JournalsrmiVol. 36, No. 2pp. 611–639

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

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Abstract

We establish endpoint bounds on a Hardy space H1H^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 H1H^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 L2L^2 (or more generally LpL^p, 1<p<1 < p < \infty) bounds. Here we initiate an investigation of endpoint bounds on the rectangular Hardy space H1H^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 LpL^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