On the Hardy space \(H^ 1\) on product of half-spaces (Q1096040)

We show that the Hardy space \(H^ 1_{anal}({\mathbb{R}}^ 2_ +\times {\mathbb{R}}^ 2_ +)\) can be identified with the class of functions f such that f and all its double and partial Hilbert transforms \(H_ kf\) belong to \(L^ 1({\mathbb{R}}^ 2)\). A basic tool used in the proof is the bi- subharmonicity of \(| F| ^ q\), where F is a vector that satisfies a generalized conjugate system of Cauchy-Riemann type.