Nodal intersections and Lp restriction theorems on the torus
Nodal intersections and Lp restriction theorems on the torus
复制标题
环面上的节点交点和 Lp 限制定理
DOI：
10.1007/s11856-015-1183-7
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发表时间：
2013
影响因子：
1
通讯作者：
Z. Rudnick
中科院分区：
文献类型：
--
作者：
J. Bourgain;Z. Rudnick
研究了标准环面上拉普拉斯算子的本征函数的节线与固定参考曲线的交点数，即本征函数限制在参考曲线上的零点个数。上界是波数k。当曲线无处零曲率，我们推测，直到一个常数倍，这也应该是正确的下限。我们给出了一个下界，它与此不同的算术量，给出的最大数量的格点在弧的大小平方根的波数k的半径为k的圆。根据Cilleruelo和Granville的猜想，这个量是有界的，在这种情况下，我们恢复了我们的猜想。为了得到下界，我们将问题简化为给出特征函数对曲线的限制的L1范数的下界，然后给出L4限制范数的上界。
We study the number of intersections of the nodal lines of an eigenfunction of the Laplacian on the standard torus with a fixed reference curve, that is, the number of zeros of the eigenfunction restricted to the curve. An upper bound is the wave number k. When the curve has nowhere zero curvature, we conjecture that, up to a constant multiple, this should also be the correct lower bound. We give a lower bound which differs from this by an arithmetic quantity, given in terms of the maximal number of lattice points in arcs of size square root of the wave number k on a circle of radius k. According to a conjecture of Cilleruelo and Granville, this quantity is bounded, in which case we recover our conjecture. To get at the lower bound, we reduce the problem to giving a lower bound for the L1 norm of the restriction of the eigenfunction to the curve, and then to an upper bound for the L4 restriction norm.