Littlewood-Paley theory for the differential operator \(\frac {\partial ^{2}}{\partial x_{1}^{2}} \frac{\partial ^{2}}{\partial x_{2}^{2}} - \frac{\partial ^{2}}{\partial x_{3}^{2}}\) (Q961990)

Let \(\mathbb D\) denote the group of all dilations \((x_1,x{}_2,x{}_3)\mapsto (2^{\nu_1}x_1,2^{\nu_2}x_2,2^{\nu_1+\nu_2}x_3)\) (\(\nu_1,\nu_2\in\mathbb Z\)) and consider the operator \[ \triangle_{\mathbb D}:= \partial^2_{x_1}\partial^2_{x_2}-\partial^2_{x_3}. \] In this article, a Littlewood-Paley theory and new Triebel-Lizorkin spaces \({\dot F}_p^{\alpha q}({\mathbb D})\) associated to \({\mathbb D}\) are studied to obtain estimates \(\| \triangle_{\mathbb D}^m f\| _{{\dot F}_p^{\alpha-2m q}({\mathbb D})}\simeq \| f\|_{{\dot F}_p^{\alpha q}({\mathbb D)}}\), which correspond to the similar estimates for the Laplacian with the usual Triebel-Lizorkin norms. The paper contains a rather technical work devoted to molecular decompositions, new Littlewood-Paley functions, and generalized \(\varphi-\psi\) transforms in the sense of Frazier and Jawerth.