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

Qi Sun, Ciqiang Zhuo

Abstract

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.

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

Abstract

In this article, we introduce inhomogeneous variable Triebel-Lizorkin spaces, , associated with the Hermite operator , where is the Laplace operator on , and mainly establish the molecular characterization of this space. As applications, we obtain some regularity results to fractional Hermite equations and the boundedness of spectral multiplier associated to the operator on the variable Triebel-Lizorkin space . Furthermore, we explain the relationship between and the variable Triebel-Lizorkin spaces (introduced in Diening t al. J. Funct. Anal. 256(2009), 1731-1768.) via the atomic decomposition.
Paper Structure (9 sections, 17 theorems, 102 equations)

This paper contains 9 sections, 17 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Hermite Operators and Kernels
  4. Variable Function Spaces
  5. Molecular Characterizations
  6. Boundedness of Singular Integrals on $F_{p(\cdot),q(\cdot)}^{\alpha(\cdot),H}({{{\mathbb R}}^n})$
  7. Regularity of Fractional Hermite Equation
  8. Boundedness of Spectral Multipliers
  9. Relationships between Spaces $F_{p(\cdot),q(\cdot)}^{\alpha(\cdot),H}({{{\mathbb R}}^n})$ and $F_{p(\cdot),q(\cdot)}^{\alpha(\cdot)}({{{\mathbb R}}^n})$

Key Result

Lemma 2.1

For any given $k\in{\mathbb N}_0$, there exist positive constants $C$ and $\delta$ such that, for any $t\in(0,\infty)$ and any $x,\ y\in {{{\mathbb R}}^n}$,

Theorems & Definitions (37)

  • Lemma 2.1
  • Proposition 2.2
  • Remark 2.3
  • Remark 2.4
  • Lemma 2.5
  • Definition 2.6
  • Remark 2.7
  • Definition 3.1
  • Remark 3.2
  • Definition 3.3
  • ...and 27 more