[Paper Review] A Simple Approach to Functional Inequalities for Non-local Dirichlet Forms

Jian Wang|arXiv (Cornell University)|Jun 12, 2013

Analytic and geometric function theory7 references3 citations

TL;DR

This paper presents a direct and general method to derive Poincaré-type inequalities, entropy inequalities, and Beckner-type inequalities for non-local Dirichlet forms with general jump kernels. By establishing sufficient conditions on the potential function $V$ and jump density $\rho$, it proves functional inequalities in $L^p$ settings, yielding new criteria for fractional Poincaré inequalities and extending results beyond Lévy measures.

ABSTRACT

With direct and simple proofs, we establish Poincaré type inequalities (including Poincaré inequalities, weak Poincaré inequalities and super Poincaré inequalities), entropy inequalities and Beckner-type inequalities for non-local Dirichlet forms. The proofs are efficient for non-local Dirichlet forms with general jump kernel, and also work for $L^p (p>1)$ settings. Our results yield a new sufficient condition for fractional Poincaré inequalities, which were recently studied in \cite{MRS,Gre}. To our knowledge this is the first result providing entropy inequalities and Beckner-type inequalities for measures more general than Lévy measures.

Motivation & Objective

• To establish a simple, direct method for deriving functional inequalities—Poincaré, weak Poincaré, super Poincaré, entropy, and Beckner-type—for non-local Dirichlet forms.
• To provide sufficient conditions on the potential $V$ and jump kernel $\rho$ that ensure these inequalities hold, even in $L^p$ settings for $p>1$.
• To extend the scope of functional inequalities beyond Lévy measures, offering new criteria for fractional Poincaré inequalities.
• To apply the results to porous media equations, deriving convergence rates via $L^p$ functional inequalities.

Proposed method

• The method relies on direct estimation of the Bergman distance $D_\Phi(a,b) = \Phi(a) - \Phi(b) - \Phi'(b)(a-b)$ associated with a convex function $\Phi$, enabling unified treatment of various functional inequalities.
• It introduces a general non-local Dirichlet form $D_{\rho,V}(f,f) = \iint_{x\neq y} (f(x)-f(y))^2 \rho(|x-y|) \, dy \, \mu_V(dx)$, where $\mu_V(dx) = e^{-V(x)}dx$.
• Sufficient conditions are derived via pointwise lower bounds on $ (e^{V(x)} + e^{V(y)}) \rho(|x-y|) $, ensuring Poincaré-type inequalities.
• The approach uses Hölder's inequality and integral estimates to bound $\mu_V(f^p) - \mu_V(f)^p$ in terms of $D_{\rho,V}(f,f^{p-1})$, leading to Beckner-type inequalities.
• The method avoids harmonic analysis techniques used in prior works, offering a more accessible and general framework.
• Applications to porous media equations are derived by linking the inequality $\mu_V(f^{m+1}) \leq c^{-1} D_{\rho,V}(f,f^m)$ to the decay rate of solutions.

Experimental results

Research questions

• RQ1 Under what conditions on $V$ and $\rho$ does the non-local Dirichlet form $D_{\rho,V}$ satisfy a Poincaré inequality with explicit constant?
• RQ2 Can Beckner-type inequalities be established for non-local Dirichlet forms with general jump kernels and in $L^p$ spaces for $p>1$?
• RQ3 What is a sufficient condition for super Poincaré inequalities in non-local settings, and how does it relate to the potential $V$ and jump density $\rho$?
• RQ4 How can functional inequalities for non-local Dirichlet forms be used to analyze convergence rates in porous media equations?
• RQ5 Can the results be extended to measures more general than Lévy measures, particularly in the context of fractional Poincaré inequalities?

Key findings

• A Poincaré inequality holds if $ (e^{V(x)} + e^{V(y)}) \rho(|x-y|) \geq c > 0 $ for all $x \neq y$, with the optimal constant $c^{-1}$.
• A weak Poincaré inequality holds with a rate function $\alpha(r)$ depending on the tail behavior of $\mu_V$ and the infimum of $ (e^{V(x)} + e^{V(y)}) \rho(|x-y|) $ over small distances.
• A super Poincaré inequality holds if $ e^{V(x)} + e^{V(y)} \geq \frac{w(x) + w(y)}{\rho(|x-y|)} $ for some $w$ with $\lim_{|x|\to\infty} w(x) = \infty$, with explicit bounds on the constant $\beta(r)$.
• Beckner-type inequalities for $p \in (1,2]$ are established via Hölder's inequality and the Poincaré condition, yielding $ \mu_V(f^p) - \mu_V(f)^p \leq c^{-1} D_{\rho,V}(f, f^{p-1}) $.
• For the example with $ e^{-V(x)} = C_{d,\varepsilon}(1+|x|)^{-(d+\varepsilon)} $ and $ \rho(r) = r^{-d-\alpha} $, the paper shows that: (i) fractional Poincaré inequality holds if $ \varepsilon \geq \alpha $, (ii) weak Poincaré holds with rate $ r^{-\varepsilon/\alpha} $ if $ \varepsilon < \alpha $, and (iii) super Poincaré holds with rate $ r^{-1/\varepsilon} $ if $ \varepsilon > \alpha $.
• For the porous media equation $ \partial_t u = L_{\rho,V}(u^m) $, the solution satisfies $ \mu_V((T_t f)^2) \leq \left[ \mu_V(f^2)^{-(m-1)/2} + c^{-1}(m-1)t \right]^{-2/(m-1)} $ under the Poincaré condition, implying algebraic decay.