On regularization of Mellin PDO's with slowly oscillating symbols of limited smoothness

We study Mellin pseudodifferential operators (shortly, Mellin PDO’s) with symbols in the algebra $\tilde{\mathcal{E}}({\mathbb{R}}_{+},V(\mathbb{R}))$ of slowly oscillating functions of limited smoothness introduced in \cite{K09}. We show that if $\mathfrak{a}\in \tilde{\mathcal{E}}({\mathbb{R}}_{+},V(\mathbb{R}))$ does not degenerate on the “boundary” of ${\mathbb{R}}_{+}\times \mathbb{R}$ in a certain sense, then the Mellin PDO $\mathrm{O}\mathrm{p}(\mathfrak{a})$ is Fredholm on the space $L^p$ for $p\in (1,\mathrm{\infty })$ and each its regularizer is of the form ${\rm Op}(\mathfrak{b})+K$where$K$isacompactoperatoron$L^p$and$\mathfrak{b}$isacertainexplicitlyconstructedfunctioninthesamealgebra$\widetilde{$\mathcal{E}$}($\mathbb{R}$+,V($\mathbb{R}$))$suchthat$\mathfrak{b}=1/\mathfrak{a}$onthe\mathrm{"}boundary\mathrm{"}of$$\mathbb{R}$+\times$\mathbb{R}$$. This result complements a known Fredholm criterion from \cite{K09} for Mellin PDO’s with symbols in the closure of $\tilde{\mathcal{E}}({\mathbb{R}}_{+},V(\mathbb{R}))$.