论文标题
具有改进光谱簇和WEYL剩余估计的产品歧管
Product Manifolds with Improved Spectral Cluster and Weyl Remainder Estimates
论文作者
论文摘要
我们表明，如果$ y $是一种紧凑的Riemannian歧管，则具有改进的$ l^q $ eigenFunction估计值，至少对于足够大的指数，如果$ x $ x $是另一个紧凑型折线，则总是会在产品歧管$ x \ times y $上获得改进的$ l^q $ bunds。同样，在$ y $的频谱计数函数上改进的Weyl剩余项范围可导致$ x \ times y $的相应改进。后者的结果部分概括了涉及球体产品的Iosevich和Wyman [14]的最新结果。 Also, if $Y$ is a product of five or more spheres, we are able to obtain optimal $L^q(Y)$ and $L^q(X\times Y)$ eigenfunction and spectral cluster estimates for large $q$, which partly addresses a conjecture from [14] and is related to (and is partly based on) classical bounds for the number of integer lattice point on $λ\cdot $ n \ ge5 $的s^{n-1} $。
We show that if $Y$ is a compact Riemannian manifold with improved $L^q$ eigenfunction estimates then, at least for large enough exponents, one always obtains improved $L^q$ bounds on the product manifold $X\times Y$ if $X$ is another compact manifold. Similarly, improved Weyl remainder term bounds on the spectral counting function of $Y$ lead to corresponding improvements on $X\times Y$. The latter results partly generalize recent ones of Iosevich and Wyman [14] involving products of spheres. Also, if $Y$ is a product of five or more spheres, we are able to obtain optimal $L^q(Y)$ and $L^q(X\times Y)$ eigenfunction and spectral cluster estimates for large $q$, which partly addresses a conjecture from [14] and is related to (and is partly based on) classical bounds for the number of integer lattice point on $λ\cdot S^{n-1}$ for $n\ge5$.
