Let G be an infinite connected graph. We study the Sobolev regularity for the Hardy–Littlewood maximal operator and its fractional variants on G. Under certain geometric conditions on G, the endpoint Sobolev regularity properties for the above maximal operators are established. In addition, we introduce Hajłasz–Sobolev spaces on G and show that the above operators are bounded on the above function spaces.

中文翻译：

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无限连通图上极大算子的Sobolev正则性

令G为无限连通图。我们研究了Hardy–Littlewood极大算子的Sobolev正则性及其在G上的分数变体。在G上的某些几何条件下，建立上述最大算子的端点Sobolev正则性。另外，我们在G上引入Hajłasz-Sobolev空间，并证明上述算子在上述函数空间上是有界的。