On a class of non-local operators in conformal geometry

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.

Keywords

High order fractional GJMS operator / Generalized boundary Yamabe problem / Sobolov trace extension

References

[1] Ache A., Chang S. Y. A.. Sobolev-trace inequalities of order four, 2015

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

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

[4] Case J.. Boundary operators associated to the Paneitz operator, 2015

[5] Case J.. Some energy inequalities involving fractional GJMS operators, 2015

[6] Case J., Chang S. Y. A.. On fractional GJMS operators, 2014

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

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

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

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

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

[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., Malchiodi A.. A strong maximum principle for the Paneitz operator and a non-local flow for the Q-curvature, 2014

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

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

[16] Graham C. R., Zworski M.. Scattering matrix in conformal geometry. Invent. Math., 2003, 152(1): 89-118

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

[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., Yang P.. Sign of Green’s function of Paneitz operators and the Q curvature, 2014

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

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

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

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

[24] Osgood B., Phillips R., Sarnak P.. Extremals of determinants of Laplacians. J. Funct. Anal., 1988, 80(1): 148-211

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

[26] Yang R.. On higher order extensions for the fractional Laplacian, 2013

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