Uniform Sobolev Estimates on compact manifolds involving singular potentials
Abstract
We obtain generalizations of the uniform Sobolev inequalities of Kenig, Ruiz and the fourth author \cite{KRS} for Euclidean spaces and Dos Santos Ferreira, Kenig and Salo \cite{DKS} for compact Riemannian manifolds involving critically singular potentials $V\in L^{n/2}$. We also obtain the analogous improved quasimode estimates of the the first, third and fourth authors \cite{BSS} , Hassell and Tacy \cite{HassellTacy}, the first and fourth author \cite{SBLog}, and Hickman \cite{Hickman} as well as analogues of the improved uniform Sobolev estimates of \cite{BSSY} and \cite{Hickman} involving such potentials. Additionally, on $S^n$, we obtain sharp uniform Sobolev inequalities involving such potentials for the optimal range of exponents, which extend the results of S. Huang and the fourth author \cite{SHSo}. For general Riemannian manifolds we improve the earlier results in \cite{BSS} by obtaining quasimode estimates for a larger (and optimal) range of exponents under the weaker assumption that $V\in L^{n/2}$.
- Publication:
-
arXiv e-prints
- Pub Date:
- September 2020
- DOI:
- 10.48550/arXiv.2009.06075
- arXiv:
- arXiv:2009.06075
- Bibcode:
- 2020arXiv200906075B
- Keywords:
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- Mathematics - Analysis of PDEs;
- Mathematics - Classical Analysis and ODEs;
- Mathematics - Spectral Theory;
- 58J50;
- 35P15
- E-Print:
- Revised version to appear in Revista Matematica Iberoamericana