论文标题
在砂纸和蒙哥马利的尖锐引理上
On a sharp lemma of Cassels and Montgomery on manifolds
论文作者
论文摘要
Let $\left( \mathcal{M},g\right) $ be a $d$-dimensional compact connected Riemannian manifold and let $\left\{ φ_{m}\right\}_{m=0}^{+\infty}$ be a complete sequence of orthonormal eigenfunctions of the Laplace-Beltrami $ \ MATHCAL {M} $上的操作员。我们表明，存在一个正常数$ c $，使得对于所有整数$ n $ and $ x $，以及对于$ \ Mathcal {m} $中的所有有限序列，$ n $点，$ \ weft \ left \ {x \ left（j \ oyt（j \ oright）\ right） a_ {j} \ right \} _ {j = 1}^{n} $我们有\ [\ sum_ {m = 0}^{x} | \ sum_ {j = 1}^{n} a_ {j}φ_{m}（x（j））| ^{2} \ geq \ max \ {cx \ sum_ {j = 1}^{n} a_ {j}^{2} {2} {2}，（\ sum_ {j = 1}^{n}^{n} a_ {j {j}）
Let $\left( \mathcal{M},g\right) $ be a $d$-dimensional compact connected Riemannian manifold and let $\left\{ φ_{m}\right\}_{m=0}^{+\infty}$ be a complete sequence of orthonormal eigenfunctions of the Laplace-Beltrami operator on $\mathcal{M}$. We show that there exists a positive constant $C$ such that for all integers $N$ and $X$ and for all finite sequences of $N$ points in $\mathcal{M}$, $\left\{ x\left( j\right) \right\}_{j=1}^{N}$, and positive weights $\left\{ a_{j}\right\}_{j=1}^{N}$ we have \[ \sum_{m=0}^{X} | \sum_{j=1}^{N} a_{j} φ_{m} ( x( j) ) | ^{2}\geq \max \{ CX\sum_{j=1}^{N}a_{j}^{2},( \sum_{j=1}^{N}a_{j}) ^{2}\}.\]
