AI 中文总结
本文研究黎曼流形上Laplace-Beltrami算子谱函数的非对角增长界，对n≥3给出Λ^{n/2}量级界，对n=2改进为Λ^{5/6}量级界，均在全测度子集上成立。
AI 中文摘要
我们研究n维黎曼流形M上Laplace-Beltrami算子谱函数的增长。对于n≥3，我们证明对任意x∈M，存在一个全测度子集Z_x⊂M，使得对所有y∈Z_x，有E_Λ(x,y)=O_{M,x,y,ε}(Λ^{n/2}(logΛ)^{3/2}(loglogΛ)^{1/2+ε})。对于n=2，我们证明了一个更强的结论：对任意x∈M，存在一个全测度子集Z_x⊂M，使得对所有y∈Z_x，有E_Λ(x,y)=O_{M,x,y,ε}(Λ^{5/6}(logΛ)^{1/6}(loglogΛ)^{1/6+ε})。
英文摘要
We study the growth of the spectral function of the Laplace-Beltrami operator on a Riemannian manifold $M$ of dimension $n$. For $n\geq 3$ we show that for any $x\in M$ there is a full measure subset $Z_x\subset M$ such that for all $y\in Z_x$ \[E_Λ(x,y) = O_{M,x,y,\varepsilon}(Λ^{n/2}(\logΛ)^{3/2}(\log\logΛ)^{1/2+\varepsilon}). \] For $n=2$ we prove a stronger statement. For any $x\in M$ there is a full measure subset $Z_x\subset M$ such that for all $y\in Z_x$ \[E_Λ(x,y) = O_{M,x,y,\varepsilon}(Λ^{5/6}(\logΛ)^{1/6}(\log\logΛ)^{1/6+\varepsilon}) .\]
CommentsThis is a first draft. Comments are welcome
