Step refinable functions and orthogonal MRA on $p$-adic Vilenkin groups

We find the necessary and sufficient conditions for refinable step function under which this function generates an orthogonal MRA in the $L_2(\mathfrak{G})$ -spaces on Vilenkin groups $\mathfrak{G}$. We consider a class of refinable step functions for which the mask $m_0(\chi )$ is constant on cosets $\mathfrak G_{-1}^\bot$anditsmodulus$|m_0(\chi)|$ takes two values only: 0 and 1. We will prove that any refinable step function $\phi$ from this class that generates an orthogonal MRA on $p$-adic Vilenkin group $\mathfrak{G}$ has Fourier transform with condition ${\rm supp} \hat \varphi(\chi)\subset \mathfrak G_{p-2}^\bot$. We show the sharpness of this result too.