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 $\widetilde{\mathcal E}({\mathbb R}_+,V({\mathbb R}))$ of slowly oscillating functions of limited smoothness introduced in [12]. We show that if ${\mathcal a} \in\widetilde{\mathcal E}({\mathbb R}_+,V({\mathbb R}))$ does not degenerate on the ``boundary" of ${\mathbb R}_+\times {\mathbb R}$ in a certain sense, then the Mellin PDO $Op({\mathcal a})$ is Fredholm on the space $L^p$ for $p\in(1,\infty)$ and each its regularizer is of the form $Op({\mathcal b})+K$ where $K$ is a compact operator on $L^p$ and ${\mathcal b}$ is a certain explicitly constructed function in the same algebra $\widetilde{\mathcal E}({\mathbb R}_+,V({\mathbb R}))$ such that ${\mathcal b}=1/{\mathcal a}$ on the ``boundary" of ${\mathbb R}_+\times {\mathbb R}$. This result complements the known Fredholm criterion from [12] for Mellin PDO's with symbols in the closure of $\widetilde{\mathcal E}({\mathbb R}_+,V({\mathbb R}))$ and extends the corresponding result by V.S. Rabinovich (see [16]) on Mellin PDO's with slowly oscillating symbols in $C^\infty({\mathbb R}_+\times {\mathbb R})$.