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

2024/01/03 by Qi Sun, Sun, Qi, Ciqiang Zhuo +1
Mathematics · #42B35 #47B15 #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2401.01768

openalex publication_date 2024/01/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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

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