Meyers inequality and strong stability for stable-like operators

2012/07/11 by Richard F. Bass, Bass, Richard F., Hua Ren +1
Computer Science · Mathematics · #45K05 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Functional Analysis (math.FA) #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1207.2715

openalex publication_date 2012/07/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let α∈ (0,2), let \cal E(u,u)=∫_{\Bbb R^d}∫_{\Bbb R^d} (u(y)-u(x))²\fracA(x,y)|x-y|^d+α dy dx be the Dirichlet form for a stable-like operator, let Γu(x)=∫_{\Bbb R^d} (u(y)-u(x))²\fracA(x,y)|x-y|^d+α dy, let L be the associated infinitesimal generator, and suppose A(x,y) is jointly measurable, symmetric, bounded, and bounded below by a positive constant. We prove that if u is the weak solution to Lu=h, then Γu∈ L^p for some p>2. This is the analogue of an inequality of Meyers for solutions to divergence form elliptic equations. As an application, we prove strong stability results for stable-like operators. If A is perturbed slightly, we give explicit bounds on how much the semigroup and fundamental solution are perturbed.

Citations

• Transition Probabilities for Symmetric Jump Processes

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