Eigenfunction and Bochner Riesz estimates on manifolds with boundary

2002/02/04 by Christopher D. Sogge, Sogge, Christopher D. · 2 citations
Mathematics · #35L70 #42B99 #Advanced Harmonic Analysis Research #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematical Analysis and Transform Methods #math.AP #math.CA #msc:35L70 #msc:42B99

paper · pdf · doi:10.48550/arxiv.math/0202032

10 pages, corrected a few typos

openalex publication_date 2002/02/04 · arxiv created 2002/03/06 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The purpose of this paper is to give a simple proof of sharp L^∞ estimates for the eigenfunctions of the Dirichlet Laplacian on smooth compact Riemannian manifolds (M,g) of dimension n≥ 2 with boundary ∂ M and then to use these estimates to prove new estimates for Bochner Riesz means in this setting. Our eigenfunction estimates involve estimating the L²→ L^∞ mapping properties of the operators χ_λ which project onto unit bands of the spectrum of the square root of the Laplacian. These generalize the recent estimates of D. Grieser for individual eigenfunctions. We use an idea of Grieser of estimating away from the boundary using earlier estimates of Seeley, Pham The Lai, and Hörmander, and then proving estimates near the boundary using a maximum principle. Our proof of the Bochner Riesz estimates is related to an argument for the Euclidean case of C. Fefferman, and one for the boundaryless Riemannian case that is due to the author. Here, unlike in the boundaryless Riemannian case we cannot use parametrices for the wave equation, and instead lie on a more economical argument that only uses the eigenfunction estimates and the finite propagation speed of the wave equation.

Cited by

• First eigenvalue estimates on complete balanced Hermitian manifolds
• Bochner-Riesz means on a conical singular manifold

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