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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.

MSC:
35R11 Fractional partial differential equations
35B65 Smoothness and regularity of solutions to PDEs
60G52 Stable stochastic processes
47G30 Pseudodifferential operators
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[1] Bass, R.; Chen, Z.-H., Regularity of harmonic functions for a class of singular stable-like processes, Math. Z., 266, 489-503, (2010) · Zbl 1201.60055
[2] Bass, R.; Levin, D., Harnack inequalities for jump processes, Potential Anal., 17, 375-388, (2002) · Zbl 0997.60089
[3] Bass, R., Regularity results for stable-like operators, J. Funct. Anal., 257, 2693-2722, (2009) · Zbl 1177.45013
[4] Bertoin, J., Lévy processes, Cambridge Tracts in Mathematics, vol. 121, (1996), Cambridge University Press Cambridge · Zbl 0861.60003
[5] Bjorland, C.; Caffarelli, L.; Figalli, A., Non-local gradient dependent operators, Adv. Math., 230, 1859-1894, (2012) · Zbl 1252.35099
[6] Bogdan, K.; Sztonyk, P., Harnack’s inequality for stable Lévy processes, Potential Anal., 22, 133-150, (2005) · Zbl 1081.60055
[7] Bogdan, K.; Sztonyk, P., Estimates of the potential kernel and Harnack’s inequality for the anisotropic fractional Laplacian, Studia Math., 181, 101-123, (2007) · Zbl 1223.47038
[8] Byczkowski, T.; Nolan, J. P.; Rajput, B., Approximation of multidimensional stable densities, J. Multivariate Anal., 46, 13-31, (1993) · Zbl 0790.60020
[9] Caffarelli, L.; Silvestre, L., Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math., 62, 597-638, (2009) · Zbl 1170.45006
[10] W. Chen, C. Li, L. Zhang, T. Cheng, A Liouville theorem for α-harmonic functions in \(\mathbb{R}_+^n\), preprint, Sep. 2014.
[11] Cheng, B. N.; Rachev, S. T., Multivariate stable future prices, Math. Finance, 5, 133-153, (1995) · Zbl 0862.62089
[12] Dziubanski, J., Asymptotic behaviour of densities of stable semigroups of measures, Probab. Theory Related Fields, 87, 459-467, (1991) · Zbl 0695.60013
[13] Fall, M. M., Entire s-harmonic functions are affine, Proc. Amer. Math. Soc., (2016), in press · Zbl 1336.35355
[14] M.M. Fall, T. Weth, Liouville theorems for a general class of nonlocal operators, preprint, Apr. 2015. · Zbl 1346.35210
[15] Gilbarg, D.; Trudinger, N. S., Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften, vol. 224, (1977), Springer-Verlag Berlin-New York · Zbl 0361.35003
[16] Glowacki, P.; Hebisch, W., Pointwise estimates for densities of stable semigroups of measures, Studia Math., 104, 243-258, (1993) · Zbl 0812.43005
[17] Grubb, G.; Kokholm, N. J., A global calculus of parameter-dependent pseudodifferential boundary problems in \(L_p\) Sobolev spaces, Acta Math., 171, 165-229, (1993) · Zbl 0811.35176
[18] Grubb, G., Fractional Laplacians on domains, a development of Hörmander’s theory of μ-transmission pseudodifferential operators, Adv. Math., 268, 478-528, (2015) · Zbl 1318.47064
[19] Grubb, G., Local and nonlocal boundary conditions for μ-transmission and fractional elliptic pseudodifferential operators, Anal. PDE, 7, 1649-1682, (2014) · Zbl 1317.35310
[20] Janicki, A.; Weron, A., Simulation and chaotic behavior of α-stable stochastic processes, Monographs and Textbooks in Pure and Applied Mathematics, vol. 178, (1994), New York
[21] Kassmann, M.; Mimica, A., Analysis of jump processes with nondegenerate jumping kernels, Stochastic Process. Appl., 123, 629-650, (2013) · Zbl 1259.60100
[22] Kassmann, M.; Rang, M.; Schwab, R. W., Hölder regularity for integro-differential equations with nonlinear directional dependence, Indiana Univ. Math. J., (2016), in press
[23] Kassmann, M.; Schwab, R. W., Regularity results for nonlocal parabolic equations, Riv. Math. Univ. Parma, 5, 183-212, (2014) · Zbl 1329.35095
[24] Landkof, N. S., Foundations of modern potential theory, (1972), Springer New York · Zbl 0253.31001
[25] Lizorkin, P. I., Multipliers of Fourier integrals in the space \(L_p\), Tr. Mat. Inst. Steklova, Proc. Steklov Inst. Math., 89, 269-290, (1967), English translation in · Zbl 0167.12404
[26] Mantegna, R. N.; Stanley, H. E., Scaling behaviour in the dynamics of an economic index, Nature, 376, 46-49, (1995)
[27] Nolan, J. P., Multivariate stable distributions: approximation, estimation, simulation and identification, (Adler, R. J.; Feldman, R. E.; Taqqu, M. S., A Practical Guide to Heavy Tails, (1998), Birkhäuser Boston), 509-526 · Zbl 0927.60021
[28] Nolan, J. P., Fitting data and assessing goodness-of-fit with stable distributions, (Applications of Heavy Tailed Distributions in Economics, Engineering and Statistics, Washington, DC, (1999))
[29] Pivato, M.; Seco, L., Estimating the spectral measure of a multivariate stable distribution via spherical harmonic analysis, J. Multivariate Anal., 87, 219-240, (2003) · Zbl 1041.60019
[30] Ros-Oton, X.; Serra, J., The Dirichlet problem for the fractional Laplacian: regularity up to the boundary, J. Math. Pures Appl., 101, 275-302, (2014) · Zbl 1285.35020
[31] Ros-Oton, X.; Serra, J., Boundary regularity for fully nonlinear integro-differential equations, Duke Math. J., (2016), in press · Zbl 1351.35245
[32] Samorodnitsky, G.; Taqqu, M. S., Stable non-Gaussian random processes: stochastic models with infinite variance, (1994), Chapman and Hall New York · Zbl 0925.60027
[33] Schwab, R. W.; Silvestre, L., Regularity for parabolic integro-differential equations with very irregular kernels, Anal. PDE, (2016), in press · Zbl 1349.47079
[34] Serra, J., Regularity for fully nonlinear nonlocal parabolic equations with rough kernels, Calc. Var. Partial Differential Equations, 54, 615-629, (2015) · Zbl 1327.35170
[35] Serra, J., \(C^{\sigma + \alpha}\) regularity for concave nonlocal fully nonlinear elliptic equations with rough kernels, Calc. Var. Partial Differential Equations, (2016), in press
[36] Silvestre, L., Hölder estimates for solutions of integro differential equations like the fractional Laplacian, Indiana Univ. Math. J., 55, 1155-1174, (2006) · Zbl 1101.45004
[37] Stein, E., Singular integrals and differentiability properties of functions, Princeton Mathematical Series, vol. 30, (1970) · Zbl 0207.13501
[38] Sztonyk, P., Regularity of harmonic functions for anisotropic fractional Laplacians, Math. Nachr., 283, 289-311, (2010) · Zbl 1194.47044
[39] Watanabe, T., Asymptotic estimates of multi-dimensional stable densities and their applications, Trans. Amer. Math. Soc., 359, 2851-2879, (2007) · Zbl 1124.62063
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