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

### Odysseas Bakas

Stockholm University, Sweden### Eric Latorre

Instituto de Ciencias Matemáticas (ICMAT), Madrid, Spain### Diana C. Rincón M.

Universidad Nacional Autónoma de México, Mexico### James Wright

University of Edinburgh, UK

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## 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 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.

## Cite this article

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 (2019), no. 2, pp. 611–639

DOI 10.4171/RMI/1144