Factorizations for variable exponent Muckenhoupt weights

Given two variable exponent Muckenhoupt weights $w\in A_{p(\cdot)}$ and $w_1\in A_{p_1(\cdot)}$, we prove that for all small enough $\theta>0,$ there holds that $w_0\in A_{p_0(\cdot)},$ where the weight is determined by $w = w_0^{1-\theta}w_1^{\theta}$ and exponent of the weight class by $1/p(\cdot) = (1-\theta)/p_0(\cdot) + \theta/p_1(\cdot).$ The proof is based on a recent reverse H\"older's inequality for variable exponent Muckenhoupt weights by Cruz-Uribe and Penrod. We upgrade these factorizations to the restricted range context by using a recent transformation formula due to Nieraeth. Then, following an extrapolation of compactness scheme by Hyt\"onen and Lappas, we provide an alternative proof of the recent extrapolation of compactness results of Lorist and Nieraeth in the context of weighted variable exponent Lebesgue spaces.