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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.

MSC:

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
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References:

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