Logarithmic improvements in Lp bounds for eigenfunctions at the critical exponent in the presence of nonpositive curvature

• 成果类型: Article
• 署名作者: Blair, Matthew D.; Sogge, Christopher D.
• 署名单位: University of New Mexico; Johns Hopkins University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-019-00873-6
• 发表日期: 2019
• 页码: 703-748
• 关键词: l-p-norms; kakeya-nikodym averages; oscillatory integrals; NODAL SETS; RESTRICTION; MANIFOLDS; Operators

摘要

We consider the problem of proving Lp bounds for eigenfunctions of the Laplacian in the high frequency limit in the presence of nonpositive curvature and more generally, manifolds without conjugate points. In particular, we prove estimates at the critical exponent pc=2(d+1)d-1, where a spectrum of scenarios for phase space concentration must be ruled out. Our work establishes a gain of an inverse power of the logarithm of the frequency in the bounds relative to the classical Lp bounds of the second author.