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.
