A Class of Multiparameter Oscillatory Singular Integral Operators: Endpoint Hardy Space Bounds

We establish endpoint bounds on a Hardy space H1 for a natural class of multiparameter singular integral operators which do not decay away from the support of rectangular atoms. Hence the usual argument via a Journ\'e-type covering lemma to deduce bounds on product H1 is not valid. We consider the class of multiparameter oscillatory singular integral operators given by convolution with the classical multiple Hilbert transform kernel modulated by a general polynomial oscillation. Various characterisations are known which give L2 (or more generally Lp,1<p<∞) bounds. Here we initiate an investigation of endpoint bounds on the rectangular Hardy space H1 in two dimensions; we give a characterisation when bounds hold which are uniform over a given subspace of polynomials and somewhat surprisingly, we discover that the Hardy space and Lp theories for these operators are very different.