The Molecular Characterizations of Variable Triebel-Lizorkin Spaces Associated with the Hermite Operator and Its Applications

In this article, we introduce inhomogeneous variable Triebel-Lizorkin spaces, $F_{p(\cdot),q(\cdot)}^{\alpha(\cdot),H}(\mathbb R^n)$, associated with the Hermite operator $H:=-\Delta+|x|^2$, where $\Delta$ is the Laplace operator on $\mathbb R^n$, and mainly establish the molecular characterization of this space. As applications, we obtain some regularity results to fractional Hermite equations $$(-\Delta+|x|^2)^\sigma u=f,\quad (-\Delta+|x|^2+I)^\sigma u=f,$$ and the boundedness of spectral multiplier associated to the operator $H$ on the variable Triebel-Lizorkin space $F_{p(\cdot),q(\cdot)}^{\alpha(\cdot),H}(\mathbb R^n)$. Furthermore, we explain the relationship between $F_{p(\cdot),q(\cdot)}^{\alpha(\cdot),H}(\mathbb R^n)$ and the variable Triebel-Lizorkin spaces $F_{p(\cdot),q(\cdot)}^{\alpha(\cdot)}(\mathbb R^n)$ (introduced in Diening t al. J. Funct. Anal. 256(2009), 1731-1768.) via the atomic decomposition.