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

Sogge, Christopher D.

J. Funct. Anal. 77, No. 1, 123-138 (1988).

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.

Reviewer: Neculai Papaghiuc (Iaşi)

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

Keywords:

mapping properties of a spectral projection operator; smooth connected compact manifold without boundary of dimension \(\geq 2\); harmonic projection operator; Sobolev inequality; Fourier transform; Riemannian manifolds

Citations:

Zbl 0298.42011; Zbl 0636.42018

References:

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