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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 $\widetilde{\mathcal{E}}(\mathbb{R}_+,V(\mathbb{R}))$ of slowly oscillating functions of limited smoothness introduced in \cite{K09}. We show that if $\mathfrak{a}\in\widetilde{\mathcal{E}}(\mathbb{R}_+,V(\mathbb{R}))$ does not degenerate on the "boundary" of $\mathbb{R}_+\times\mathbb{R}$ in a certain sense, then the Mellin PDO ${\rm Op}(\mathfrak{a})$ is Fredholm on the space $L^p$ for $p\in(1,\infty)$ and each its regularizer is of the form ${\rm Op}(\mathfrak{b})+K$ where $K$ is a compact operator on $L^p$ and $\mathfrak{b}$ is a certain explicitly constructed function in the same algebra $\widetilde{\mathcal{E}}(\mathbb{R}_+,V(\mathbb{R}))$ such that $\mathfrak{b}=1/\mathfrak{a}$ on the "boundary" 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 $\widetilde{\mathcal{E}}(\mathbb{R}_+,V(\mathbb{R}))$.