In this paper, we develop a sliding method for the higher order fractional Laplacians. We first obtain the key ingredients to obtain monotonicity of solutions, such as narrow region maximum principles in bounded or unbounded domains. Then we introduce a new idea of estimating the singular integrals defining the fractional Laplacian along a sequence of approximate maximum points and illustrate how this sliding method can be employed to obtain monotonicity of solutions. We believe that the narrow region maximum principles will become useful tools in analyzing higher order fractional equations.

中文翻译：

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高阶分数拉普拉斯算子的滑动方法

在本文中，我们为高阶分数拉普拉斯算子开发了一种滑动方法。我们首先获得获得解的单调性的关键成分，例如有界或无界域中的窄区域最大值原则。然后，我们介绍了一种新的想法，即沿一系列近似最大值点估计定义分数拉普拉斯算子的奇异积分，并说明如何使用这种滑动方法来获得解的单调性。我们相信窄域最大值原理将成为分析高阶分数方程的有用工具。