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 $H:=-\mathrm{\Delta }+\mathrm{\mid }x{\mathrm{\mid }}^2$, where $\mathrm{\Delta }$ 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 $$(-\mathrm{\Delta }+\mathrm{\mid }x{\mathrm{\mid }}^2{)}^{\sigma }u=f,(-\mathrm{\Delta }+\mathrm{\mid }x{\mathrm{\mid }}^2+I{)}^{\sigma }u=f,$$ 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.