Communications in Mathematical Analysis

On Regularization of Mellin PDO’s with Slowly Oscillating Symbols of Limited Smoothness

A. Karlovich, Yu. Karlovich, and A. Lebre

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

Article information

Commun. Math. Anal., Volume 17, Number 2 (2014), 189-208.

First available in Project Euclid: 18 December 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47G30: Pseudodifferential operators [See also 35Sxx, 58Jxx]

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


Karlovich, A.; Karlovich, Yu.; Lebre, A. On Regularization of Mellin PDO’s with Slowly Oscillating Symbols of Limited Smoothness. Commun. Math. Anal. 17 (2014), no. 2, 189--208.

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