# American Institute of Mathematical Sciences

2011, 31(3): 975-983. doi: 10.3934/dcds.2011.31.975

## A Harnack inequality for fractional Laplace equations with lower order terms

 1 Departamento de Matemática, Universidad Técnica Federico Santa María, Avda. España 1680, Valparaíso, Chile 2 School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

Received  February 2010 Revised  May 2011 Published  August 2011

We establish a Harnack inequality of fractional Laplace equations without imposing sign condition on the coefficient of zero order term via the Moser's iteration and John-Nirenberg inequality.
Citation: Jinggang Tan, Jingang Xiong. A Harnack inequality for fractional Laplace equations with lower order terms. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 975-983. doi: 10.3934/dcds.2011.31.975
##### References:
 [1] R. F. Bass and D. A. Levin, Harnack inequalities for jump processes,, Potential Anal., 17 (2002), 375. doi: 10.1023/A:1016378210944. [2] X. Cabre and Y. Sire, Nonlinear equations for fractional laplacians I: Regularity, maximum principles, and hamiltonian estimates,, preprint, (). [3] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, Comm. Partial Differential Equations, 32 (2007), 1245. [4] Z.-Q. Chen and R. Song, Estimates on Green functions and Poisson kernels for symmetric stable processes,, Math. Ann., 312 (1998), 465. doi: 10.1007/s002080050232. [5] E. Fabes, C. Kenig and R. Serapioni, The local regularity of solutions of degenerate elliptic equations,, Comm. Partial Differential Equations, 7 (1982), 77. [6] Q. Han and F.-H. Lin, "Elliptic Partial Differential Equations,", Courant Lecture Notes in Mathematics, 1 (1997). [7] Z.-C. Han and Y. Y. Li, The Yamabe problem on manifolds with boundary: Existence and compactness results,, Duke Math. J., 99 (1999), 489. doi: 10.1215/S0012-7094-99-09916-7. [8] F. John and L. Nirenberg, On functions of bounded mean oscillation,, Comm. Pure Appl. Math., 14 (1961), 415. doi: 10.1002/cpa.3160140317. [9] M. Kassmann, The classical Harnack inequality fails for non-local operators,, preprint., (). [10] B. Muckenhoupt and R. Wheeden, Weighted norm inequalities for fractional integrals,, Trans. Amer. Math. Soc., 192 (1974), 261. doi: 10.1090/S0002-9947-1974-0340523-6. [11] E. Stein, "Singular Integrals and Differentiability Properties of Function,", Princeton Mathematical Series, 30 (1970).

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##### References:
 [1] R. F. Bass and D. A. Levin, Harnack inequalities for jump processes,, Potential Anal., 17 (2002), 375. doi: 10.1023/A:1016378210944. [2] X. Cabre and Y. Sire, Nonlinear equations for fractional laplacians I: Regularity, maximum principles, and hamiltonian estimates,, preprint, (). [3] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, Comm. Partial Differential Equations, 32 (2007), 1245. [4] Z.-Q. Chen and R. Song, Estimates on Green functions and Poisson kernels for symmetric stable processes,, Math. Ann., 312 (1998), 465. doi: 10.1007/s002080050232. [5] E. Fabes, C. Kenig and R. Serapioni, The local regularity of solutions of degenerate elliptic equations,, Comm. Partial Differential Equations, 7 (1982), 77. [6] Q. Han and F.-H. Lin, "Elliptic Partial Differential Equations,", Courant Lecture Notes in Mathematics, 1 (1997). [7] Z.-C. Han and Y. Y. Li, The Yamabe problem on manifolds with boundary: Existence and compactness results,, Duke Math. J., 99 (1999), 489. doi: 10.1215/S0012-7094-99-09916-7. [8] F. John and L. Nirenberg, On functions of bounded mean oscillation,, Comm. Pure Appl. Math., 14 (1961), 415. doi: 10.1002/cpa.3160140317. [9] M. Kassmann, The classical Harnack inequality fails for non-local operators,, preprint., (). [10] B. Muckenhoupt and R. Wheeden, Weighted norm inequalities for fractional integrals,, Trans. Amer. Math. Soc., 192 (1974), 261. doi: 10.1090/S0002-9947-1974-0340523-6. [11] E. Stein, "Singular Integrals and Differentiability Properties of Function,", Princeton Mathematical Series, 30 (1970).
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