  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.
