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On a class of non-local operators in conformal geometry

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Abstract

In this expository article, the authors discuss the connection between the study of non-local operators on Euclidean space to the study of fractional GJMS operators in conformal geometry. The emphasis is on the study of a class of fourth order operators and their third order boundary operators. These third order operators are generalizations of the Dirichlet-to-Neumann operator.

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References

  1. Ache, A. and Chang, S. Y. A., Sobolev-trace inequalities of order four, 2015. arXiv: 1509.06069v1

    Google Scholar 

  2. Beckner, W., Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality, Ann. of Math. (2), 138(1), 1993, 213–242.

    Article  MathSciNet  MATH  Google Scholar 

  3. Branson, T. P., Sharp inequalities, the functional determinant, and the complementary series, Trans. Amer. Math. Soc., 347(10), 1995, 3671–3742.

    Article  MathSciNet  MATH  Google Scholar 

  4. Case, J., Boundary operators associated to the Paneitz operator, 2015. arXiv: 1509.08347

    Google Scholar 

  5. Case, J., Some energy inequalities involving fractional GJMS operators, 2015. arXiv: 1509.08342

    Google Scholar 

  6. Case, J. and Chang, S. Y. A., On fractional GJMS operators, 2014. arXiv: 1406.1846

    MATH  Google Scholar 

  7. Chang, S. Y. A. and González, S., María del Mar, Fractional Laplacian in conformal geometry, Adv. Math., 226(2), 2011, 1410–1432.

    Article  MathSciNet  MATH  Google Scholar 

  8. Caffarelli, L. and Silvestre, L., An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32(7–9), 2007, 1245–1260.

    Article  MathSciNet  MATH  Google Scholar 

  9. Escobar, J. F., The Yamabe problem on manifolds with boundary, J. Differential Geom., 35(1), 1992, 21–84.

  10. Fefferman, C. and Graham, C. R., Q-curvature and Poincaré metrics, Math. Res. Lett., 9(2–3), 2002, 139–151.

    Article  MathSciNet  MATH  Google Scholar 

  11. Gursky, M., Hang, F. B. and Lin, Y. J., Riemannian manifolds with positive Yamabe invariant and Paneitz operator, International Mathmeatics Notices, 2015.

    Google Scholar 

  12. Graham, C. R., Jenne, R., Mason, L. J. and Sparling, G. A. J., Conformally invariant powers of the Laplacian, I. Existence, J. London Math. Soc. (2), 46(3), 1992, 557–565.

  13. Matthew, G. and Malchiodi, A., A strong maximum principle for the Paneitz operator and a non-local flow for the Q-curvature, 2014. arXiv: 1401.3216v5

    MATH  Google Scholar 

  14. González, M. M. and Qing, J., Fractional conformal Laplacians and fractional Yamabe problems, Anal. PDE, 6(7), 2013, 1535–1576.

    Article  MathSciNet  MATH  Google Scholar 

  15. Gursky, M. J., The principal eigenvalue of a conformally invariant differential operator, with an application to semilinear elliptic PDE, Comm. Math. Phys., 207(1), 1999, 131–143.

    Article  MathSciNet  MATH  Google Scholar 

  16. Graham, C. R. and Zworski, M., Scattering matrix in conformal geometry, Invent. Math., 152(1), 2003, 89–118.

    Article  MathSciNet  MATH  Google Scholar 

  17. Hang, F. B. and Yang, P., Q curvature on a class of 3 manifolds, Comm. Pure Appl. Math., 69(4), 2016, 734–744.

    Article  MathSciNet  MATH  Google Scholar 

  18. Hang, F. B. and Yang, P., Q curvature on a class of manifolds with dimension at least 5, 2014. arXiv: 1411.3926

  19. Hang, F. B. and Yang, P., Sign of Green’s function of Paneitz operators and the Q curvature, 2014. arXiv: 1411.3924

    MATH  Google Scholar 

  20. Hang, F. B. and Yang, P., Lecture on the fourth order Q curvature equation, 2015, preprint.

    Google Scholar 

  21. Lee, J. M., The spectrum of an asymptotically hyperbolic Einstein manifold, Comm. Anal. Geom., 3(1–2), 1995, 253–271.

    Article  MathSciNet  MATH  Google Scholar 

  22. Lebedev, N. A. and Milin, I. M., On the coefficients of certain classes of analytic functions, Mat. Sbornik N. S., 28(70), 1951, 359–400.

    MathSciNet  Google Scholar 

  23. Mazzeo, R. R. and Melrose, R. B., Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature, J. Funct. Anal., 75(2), 1987, 260–310.

    Article  MathSciNet  MATH  Google Scholar 

  24. Osgood, B., Phillips, R. and Sarnak, P., Extremals of determinants of Laplacians, J. Funct. Anal., 80(1), 1988, 148–211.

    Article  MathSciNet  MATH  Google Scholar 

  25. Paneitz, S. M., A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds (summary), SIGMA Symmetry Integrability Geom. Methods Appl., 4, 2008, Paper 036, 3.

    MathSciNet  MATH  Google Scholar 

  26. Yang, R., On higher order extensions for the fractional Laplacian, 2013. arXiv: 1302.4413

    Google Scholar 

  27. Yang, P. and King, D. and Chang, S. Y. A., Renormalized volumes for conformally compact Einstein manifolds, Sovrem. Mat. Fundam. Napravl., 17, 2006, 129–142.

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Mathematics Department, Princeton University, Fine Hall, Washington Road, Princeton, New Jersey, 08544-1000, USA

    Sun Yung Alice Chang & Ray A. Yang

Authors
  1. Sun Yung Alice Chang
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  2. Ray A. Yang
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Corresponding author

Correspondence to Sun Yung Alice Chang.

Additional information

Dedicated to Professor Haim Brezis on his 70th birthday

This work was supported by NSF grant DMS-1509505 and a postdoctoral fellowship of the National Science Foundation (No.DMS-1103786).

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Chang, S.Y.A., Yang, R.A. On a class of non-local operators in conformal geometry. Chin. Ann. Math. Ser. B 38, 215–234 (2017). https://doi.org/10.1007/s11401-016-1068-z

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  • DOI: https://doi.org/10.1007/s11401-016-1068-z

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