151--168 (1993; Zbl 0794.42006)], not only in the one-dimensional case but for any dimension.