# American Institute of Mathematical Sciences

July  2013, 33(7): 3153-3170. doi: 10.3934/dcds.2013.33.3153

## Harnack's inequality for fractional nonlocal equations

 1 Department of Mathematics, The University of Texas at Austin, 1 University Station, C1200, Austin, TX 78712-1202, United States 2 Department of Mathematics, Sun Yat-sen (Zhongshan) University, 510275 Guangzhou, China

Received  March 2012 Revised  June 2012 Published  January 2013

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.
Citation: Pablo Raúl Stinga, Chao Zhang. Harnack's inequality for fractional nonlocal equations. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 3153-3170. doi: 10.3934/dcds.2013.33.3153
##### References:
 [1] I. Abu-Falahah, R. A. Macías, C. Segovia and J. L. Torrea, Transferring strong boundedness among Laguerre orthogonal systems, Proc. Indian Acad. Sci. Math. Sci., 119 (2009), 203-220. doi: 10.1007/s12044-009-0021-4. [2] I. Abu-Falahah and J. L. Torrea, Hermite function expansions versus Hermite polynomial expansions, Glasgow Math. J., 48 (2006), 203-215. doi: 10.1017/S0017089506003004. [3] J. J. Betancor, J. Dziubański and J. L. Torrea, On Hardy spaces associated with Bessel operators, J. Anal. Math., 107 (2009), 195-219. doi: 10.1007/s11854-009-0008-1. [4] S. Bochner, Sturm-Liouville and heat equations whose eigenfunctions are ultraspherical polynomials or associated Bessel functions, in "Proceedings of the Conference on Differential Equations (dedicated to A. Weinstein)", University of Maryland Book Store, College Park, Md., (1956), 23-48. [5] L. Caffarelli, Some nonlinear problems involving non-local diffusions, in "ICIAM 07-6th International Congress on Industrial and Applied Mathematics," Eur. Math. Soc., Zürich, (2009), 43-56. doi: 10.4171/056-1/3. [6] L. Caffarelli, S. Salsa and L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math., 171 (2008), 425-461. doi: 10.1007/s00222-007-0086-6. [7] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306. [8] L. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math. (2), 171 (2010), 1903-1930. doi: 10.4007/annals.2010.171.1903. [9] E. B. Davies, "Heat Kernels and Spectral Theory," Cambridge Tracts in Mathematics 92, Cambridge Univ. Press, Cambridge, 1989. doi: 10.1017/CBO9780511566158. [10] E. B. Fabes, C. E. Kenig and R. P. Serapioni, The local regularity of solutions to degenerate elliptic equations, Comm. Partial Differential Equations, 7 (1982), 77-116. doi: 10.1080/03605308208820218. [11] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Classics in Mathematics, Springer-Verlag, Berlin, 2001. [12] C. E. Gutiérrez, Harnack's inequality for degenerate Schrödinger operators, Trans. Amer. Math. Soc., 312 (1989), 403-419. doi: 10.2307/2001222. [13] C. E. Gutiérrez, On the Riesz transforms for Gaussian measures, J. Funct. Anal., 120 (1994), 107-134. doi: 10.1006/jfan.1994.1026. [14] C. E. Gutiérrez, A. Incognito and J. L. Torrea, Riesz transforms, $g$-functions, and multipliers for the Laguerre semigroup, Houston J. Math., 27 (2001), 579-592. [15] T. Lin, Y. Y. Li and J. Xiong, On a fractional Nirenberg problem, part I: Blow up analysis and compactness of solutions, preprint (2011), 59 pp, [16] N. S. Landkof, "Foundations of Modern Potential Theory," Die Grundlehren der mathematischen Wissenschaften, Band 180, Springer-Verlag, New York-Heidelberg, 1972. [17] N. N. Lebedev, "Special Functions and Their Applications," Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965. [18] B. Muckenhoupt and E. M. Stein, Classical expansions and their relation to conjugate harmonic functions, Trans. Amer. Math. Soc., 118 (1965), 17-92. [19] L. Roncal and P. R. Stinga, Fractional Laplacian on the torus, preprint (2012), 16 pp, [20] W. Rudin, "Functional Analysis," McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. [21] M. F. Shlesinger, G. M. Zaslavsky and J. Klafter, Strange kinetics, Nature, 363 (1993), 31-37. [22] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112. doi: 10.1002/cpa.20153. [23] P. R. Stinga and J. L. Torrea, Extension problem and Harnack's inequality for some fractional operators, Comm. Partial Differential Equations, 35 (2010), 2092-2122. doi: 10.1080/03605301003735680. [24] G. Szegö, "Orthogonal Polynomials," Fourth edition, American Mathematical Society Colloquium Publications XXIII, American Mathematical Society, Providence, R.I., 1975. [25] J. Tan and J. Xiong, A Harnack inequality for fractional Laplace equations with lower order terms, Discrete Contin. Dyn. Syst., 31 (2011), 975-983. doi: 10.3934/dcds.2011.31.975. [26] S. Thangavelu, "Lectures on Hermite and Laguerre Expansions," Mathematical Notes 42, Princeton University Press, Princeton, NJ, 1993. [27] E. C. Titchmarsh, "Intoduction to the Theory of Fourier Integrals," Third edition, Chelsea Publishing Co., New York, 1986. [28] N. S. Trudinger, Linear elliptic operators with measurable coefficients, Ann. Scuola Norm. Sup. Pisa (3), 27 (1973), 265-308. [29] K. Yosida, "Functional Analysis," Classics in Mathematics, Springer-Verlag, Berlin, 1995.

