Littlewood–Paley Theory for the Differential Operator $\frac{{\partial }^2}{\partial x_1^2}\frac{{\partial }^2}{\partial x_2^2}-\frac{{\partial }^2}{\partial x_3^2}$

Abstract

Littlewood–Paley theory for the differential operator, ${\Delta }_{\mathbf{D}}={\partial }_{x_1}^2{\partial }_{x_2}^2-{\partial }_{x_3}^2$, is developed. This study leads to the introduction of a new class of Triebel–Lizorkin spaces ${\dot{F}}_p^{\alpha ,q}(\mathbb{D})$ associated with the dilation $(x_1,x_2,x_3)\to (2^{{\nu }_1}{x}_1,2^{{\nu }_2}{x}_2,2^{{\nu }_1+{\nu }_2}{x}_3)$, $({\nu }_1,{\nu }_2)\in {\mathbb{Z}}^2$. The corresponding atomic and molecular decompositions are obtained. A frame generated by modulations, dilations and translations is also studied. Using this result, we show that ${\Delta }_{\mathbb{D}}$ is a linear isomorphism from ${\dot{F}}_p^{\alpha ,q}$ to ${\dot{F}}_p^{\alpha -2,q}$.