Factorizations for variable exponent Muckenhoupt weights

Given two variable exponent Muckenhoupt weights $w\in \mathcal{A}_{p(\cdot)}$ and $w_1\in \mathcal{A}_{p_1(\cdot)}$, we prove that for all small enough $θ>0,$ there holds that $w_0\in \mathcal{A}_{p_0(\cdot)},$ where the weight is determined by $w = w_0^{1-θ}w_1^θ$ and exponent of the weight class b