A Sharp Higher Order Sobolev Inequality on Riemannian Manifolds

2024/09/13 by Samuel Zeitler, Zeitler, Samuel · 2 citations
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2409.08920

openalex publication_date 2024/09/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let m, n be integers such that (n)/(2) > m ≥ 1 and let (M, g) be a closed n-dimensional Riemannian manifold. We prove there exists some B ∈ ℝ depending only on (M, g) , m , and n such that for all u ∈ H_m²(M) , ‖ u ‖_2^#² ≤ K(m,n) ∫_M (Δ^(m)/(2) u)² dv_g + B ‖ u ‖_H_m-1²(M)² where 2^# = (2n)/(n-2m) , K(m,n) is the square of the best constant for the embedding W^m,2(ℝⁿ) ⊂ L^{2^#}(ℝⁿ) , H_m²(M) is the Sobolev space consisting of functions on M with m weak derivatives in L²(M) , and Δ^(m)/(2) = ∇ Δ^(m-1)/(2) if m is odd. This inequality is sharp in the sense that K(m,n) cannot be lowered to any smaller constant. This extends the work of Hebey-Vaugon and Hebey which correspond respectively to the cases m=1 and m=2 .

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• Sharp Fractional Sobolev Embeddings on Closed Manifolds
• Optimal Sobolev inequalities of high order with L²-remainder

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