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Symmetry and functional inequalities for stable Lévy-type operators

Lu-Jing Huang, Tao Wang

Abstract

In this paper, we provide the sufficient and necessary conditions for the symmetry of the following stable Lévy-type operator $\mathcal{L}$ on $\mathbb{R}$: $$\mathcal{L}=a(x){Δ^{α/2}}+b(x)\frac{\d}{\d x},$$ where $a,b$ are the continuous positive and differentiable functions, respectively. Under the assumption of symmetry, we further study the criteria for functional inequalities, including Poincaré inequalities, logarithmic Sobolev inequalities and Nash inequalities. Our proofs rely on the Orlicz space theory and the estimates of the Green functions.

Symmetry and functional inequalities for stable Lévy-type operators

Abstract

In this paper, we provide the sufficient and necessary conditions for the symmetry of the following stable Lévy-type operator on : where are the continuous positive and differentiable functions, respectively. Under the assumption of symmetry, we further study the criteria for functional inequalities, including Poincaré inequalities, logarithmic Sobolev inequalities and Nash inequalities. Our proofs rely on the Orlicz space theory and the estimates of the Green functions.
Paper Structure (8 sections, 15 theorems, 202 equations)

This paper contains 8 sections, 15 theorems, 202 equations.

Table of Contents

  1. Introduction and main results
  2. Symmetry for stable Lévy-type operators
  3. Estimates for Green function
  4. Monotonicity of Green function $G_X^{[-1,1]}(\cdot,\cdot)$
  5. Upper bound for Green operator $G^{[-n,n]}$
  6. Orlicz-Poincaré inequality
  7. Proofs of Theorems \ref{['thm-logS']}, \ref{['thm-Nash']} and \ref{['interp']}
  8. Super-Poincaré inequality

Key Result

Theorem 1.1

Let $\mathscr{P}$ be the set of measures on $\mathbb{R}$ that are absolute continuous with respect to Lebesgue measure. Then the Lévy-type operator fractional-drift is symmetric with respect to a measure $\mu\in\mathscr{P}$ if and only if $b\equiv 0$. In this case, its reversible measure is $\mu({\h

Theorems & Definitions (23)

  • Theorem 1.1
  • Theorem 1.2: W23
  • Theorem 1.3: Logarithmic Sobolev inequality
  • Theorem 1.4: Nash inequality
  • Theorem 1.5: Super-Poincaré inequality
  • Theorem 1.6: Interpolations of Poincaré and logarithmic Sobolev inequalities
  • Remark 1.7
  • Example 1.8
  • Example 1.9
  • Remark 1.10
  • ...and 13 more