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We establish endpoint bounds on a Hardy space 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 Journé-type covering lemma to deduce bounds on product 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 (or more generally , ) bounds. Here we initiate an investigation of endpoint bounds on the rectangular Hardy space 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 theories for these operators are very different.
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Odysseas Bakas, Eric Latorre, Diana C. Rincón M., James Wright, A class of multiparameter oscillatory singular integral operators: endpoint Hardy space bounds. Rev. Mat. Iberoam. 36 (2020), no. 2, pp. 611–639DOI 10.4171/RMI/1144