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2014 On Regularization of Mellin PDO’s with Slowly Oscillating Symbols of Limited Smoothness
A. Karlovich, Yu. Karlovich, A. Lebre
Commun. Math. Anal. 17(2): 189-208 (2014).


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 $C^\infty({\mathbb R}_+\times {\mathbb R})$.


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A. Karlovich. Yu. Karlovich. A. Lebre. "On Regularization of Mellin PDO’s with Slowly Oscillating Symbols of Limited Smoothness." Commun. Math. Anal. 17 (2) 189 - 208, 2014.


Published: 2014
First available in Project Euclid: 18 December 2014

zbMATH: 1332.47025
MathSciNet: MR3292968

Primary: 47G30

Keywords: Fredholmness , maximal ideal space. , Mellin pseudodifferential operator , regularizer , slowly oscillating symbol

Rights: Copyright © 2014 Mathematical Research Publishers

Vol.17 • No. 2 • 2014
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