This work extends the theory of Rychkov, who developed the theory of $A_{p}^{\mathrm{loc}}$ weights. It also extends the work by Cruz-Uribe SFO, Fiorenza, and Neugebauer. The class $A_{p(\cdot )}^{\mathrm{loc}}$ is defined. The weighted inequality for the local Hardy–Littlewood maximal operator on Lebesgue spaces with variable exponents is proven. Cruz-Uribe SFO, Fiorenza, and Neugebauer considered the Muckenhoupt class for Lebesgue spaces with variable exponents. However, due to the setting of variable exponents, a new method for extending weights is needed. The proposed extension method differs from that by Rychkov. A passage to the vector-valued inequality is realized by means of the extrapolation technique. This technique is an adaptation of the work by Cruz-Uribe and Wang. Additionally, a theory of extrapolation adapted to our class of weights is also obtained.

中文翻译：

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用于变量指数的局部Muckenhoupt类

这项工作扩展了Rychkov的理论，后者开发了$ A_ {p} ^ {\ mathrm {loc}} $权重理论。它还扩展了Cruz-Uribe SFO，Fiorenza和Neugebauer的工作。定义了类$ A_ {p（\ cdot）} ^ {\ mathrm {loc}} $。证明了具有可变指数的Lebesgue空间上局部Hardy–Littlewood极大算子的加权不等式。Cruz-Uribe SFO，Fiorenza和Neugebauer考虑了具有可变指数的Lebesgue空间的Muckenhoupt类。但是，由于设置了可变指数，因此需要一种新的扩展权重的方法。提出的扩展方法与Rychkov的扩展方法不同。通过外推技术可以实现向量值不等式的传递。该技术是对Cruz-Uribe和Wang的作品的改编。此外，