The molecular characterizations of variable Triebel–Lizorkin spaces associated with the Hermite operator and its applications

Abstract

In this article, we introduce inhomogeneous variable Triebel–Lizorkin spaces, $F_{p\left(\cdot \right),q\left(\cdot \right)}^{\alpha \left(\cdot \right),H}\left(R^n\right)$, associated with the Hermite operator $H≔-\Delta +{\left|x\right|}^2$, where $\Delta$ is the Laplace operator on $R^n$, and mainly establish the molecular characterization of these spaces. As applications, we obtain some regularity results to fractional Hermite equations ${(-\Delta +{\left|x\right|}^2)}^{\sigma }u=f,{(-\Delta +{\left|x\right|}^2+I)}^{\sigma }u=f,$ where $\sigma \in (0,\infty )$, and the boundedness of spectral multiplier associated to the operator $H$ on the variable Triebel–Lizorkin space $F_{p\left(\cdot \right),q\left(\cdot \right)}^{\alpha \left(\cdot \right),H}\left(R^n\right)$. Furthermore, we explain the relationship between $F_{p\left(\cdot \right),q\left(\cdot \right)}^{\alpha \left(\cdot \right),H}\left(R^n\right)$ and the variable Triebel–Lizorkin spaces $F_{p\left(\cdot \right),q\left(\cdot \right)}^{\alpha \left(\cdot \right)}\left(R^n\right)$ (introduced in Diening et al. (2009).) via the atomic decomposition.