Uniform Sobolev estimates on compact manifolds involving singular potentials

Matthew D Blair,Xiaoqi Huang,Christopher D Sogge,Yannick Sire

doi:10.4171/rmi/1300

Matthew D Blair, Xiaoqi Huang + Show 2 more

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https://doi.org/10.4171/rmi/1300

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Abstract

We obtain generalizations of the uniform Sobolev inequalities of Kenig, Ruiz and the fourth author (1986) for Euclidean spaces and Dos Santos Ferreira, Kenig and Salo (2014) for compact Riemannian manifolds involving critically singular potentials $V\\in L^{n/2}$. We also obtain the analogous improved quasimode estimates of the first, third and fourth author (2021), Hassell and Tacy (2015), the first and fourth author (2019), and Hickman (2020), as well as analogues of the improved uniform Sobolev estimates of Bourgain, Shao, the fourth author and Yao (2015), and Hickman (2020), 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 (2014). For general Riemannian manifolds, we improve the earlier results in of the first, third and fourth authors (2021) by obtaining quasimode estimates for a larger (and optimal) range of exponents under the weaker assumption that $V\\in L^{n/2}$.

Highlights

Introduction and main resultsThe main purpose of this paper is to extend the uniform Sobolev inequalities on compact Riemannian manifolds (M, g) of [10], [9], and [24] to include Schrodinger operators, (1.1)HV = −∆g + V (x), with critically singular potentials V, which are always assumed to be real-valued
Notwithstanding, for Sn, we can get an improvement over Theorems 1.3 and 1.4 for the uniform Sobolev estimates by obtaining bounds for the optimal range of exponents satisfying (1.12)
For both of the exponents in (2.1), we shall assume that we have improvements of the classical quasimode estimates of the fourth author [27] of the form

Summary

Since these two facts yield the desired

Lp(M ) → Lq(M ) bounds for (HV + 1)−α/2 the proof is complete. As in [9], in certain geometries we can obtain improved uniform Sobolev estimates and quasimode estimates using improved bounds for the unperturbed operator H0. Notwithstanding, for Sn, we can get an improvement over Theorems 1.3 and 1.4 for the uniform Sobolev estimates by obtaining bounds for the optimal range of exponents satisfying (1.12). For both of the exponents in (2.1), we shall assume that we have improvements of the classical quasimode estimates of the fourth author [27] of the form (2.2). To obtain the quasimode estimate (2.9) in the Corollary we need to see that the bounds in (2.29) are valid when 1 ≤ λ < Λ, with Λ = Λ(M, q, V ) ≥ 1 being the fixed constant in Theorem 2.1 This just follows from the fact that δ(λ, q) and ε(λ) are assumed to be nonzero and continuous, and by the spectral theorem (2.33). Assume (M, g) is a compact Riemannian manifold of dimension n ≥ 5

Assume further holds for some

By we can fix N large enough so that C

Since the remaining larger exponents q follows from the case

We noted before that

Recall that σ

Sobolev inequality at