A Class of Multiparameter Oscillatory Singular Integral Operators: Endpoint Hardy Space Bounds

TL;DR

This paper establishes endpoint Hardy space bounds for a class of multiparameter oscillatory singular integral operators, revealing differences from classical $L^p$ theories and addressing challenges due to non-decaying kernels.

Contribution

It introduces endpoint bounds on $H^1$ for multiparameter oscillatory operators, a case not covered by traditional methods, and characterizes when these bounds are uniform over polynomial subspaces.

Findings

Endpoint $H^1$ bounds are established for certain multiparameter oscillatory operators.

The Hardy space and $L^p$ theories for these operators differ significantly.

A characterization of polynomial subspaces where bounds hold uniformly is provided.

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 Journ\'e-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.