A multidimensional Paley-Wiener theorem

The Paley-Wiener theorem states that the Hilbert transform of an integrable odd function, which is monotone on R + , is integrable.There exists an extension of this result for functions with generalized monotonicity.In this paper, we extend the latter result to the multivariate case.What is proved under a multidimensional condition of generalized monotonicity, is the integrability of the Hilbert transforms with respect to separate variables and their combinations for the groups of the variables.In other words, the main result ensures the belonging of an integrable function odd in each variable to the product Hardy space.