Concerning the \(L^ p\) norm of spectral clusters for second-order elliptic operators on compact manifolds. (English) Zbl 0641.46011

The purpose of the present paper is to study the \(L^p(M)\to L^q(M)\) mapping properties of a spectral projection operator, where \(M\) is a smooth connected compact manifold without boundary of dimension \(\geq 2\). This operator is a generalization of the harmonic projection operator for spherical harmonics on \(S^n\) [see C. D. Sogge, “Oscillatory integrals and spherical harmonics”, Duke Math. J. 53, 43–65 (1986; Zbl 0636.42018)].
As a corollary of a certain “Sobolev inequality” the author generalizes the \(L^2\) restriction theorems of C. Fefferman, E. M. Stein and P. Tomas [see P. Tomas, Bull. Am. Math. Soc. 81, 477–478 (1975; Zbl 0298.42011)] for the Fourier transform in \(\mathbb R^n\) to the setting of Riemannian manifolds.


46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35J25 Boundary value problems for second-order elliptic equations
58J32 Boundary value problems on manifolds
Full Text: DOI


[1] Bonami, A.; Clerc, J. L., Sommes de Cesaro et multiplicateurs des developments en harmonics spheriques, Trans. Amer. Math. Soc., 183, 223-263 (1973) · Zbl 0278.43015
[2] Besse, A. L., Manifolds All of Whose Geodesics Are Closed (1971), Springer-Verlag: Springer-Verlag New York · Zbl 0387.53010
[3] Carleson, L.; Sjölin, P., Oscillatory integrals and a multiplier problem for the disc, Studia Math., 44, 287-299 (1972) · Zbl 0215.18303
[4] Clerc, J. L., Multipliers on symmetric spaces, (Proc. Sympos. Pure Math., 35 (1979)), 345-353, Part 2
[5] Cooke, R., A Cantor-Lebesgue theorem in two dimensions, (Proc. Amer. Math. Soc., 30 (1971)), 547-550 · Zbl 0222.42014
[6] Fefferman, C., Inequalities for strongly singular convolution operators, Acta Math., 124, 9-36 (1970) · Zbl 0188.42601
[7] Fefferman, C., A note on spherical summation operators, Israel J. Math., 94, 44-52 (1973) · Zbl 0262.42007
[8] Gelfand, I. M.; Shilov, G. E., (Generalized Functions, Vol. I (1964), Academic Press: Academic Press New York)
[9] Helgason, S., Differential Geometry and Symmetric Spaces (1962), Academic Press: Academic Press New York · Zbl 0122.39901
[10] Hörmander, L., The spectral function of an elliptic operator, Acta Math., 88, 341-370 (1968) · Zbl 0164.40701
[11] Hörmander, L., On the Riesz means of spectral functions and eigenfunction expansions for elliptic differential operators, (Some Recent Advances in the Basic Sciences (1969), Yeshiva Univ: Yeshiva Univ New York)
[12] Hörmander, L., Oscillatory integrals and multipliers on \(FL^n \), Ark. Math., 11, 1-11 (1971) · Zbl 0254.42010
[13] Hörmander, L., Uniqueness theorems for second order differential equations, Comm. Partial Differential Equations, 8, 21-64 (1983) · Zbl 0546.35023
[14] Hörmander, L., (The Analysis of Linear Partial Differential Equations, Vol. III (1985), Springer-Verlag: Springer-Verlag New York)
[15] Jerison, D., Carleman inequalities for the Dirac and Laplace operators and unique continuation, Adv. in Math., 62, 118-134 (1986) · Zbl 0627.35008
[16] Jerison, D.; Kenig, C., Unique continuation and absence of positive eigenvalues for Shrodinger operators, Ann. of Math., 121, 463-494 (1985) · Zbl 0593.35119
[17] Kenig, C., Carleman estimates, uniform Sobolev inequalities for second order differential operators and unique continuation, (Proceedings, I.C.M.. Proceedings, I.C.M., Berkeley, CA (1986)), to appear · Zbl 0692.35019
[18] Kenig, C.; Ruiz, A.; Sogge, C. D., Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators, Duke Math. J., 55, 329-347 (1987) · Zbl 0644.35012
[19] Seeley, R., Complex powers of an elliptic operator, (Proc. Sympos. Pure Math., 10 (1967)), 288-307 · Zbl 0159.15504
[20] Sogge, C. D., Oscillatory integrals and spherical harmonics, Duke Math. J., 53, 43-65 (1986) · Zbl 0636.42018
[21] Sogge, C. D., A sharp restriction theorem for degenerate curves in \(R^2\), Amer. J. Math., 109, 223-228 (1987) · Zbl 0621.42009
[22] Stanton, R.; Weinstein, A., On the \(L^4\) norm of spherical harmonics, (Math. Proc. Cambridge Philos. Soc., 89 (1981)), 343-358 · Zbl 0479.33010
[23] stein, E. M., Singular Integrals and Differentiability Properties of Functions (1971), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ
[25] Stein, E. M.; Weiss, G., Introduction to Fourier Analysis on Euclidean Spaces (1971), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ · Zbl 0232.42007
[26] Taylor, M., Pseudodifferential Operators (1981), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ · Zbl 0453.47026
[27] Tomas, P., A restriction theorem for the Fourier transform, Bull. Amer. Math. Soc., 81, 477-478 (1975) · Zbl 0298.42011
[28] Vilenkin, N. J., Special Functions and the Theory of Group Representations, (Amer. Math. Soc. Transl., Vol 22 (1968), Amer. Math. Soc: Amer. Math. Soc Providence, RI)
[29] Weinstein, A., Fourier integral operators, quantization and the spectra of Riemannian manifolds, (Colloque International de Geométry Symplectique et Physique Mathematic (1976), CNRS: CNRS Paris) · Zbl 0327.58013
[30] Weinstein, A., Asymptotics of eigenvalue clusters for the Laplacian plus a potential, Duke Math. J., 44, 883-892 (1977) · Zbl 0385.58013
[31] Zygmund, A., On Fourier coefficients and transforms of two variables, Studia Math., 50, 189-201 (1974) · Zbl 0278.42005
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