Boundedness criterion of Journéʼs class of singular integrals on multiparameter Hardy spaces

Abstract

This article concerns nonconvolutional type operators (also known as Journéʼs type operators) associated with a multiparameter family of dilations given by (x1,x2,…,xm)→(δ1x1,δ2x2,…,δmxm) where x1∈Rn1,x2∈Rn2,…,xm∈Rnm and m⩾3. We are especially interested in the boundedness of such operators on the multiparameter Hardy spaces. This work is motivated by Pipherʼs result on the boundedness of these operators from the multiparameter Hp spaces to Lp spaces for 0<p⩽1, and Journéʼs counter-example which shows that the number of parameter plays a crucial role in boundedness of singular integral operators on multiparameter Hardy spaces. Journéʼs work shows that there is a sharp difference between the situations for two and three or more parameters. We establish in this paper the necessary and sufficient conditions under which the singular integral operators in Journéʼs class are bounded on multiparameter Hp spaces (0<p⩽1) with arbitrary number of parameters.