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Regularity theory for general stable operators. (English) Zbl 1346.35220
The authors study the infinitesimal generator of general symmetric stable Lévy processes, namely the operator \[ Lu(x)=\int_{S^{n-1}}\int_{-\infty}^{+\infty}(u(x+\theta r)+u(x-\theta r)-2u(x))\frac{dr}{|r|^{1+2s}}d\mu(\theta), \] where \(s\in (0,1)\), and \(\mu\) is a nonnegative and finite measure on the unit sphere (called the spectral measure) which satisfies some ellipticity assumptions. First, they prove some new and sharp interior regularity results for the solutions of equation \(Lu=f\) in Hölder space \(B_1\). Namely, if \(f\in C^{\alpha}\), then \(u\in C^{\alpha+2s}\) when \(\alpha+2s\) is not an integer. If \(f\in L^{\infty}\), then \(u\) is \(C^{2s}\) when \(s\not=1/2\), and \(C^{2s-\varepsilon}\) for all \(\varepsilon>0\) when \(s=1/2\). Then they present some boundary regularity results for the solutions of equation \(Lu=f\) in \(\Omega\subset \mathbb{R}^n\), \(u=0\) in \(\mathbb{R}^n\setminus \Omega\), where \(\Omega\) is a \(C^{1,1}\) domain. More precisely, the solutions satisfy \(u/d^s\in C^{s-\varepsilon}(\bar \Omega)\) for all \(\varepsilon>0\), where \(d\) is the distance to \(\partial \Omega\). In their proofs, the authors use a Liouville-type theorem in \(\mathbb{R}^n\) or \(\mathbb{R}^n_+\), and a blow up and compactness argument for the estimation of solutions.

35R11 Fractional partial differential equations
35B65 Smoothness and regularity of solutions to PDEs
60G52 Stable stochastic processes
47G30 Pseudodifferential operators
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