This paper is devoted to interior estimates for eigenfunctions of the restricted fractional Laplacian on a bounded domain in ${\mathbb{R}}^d$. We prove that the eigenfunctions satisfy the expected $L^p$ bounds analogous to the classical results by Sogge [24]. As the fractional Laplacian is nonlocal, the standard method for Laplacian eigenfunction estimates can no longer work here. In the proof, we mainly reduce $L^p$ bounds to a kind of $L^2$ commutator estimates, which can be handled by the explicit integral expression of the restricted fractional Laplacian and its heat kernel estimates.

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有界域上分数拉普拉斯算子的本征函数的内部估计

本文致力于在有界域上限制分数拉普拉斯算子的本征函数的内部估计 ${\mathbb{电阻}}^d$. 我们证明本征函数满足期望${升}^{磷}$边界类似于 Sogge [24] 的经典结果。由于分数拉普拉斯算子是非局部的，拉普拉斯本征函数估计的标准方法在这里不再适用。在证明中，我们主要减少${升}^{磷}$ 有界于一种 ${升}^2$ 交换子估计，可以通过限制分数拉普拉斯算子的显式积分表达式及其热核估计来处理。