标题： 带有改进的谱簇和Weyl余项估计的乘积流形 显示英文标题

标题： Product Manifolds with Improved Spectral Cluster and Weyl Remainder Estimates

摘要： 我们证明，如果$Y$是一个具有改进的$L^q$特征函数估计的紧致黎曼流形，那么至少对于足够大的指数，总是可以得到乘积流形$X\times Y$上的改进的$L^q$界，如果$X$是另一个紧致流形的话。 类似地，$Y$的谱计数函数上的改进的 Weyl 余项界会导致$X\times Y$上相应的改进。 后一结果部分推广了 Iosevich 和 Wyman [14] 关于球面乘积的最新结果。 此外，如果 $Y$ 是五个或更多球面的乘积，我们能够得到当 $q$ 较大时的最优 $L^q(Y)$ 和 $L^q(X\times Y)$ 特征函数和谱簇估计，这部分解决了[14]中的一个猜想，并且与(并部分基于)关于 $\lambda \cdot S^{n-1}$ 上整数格点数量的经典界有关，对于 $n\ge5$。 显示英文摘要

摘要： 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 $\lambda \cdot S^{n-1}$ for $n\ge5$.