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

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## Abstract

We study Mellin pseudodifferential operators (shortly, Mellin PDO's) with symbolsin the algebra $\widetilde{\cE}(\R_+,V(\R))$ of slowly oscillating functions oflimited smoothness introduced in \cite{K09}. We show that if$\fa\in\widetilde{\cE}(\R_+,V(\R))$ does not degenerate on the boundary" of$\R_+\times\R$ in a certain sense, then the Mellin PDO $\Op(\fa)$ is Fredholmon the space $L^p$ for $p\in(1,\infty)$ and each its regularizer is of the form$\Op(\fb)+K$ where $K$ is a compact operator on $L^p$ and $\fb$ is a certainexplicitly constructed function in the same algebra $\widetilde{\cE}(\R_+,V(\R))$such that $\fb=1/\fa$ on the boundary" of $\R_+\times\R$. This result complementsthe known Fredholm criterion from \cite{K09} for Mellin PDO's with symbols in theclosure of $\widetilde{\cE}(\R_+,V(\R))$and extends the corresponding result by V.S. Rabinovich (see \cite{R98})on Mellin PDO's with slowly oscillating symbols in $C^\infty(\R_+\times\R)$.
Original language Unknown 189-208 Communications In Mathematical Analysis 17 2 Published - 1 Jan 2014