The Molecular Characterizations of Variable Triebel-Lizorkin Spaces Associated with the Hermite Operator and Its Applications
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arXiv
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2024
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| _version_ | 1866914628699684864 |
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| author | Sun, Qi Zhuo, Ciqiang |
| author_facet | Sun, Qi Zhuo, Ciqiang |
| contents | In this article, we introduce inhomogeneous variable Triebel-Lizorkin spaces, $F_{p(\cdot),q(\cdot)}^{α(\cdot),H}(\mathbb R^n)$, associated with the Hermite operator $H:=-Δ+|x|^2$, where $Δ$ 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 $$(-Δ+|x|^2)^σu=f,\quad (-Δ+|x|^2+I)^σu=f,$$ and the boundedness of spectral multiplier associated to the operator $H$ on the variable Triebel-Lizorkin space $F_{p(\cdot),q(\cdot)}^{α(\cdot),H}(\mathbb R^n)$. Furthermore, we explain the relationship between $F_{p(\cdot),q(\cdot)}^{α(\cdot),H}(\mathbb R^n)$ and the variable Triebel-Lizorkin spaces $F_{p(\cdot),q(\cdot)}^{α(\cdot)}(\mathbb R^n)$ (introduced in Diening t al. J. Funct. Anal. 256(2009), 1731-1768.) via the atomic decomposition. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_01768 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Molecular Characterizations of Variable Triebel-Lizorkin Spaces Associated with the Hermite Operator and Its Applications Sun, Qi Zhuo, Ciqiang Functional Analysis Analysis of PDEs Classical Analysis and ODEs 42B35, 47B15 In this article, we introduce inhomogeneous variable Triebel-Lizorkin spaces, $F_{p(\cdot),q(\cdot)}^{α(\cdot),H}(\mathbb R^n)$, associated with the Hermite operator $H:=-Δ+|x|^2$, where $Δ$ 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 $$(-Δ+|x|^2)^σu=f,\quad (-Δ+|x|^2+I)^σu=f,$$ and the boundedness of spectral multiplier associated to the operator $H$ on the variable Triebel-Lizorkin space $F_{p(\cdot),q(\cdot)}^{α(\cdot),H}(\mathbb R^n)$. Furthermore, we explain the relationship between $F_{p(\cdot),q(\cdot)}^{α(\cdot),H}(\mathbb R^n)$ and the variable Triebel-Lizorkin spaces $F_{p(\cdot),q(\cdot)}^{α(\cdot)}(\mathbb R^n)$ (introduced in Diening t al. J. Funct. Anal. 256(2009), 1731-1768.) via the atomic decomposition. |
| title | The Molecular Characterizations of Variable Triebel-Lizorkin Spaces Associated with the Hermite Operator and Its Applications |
| topic | Functional Analysis Analysis of PDEs Classical Analysis and ODEs 42B35, 47B15 |
| url | https://arxiv.org/abs/2401.01768 |