Local Muckenhoupt class for variable exponents

We define $A_{p(\cdot)}^{\rm loc}$ and show that the weighted inequality for local Hardy--Littlewood maximal operator on the Lebesgue spaces with variable exponent. This work will extend the theory of Rychkov, who developed the theory of $A_p^{\rm loc}$ weights. It will also extend the work by Cruz-Uribe. SFO, Fiorenza and Neugebaucer, who considered the Muckenhoupt class for Lebesgue spaces with variable exponents. Due to the setting of variable exponents, a new method of extension of weights will be needed; the extension method is different from the one by Rychkov. A passage to the vector-valued inequality is also done by means of the extrapolation technique. This technique is an adaptation of the work by Cruz-Uribe and Wang. We develop the theory of extrapolation adapted to our class of weights.