Lusin functions on product spaces (Q1107053)

The author extends to the product space setting parts of the Caldéron- Torchinsky theory of parabolic \(H^ p\) spaces. In particular he shows the equivalence of the \(L^ p\) `norms' of the Lusin functions and the nontangential maximal functions arising from certain two-parameter families of linear transformations on product spaces. An extension of the weak type estimate for the double Hilbert transform [cf. \textit{C. Fefferman}, Studia Math. 44, 1-15 (1972; Zbl 0239.44006)] is provided as an application of the theory.