Concerning the Lp norm of spectral clusters for second-order elliptic operators on compact manifolds

In this paper we study the (Lp, L2) mapping properties of a spectral projection operator for Riemannian manifolds. This operator is a generalization of the harmonic projection operator for spherical harmonics on S” (see C. D. Sogge, Duke Math. J. 53 (1986), 43-65). Among other things, we generalize the L2 restriction theorems of C. Fefferman, E. M. Stein, and P. Tomas (Bull. Amer. Math. Sot. 81 (1975), 477478) for the Fourier transform in R” to the setting of Riemannian manifolds. We obtain these results for the spectral projection operator as a corollary of a certain “Sobolev inequality” involving d +r2 for large r. This Sobolev inequality generalizes certain results for R” of C. Kenig, A. Ruiz, and the author (Duke Math. J. 55 (1987), 329-347). The main tools in the proof of the Sobolev inequalities for Riemannian manifolds are the Hadamard parametrix (cf. L. Hbrmander, Acta Mad 88 (1968), 341-370, and “The Analysis of Linear Partial Differential Equations,” Vol. III, Springer-Verlag, New York, 1985) and oscillatory integral theorems of L. Carleson and P. Sjolin (Studia Math. 44 (19X?), 287-299) and Stein (Arm. Math. Stud. 112 ( 1986), 307-357). 0 1988 Academic pnss, IW.

[1]  C. Sogge.  A SHARP RESTRICTION THEOREM FOR DEGENERATE CURVES IN R2 , 1987 .

[2]  G. Temple,et al.  GENERALIZED FUNCTIONS, VOL. I , 1965 .

[3]  A. Bonami,et al.  Sommes de Cesàro et multiplicateurs des développements en harmoniques sphériques , 1973 .

[4]  Roger Cooke.  A Cantor-Lebesgue theorem in two dimensions , 1971 .

[5]  S. Helgason.  Differential Geometry and Symmetric Spaces , 1964 .

[6]  N. Vilenkin.  Special Functions and the Theory of Group Representations , 1968 .

[7]  A. Gelbart,et al.  Some recent advances in the basic sciences , 1969 .

[8]  A. Zygmund.  On Fourier coefficients and transforms of functions of two variables , 1974 .

[9]  P. A. Tomas.  A restriction theorem for the Fourier transform , 1975 .

[10]  A. Weinstein,et al.  On the L4 norm of spherical harmonics , 1981, Mathematical Proceedings of the Cambridge Philosophical Society.

[11]  E. Stein.  Singular Integrals and Di?erentiability Properties of Functions , 1971 .

[12]  D. Jerison.  Carleman inequalities for the Dirac and Laplace operators and unique continuation , 1986 .

[13]  Charles Fefferman,et al.  A note on spherical summation multipliers , 1973 .

[14]  Robert T. Seeley,et al.  Complex powers of an elliptic operator , 1967 .

[15]  A. Weinstein.  Asymptotics of eigenvalue clusters for the Laplacian plus a potential , 1977 .

[16]  P. Sjölin,et al.  Oscillatory integrals and multiplier problem for the disc , 1972 .

[17]  L. Hörmander,et al.  The spectral function of an elliptic operator , 1968 .

[18]  Christopher D. Sogge,et al.  Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators , 1987 .

[19]  J. Cooper.  SINGULAR INTEGRALS AND DIFFERENTIABILITY PROPERTIES OF FUNCTIONS , 1973 .

[20]  L. Hörmander,et al.  Oscillatory integrals and multipliers onFLp , 1973 .

[21]  Charles Fefferman,et al.  Inequalities for strongly singular convolution operators , 1970 .

[22]  R. Bishop.  Review: Arthur L. Besse, Manifolds all of whose geodesics are closed , 1984 .

[23]  Elias M. Stein,et al.  Unique continuation and absence of positive eigenvalues for Schrodinger operators , 1985 .

[24]  Elias M. Stein,et al.  OSCILLATORY INTEGRALS IN FOURIER ANALYSIS , 1987 .

[25]  E. Stein,et al.  Introduction to Fourier Analysis on Euclidean Spaces. , 1971 .

[26]  Christopher D. Sogge,et al.  Oscillatory integrals and spherical harmonics , 1986 .