Algebras of Mellin pseudodifferential operators with quasicontinuous symbols.

The paper deals with studying a Banach algebra ${\mathfrak D}_p$ generated by Mellin pseudodifferential operators with a subclass $\widehat{\mathcal E}(\mathbb{R}_+, \!V(\mathbb{R}))$ of quasicontinuous $V(\mathbb{R})$-valued symbols on Lebesgue spaces $L^p(\mathbb{R}_+,d\mu)$, where $V(\mathbb{R})$ is the Banach algebra of absolutely continuous functions of bounded total variation on the real line $\mathbb{R}$, $p\in(1,\infty)$ and $d\mu(r)=dr/r$. Constructing approximations of symbols ${\mathfrak a}\in\widehat{\mathcal E} (\mathbb{R}_+,V(\mathbb{R}))$ by functions similar to symbols of singular integral operators with quasicontinuous coefficients, which form a Banach algebra ${\mathfrak A}_p$, the similarity of the algebras ${\mathfrak D}_p$ and ${\mathfrak A}_p$ is established, and the compactness of commutators and semicommutators of Mellin pseudodifferential operators $\operatorname{Op}({\mathfrak a})\in{\mathfrak D}_p$ with symbols ${\mathfrak a} \in\widehat{\mathcal E}(\mathbb{R}_+,V(\mathbb{R}))$ is proved. A Fredholm symbol calculus for the Banach algebra ${\mathfrak D}_p$ is constructed and a Fredholm criterion and an index formula for the operators $D\in{\mathfrak D}_p$ are obtained.