@article {6703,
	title = {On regularization of Mellin PDO{\textquoteright}s with slowly oscillating symbols of limited smoothness},
	journal = {Communications in Mathematical Analysis},
	volume = {17},
	year = {2014},
	pages = {189-208},
	abstract = {<script src={\textquoteright}https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML{\textquoteright}></script><p>We study Mellin pseudodifferential operators (shortly, Mellin PDO{\textquoteright}s) with symbols in the algebra \(\widetilde{\mathcal{E}}(\mathbb{R}_+,V(\mathbb{R}))\) of slowly oscillating functions of limited  smoothness introduced in [K09]. We show that if \(\mathfrak{a}\in\widetilde{\mathcal{E}}(\mathbb{R}_+,V(\mathbb{R}))\) does not degenerate on the {\textquoteleft}{\textquoteleft}boundary" of \(\mathbb{R}_+\times\mathbb{R}\) in a certain sense,  then the Mellin PDO \(\operatorname{Op}(\mathfrak{a})\) is Fredholm on the space \(L^p\) for \(p\in(1,\infty)\) and each its regularizer is of the form \(\operatorname{Op}(\mathfrak{b})+K\) 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 {\textquoteleft}{\textquoteleft}boundary" of \(\mathbb{R}_+\times\mathbb{R}\). This result complements the known Fredholm criterion from [K09] for Mellin PDO{\textquoteright}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 [R98]) on Mellin PDO{\textquoteright}s with slowly oscillating symbols in \(C^\infty(\mathbb{R}_+\times\mathbb{R})\). </p>
},
	url = {http://www.math-res-pub.org/cma/17/2/regularization-mellin-pdo\%E2\%80\%99s-slowly-oscillating-symbols-limited-smoothness},
	author = {Karlovich, Alexei Yu. and Karlovich, Yuri I. and Lebre, Amarino B.}
}
