We show that for every p š (1, ∞) there exists a weight w such that the Lorentz Gamma space Γ_p,w is reflexive, its lower Boyd and Zippin indices are equal to zero and its upper Boyd and Zippin indices are equal to one. As a consequence, the Hardy-Littlewood maximal operator is unbounded on the constructed reflexive space Γ_p,w and on its associate space \(\Gamma _{p,w}^\prime\).

中文翻译：

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具有平凡 Boyd 和 Zippin 指数的自反洛伦兹伽玛空间的例子

我们证明对于每个p š (1, ∞) 都存在一个权重w使得洛伦兹伽玛空间 Γ _p,w是自反的，它的下博伊德和 Zippin 指数等于零，它的上博伊德和 Zippin 指数等于一。因此，Hardy-Littlewood 极大算子在构造的自反空间 Γ _p,w及其关联空间\(\Gamma _{p,w}^\prime\)上是无界的。