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##### References:
 [1] I. Abu-Falahah, R. A. Macías, C. Segovia and J. L. Torrea, Transferring strong boundedness among Laguerre orthogonal systems, Proc. Indian Acad. Sci. Math. Sci., 119 (2009), 203-220. doi: 10.1007/s12044-009-0021-4. [2] I. Abu-Falahah and J. L. Torrea, Hermite function expansions versus Hermite polynomial expansions, Glasgow Math. J., 48 (2006), 203-215. doi: 10.1017/S0017089506003004. [3] J. J. Betancor, J. Dziubański and J. L. Torrea, On Hardy spaces associated with Bessel operators, J. Anal. Math., 107 (2009), 195-219. doi: 10.1007/s11854-009-0008-1. [4] S. Bochner, Sturm-Liouville and heat equations whose eigenfunctions are ultraspherical polynomials or associated Bessel functions, in "Proceedings of the Conference on Differential Equations (dedicated to A. Weinstein)", University of Maryland Book Store, College Park, Md., (1956), 23-48. [5] L. Caffarelli, Some nonlinear problems involving non-local diffusions, in "ICIAM 07-6th International Congress on Industrial and Applied Mathematics," Eur. Math. Soc., Zürich, (2009), 43-56. doi: 10.4171/056-1/3. [6] L. Caffarelli, S. Salsa and L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math., 171 (2008), 425-461. doi: 10.1007/s00222-007-0086-6. [7] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306. [8] L. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math. (2), 171 (2010), 1903-1930. doi: 10.4007/annals.2010.171.1903. [9] E. B. Davies, "Heat Kernels and Spectral Theory," Cambridge Tracts in Mathematics 92, Cambridge Univ. Press, Cambridge, 1989. doi: 10.1017/CBO9780511566158. [10] E. B. Fabes, C. E. Kenig and R. P. Serapioni, The local regularity of solutions to degenerate elliptic equations, Comm. Partial Differential Equations, 7 (1982), 77-116. doi: 10.1080/03605308208820218. [11] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Classics in Mathematics, Springer-Verlag, Berlin, 2001. [12] C. E. Gutiérrez, Harnack's inequality for degenerate Schrödinger operators, Trans. Amer. Math. Soc., 312 (1989), 403-419. doi: 10.2307/2001222. [13] C. E. Gutiérrez, On the Riesz transforms for Gaussian measures, J. Funct. Anal., 120 (1994), 107-134. doi: 10.1006/jfan.1994.1026. [14] C. E. Gutiérrez, A. Incognito and J. L. Torrea, Riesz transforms, $g$-functions, and multipliers for the Laguerre semigroup, Houston J. Math., 27 (2001), 579-592. [15] T. Lin, Y. Y. Li and J. Xiong, On a fractional Nirenberg problem, part I: Blow up analysis and compactness of solutions, preprint (2011), 59 pp, [16] N. S. Landkof, "Foundations of Modern Potential Theory," Die Grundlehren der mathematischen Wissenschaften, Band 180, Springer-Verlag, New York-Heidelberg, 1972. [17] N. N. Lebedev, "Special Functions and Their Applications," Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965. [18] B. Muckenhoupt and E. M. Stein, Classical expansions and their relation to conjugate harmonic functions, Trans. Amer. Math. Soc., 118 (1965), 17-92. [19] L. Roncal and P. R. Stinga, Fractional Laplacian on the torus, preprint (2012), 16 pp, [20] W. Rudin, "Functional Analysis," McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. [21] M. F. Shlesinger, G. M. Zaslavsky and J. Klafter, Strange kinetics, Nature, 363 (1993), 31-37. [22] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112. doi: 10.1002/cpa.20153. [23] P. R. Stinga and J. L. Torrea, Extension problem and Harnack's inequality for some fractional operators, Comm. Partial Differential Equations, 35 (2010), 2092-2122. doi: 10.1080/03605301003735680. [24] G. Szegö, "Orthogonal Polynomials," Fourth edition, American Mathematical Society Colloquium Publications XXIII, American Mathematical Society, Providence, R.I., 1975. [25] J. Tan and J. Xiong, A Harnack inequality for fractional Laplace equations with lower order terms, Discrete Contin. Dyn. Syst., 31 (2011), 975-983. doi: 10.3934/dcds.2011.31.975. [26] S. Thangavelu, "Lectures on Hermite and Laguerre Expansions," Mathematical Notes 42, Princeton University Press, Princeton, NJ, 1993. [27] E. C. Titchmarsh, "Intoduction to the Theory of Fourier Integrals," Third edition, Chelsea Publishing Co., New York, 1986. [28] N. S. Trudinger, Linear elliptic operators with measurable coefficients, Ann. Scuola Norm. Sup. Pisa (3), 27 (1973), 265-308. [29] K. Yosida, "Functional Analysis," Classics in Mathematics, Springer-Verlag, Berlin, 1995.
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