# Revista Matemática Iberoamericana

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**Published online first: 2019-10-21**

**DOI: 10.4171/rmi/1144**

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

Odysseas Bakas^{[1]}, Eric Latorre

^{[2]}, Diana C. Rincón M.

^{[3]}and James Wright

^{[4]}(1) Stockholm University, Sweden

(2) Instituto de Ciencias Matemáticas (ICMAT), Madrid, Spain

(3) Universidad Nacional Autónoma de México, Mexico

(4) University of Edinburgh, UK

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 Journé-type 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 $L^p$, $1 < p < \infty$) 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.

*Keywords: *Multiparameter singular integral operators, oscillatory integrals, endpoint bounds, Hardy spaces

Bakas Odysseas, Latorre Eric, Rincón M. Diana, Wright James: A class of multiparameter oscillatory singular integral operators: endpoint Hardy space bounds. *Rev. Mat. Iberoam.* Electronically published on October 21, 2019. doi: 10.4171/rmi/1144 (to appear in print)