We consider the fractional Laplacian with Hardy potential and study the scale of homogeneous $L^p$ Sobolev spaces generated by this operator. Besides generalized and reversed Hardy inequalities, the analysis relies on a Hormander multiplier theorem which is crucial to construct a basic Littlewood--Paley theory. The results extend those obtained recently in $L^2$ but do not cover negative coupling constants in general due to the slow decay of the associated heat kernel.

中文翻译：

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与广义哈代算子相关的 Sobolev 空间的尺度

我们考虑具有哈代势的分数拉普拉斯算子，并研究由该算子生成的齐次 $L^p$ Sobolev 空间的尺度。除了广义和反向哈代不等式外，该分析还依赖于霍曼德乘数定理，该定理对于构建基本的 Littlewood-Paley 理论至关重要。结果扩展了最近在 $L^2$ 中获得的结果，但由于相关热核的缓慢衰减，通常不包括负耦合常数。