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

Abstract

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 Sn (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. Soc. 81 1975, 477–478) for the Fourier transform in Rn to the setting of Riemannian manifolds. We obtain these results for the spectral projection operator as a corollary of a certain “Sobolev inequality” involving Δ + τ2 for large τ. This Sobolev inequality generalizes certain results for Rn 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. Hörmander, Acta Math. 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. Sjölin (Studia Math. 44 1972, 287–299) and Stein (Ann. Math. Stud. 112 1986, 307–357).