Continuity of Multi-parameter Paraproduct

Abstract

To study the boundedness of operators on multi-parameter local Hardy space $h^{p}({\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})$, inhomogeneous Journé class has been introduced. It is well known that operators in the Journé class are bounded on multi-parameter Hardy space $H^{p}({\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})$ if and only if $T^{*}_{1}(1)=T^{*}_{2}(1)=0$ for p near 1. Under the same conditions, operators in inhomogeneous Journé class are bounded on $h^{p}({\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})$. In this paper, We give an operator belonging to the inhomogeneous Journé class without $T^{*}_{1}(1)=T^{*}_{2}(1)=0$ and prove its boundedness from $h^{p}({\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})$ to $h^{p}({\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})$ by almost orthogonality estimates. It implies that $T^{*}_{1}(1)=T^{*}_{2}(1)=0$ is not a necessary condition for the boundedness on $h^{p}({\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})$ of a singular operator in the inhomogeneous Journé class.