A class of multiparameter oscillatory singular integral operators: endpoint Hardy space bounds

Abstract

We establish endpoint bounds on a Hardy space H 1 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 Journétype covering lemma to deduce bounds on product H 1 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 L 2 (or more generally L p , 1 < p < ∞) bounds. Here we initiate an investigation of endpoint bounds on the rectangular Hardy space H 1 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 L p theories for these operators are very different.