Uniform Sobolev Estimates on compact manifolds involving singular\n potentials

2020/09/13 by Matthew D. Blair, Blair, Matthew D., Xiaoqi Huang +5
Mathematics · Social Sciences · #35P15 #58J50 #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Nonlinear Partial Differential Equations #Soviet and Russian History #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.2009.06075

openalex publication_date 2020/09/13 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

Abstract

We obtain generalizations of the uniform Sobolev inequalities of Kenig, Ruiz\nand the fourth author citeKRS for Euclidean spaces and Dos Santos Ferreira,\nKenig and Salo citeDKS for compact Riemannian manifolds involving critically\nsingular potentials V\∈ L^n/2. We also obtain the analogous improved\nquasimode estimates of the the first, third and fourth authors citeBSS ,\nHassell and Tacy citeHassellTacy, the first and fourth author citeSBLog,\nand Hickman citeHickman as well as analogues of the improved uniform Sobolev\nestimates of citeBSSY and citeHickman involving such potentials.\nAdditionally, on Sⁿ, we obtain sharp uniform Sobolev inequalities involving\nsuch potentials for the optimal range of exponents, which extend the results of\nS. Huang and the fourth author citeSHSo. For general Riemannian manifolds we\nimprove the earlier results in citeBSS by obtaining quasimode estimates for\na larger (and optimal) range of exponents under the weaker assumption that\nV\∈ L^n/2.\n

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