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

Abstract

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 $\mathrm{Op}(\mathfrak{a})$ is Fredholm on the space $L^p$ for $p\in (1,\infty )$ and each its regularizer is of the form $\mathrm{Op}(\mathfrak{b})+K$ where $K$ is a compact operator on $L^p$ and $\mathfrak{b}$ is a certain explicitly constructed function in the same algebra $\tilde{\mathcal{E}}({\mathbb{R}}_{+},V(\mathbb{R}))$ 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 $\tilde{\mathcal{E}}({\mathbb{R}}_{+},V(\mathbb{R}))$.

20 pages