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Intrinsic Hardy inequality for fractional Kolmogorov operator

Damir Kinzebulatov

Abstract

We obtain Hardy inequality for non-local diffusion operator with singular drift, in the case when the strength of attraction to the origin by the drift takes the critical value.

Intrinsic Hardy inequality for fractional Kolmogorov operator

Abstract

We obtain Hardy inequality for non-local diffusion operator with singular drift, in the case when the strength of attraction to the origin by the drift takes the critical value.
Paper Structure (9 sections, 9 theorems, 150 equations, 2 figures)

This paper contains 9 sections, 9 theorems, 150 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Remarks
  4. Further discussion
  5. Notations and auxiliary results used in the proofs
  6. Proof of Theorem \ref{['thm1']}
  7. Proof of Theorem \ref{['thm2']}
  8. Proof of Corollary \ref{['cor1']}
  9. Proof of Corollary \ref{['cor2']}

Key Result

Theorem 1

Let $1<\alpha < 2$. There exists constant $c=c(d,\alpha)$ such that, for and every $u \in C^\infty=C^\infty(\Pi^d)$, and, furthermore, for all $\varepsilon>0$, where $b_\varepsilon:=e^{-\varepsilon (-\Delta)^{\frac{\alpha_1}{2}}} b$ (De Giorgi mollifier) with $0<\alpha_1 \leq 2$ fixed by $0 \leq \alpha \leq d-\alpha_1$ possibly after increasing $c$ to $c=c(d,\alpha,\alpha_1)$,.

Figures (2)

  • Figure 1: The dependence of the exponent in the Lyapunov function $|\cdot|^{-d+\gamma}$ on the coupling constant (strength of attraction by the drift) $\nu \in [0,\nu_\star[$. The vertical line represents the largest value of the coupling constant $\nu_{{\rm SV}}=\lim_{p \uparrow \infty} \frac{p}{d-\alpha} \frac{4(p-1)}{p^2}\kappa_{\frac{d-\alpha}{2}}=\frac{2^{\alpha+2}}{d-\alpha}\left(\frac{\Gamma(\frac{d+\alpha}{4})}{\Gamma(\frac{d-\alpha}{4})}\right)^2$that can be attained using the Stroock-Varopoulos inequality \ref{['SV']}. It is seen that nothing dramatic happens to the Lyapunov function as $\nu$ reaches and surpasses $\nu_{{\rm SV}}$.
  • Figure :

Theorems & Definitions (13)

  • Theorem 1: A priori Hardy inequality
  • Corollary 1: Contractivity in the hyperbolic Orlicz space
  • Theorem 2: A posteriori Hardy inequality
  • Corollary 2
  • Lemma 1
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Corollary 3
  • ...and 3 more