[Paper Review] Harnack's inequality for fractional nonlocal equations

Pablo Raúl Stinga, Chao Zhang|arXiv (Cornell University)|Mar 7, 2012

Differential Equations and Boundary Problems15 references3 citations

TL;DR

This paper establishes interior Harnack's inequalities for nonnegative solutions of fractional nonlocal equations involving various operators, including divergence form elliptic operators, Schrödinger operators, and Bessel operators. Using a generalized Caffarelli-Silvestre extension method and transference techniques, the authors prove that solutions to $ L^\sigma f = 0 $ satisfy $ \sup_K f \leq C \inf_K f $ for compact subsets $ K $, with $ C $ depending only on $ \sigma $, $ n $, $ K $, and operator coefficients, extending classical Harnack theory to nonlocal settings.

ABSTRACT

We prove interior Harnack's inequalities for solutions of fractional nonlocal equations. Our examples include fractional powers of divergence form elliptic operators with potentials, operators arising in classical orthogonal expansions and the radial Laplacian. To get the results we use an analytic method based on a generalization of the Caffarelli--Silvestre extension problem, the Harnack's inequality for degenerate Schrödinger operators proved by C. E. Gutiérrez, and a transference method. In this manner we apply local PDE techniques to nonlocal operators. On the way a maximum principle and a Liouville theorem for some fractional nonlocal equations are obtained.

Motivation & Objective

• To extend Harnack's inequality to fractional nonlocal equations governed by a broad class of second-order differential operators.
• To develop a general framework for proving Harnack-type estimates for nonlocal operators using local PDE techniques.
• To establish maximum principles and Liouville theorems for fractional nonlocal equations as byproducts of the Harnack inequality.
• To unify the treatment of diverse operators—divergence form, harmonic oscillator, Laguerre, Bessel, and radial Laplacian—under a single analytical method.

Proposed method

• Generalizing the Caffarelli-Silvestre extension problem to define fractional powers $ L^\sigma $ of nonlocal operators via the spectral theorem.
• Applying the theory of degenerate elliptic equations with $ A_2 $ weights, particularly Gutiérrez's Harnack inequality for degenerate Schrödinger operators.
• Using a transference method to extend results from one operator class (e.g., $ S_\lambda $) to another (e.g., $ \Delta_\lambda $) via unitary equivalence.
• Constructing explicit integral representations of solutions via heat-diffusion semigroups and Hankel or Fourier transforms.
• Proving convergence of derivatives in $ L^2 $-norm as the extension variable $ y \to 0^+ $, ensuring the trace recovers the original solution.
• Establishing local Hölder continuity and weak solution properties in the extended space to infer regularity in the original domain.

Experimental results

Research questions

• RQ1 Can Harnack's inequality be established for fractional powers $ L^\sigma $ of nonlocal operators beyond the fractional Laplacian?
• RQ2 How can local PDE techniques be adapted to nonlocal operators through extension methods?
• RQ3 What is the role of degenerate elliptic equations with $ A_2 $ weights in proving Harnack inequalities for nonlocal equations?
• RQ4 To what extent do transference methods allow the transfer of Harnack-type estimates across different classes of differential operators?
• RQ5 What are the implications of the Harnack inequality for maximum principles and Liouville-type theorems in nonlocal settings?

Key findings

• A general Harnack inequality holds for all nonnegative solutions $ f \in \mathrm{Dom}(L) $ of $ L^\sigma f = 0 $ in $ L^2(\mathcal{O}, d\eta) $, with $ \sup_K f \leq C \inf_K f $ for compact subsets $ K \subset \mathcal{O} $.
• The constant $ C $ depends only on $ \sigma $, $ n $, $ K $, and the coefficients of the operator $ L $, not on the specific solution $ f $.
• Solutions $ f $ are continuous in $ \mathcal{O} $, and the Harnack inequality implies local Hölder continuity.
• The method applies to a wide class of operators, including divergence form elliptic operators with potentials, Ornstein-Uhlenbeck and harmonic oscillators, Laguerre and ultraspherical operators, and Bessel operators.
• The transference method allows extending results from $ S_\lambda $ to $ \Delta_\lambda $ via the unitary transformation $ Uf(x) = x^\lambda f(x) $, yielding $ (\Delta_\lambda)^\sigma = U^{-1} (S_\lambda)^\sigma U $.
• A maximum principle and a Liouville theorem are derived as consequences of the Harnack inequality, confirming the absence of nontrivial nonnegative solutions under certain conditions.