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

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