## Commutators of convolution type operators with piecewise quasicontinuous data.(English)Zbl 1319.47033

Summary: Applying the theory of Calderón-Zygmund operators, we study the compactness of the commutators $$[aI,W^0(b)]$$ of multiplication operators $$aI$$ and convolution operators $$W^0(b)$$ on weighted Lebesgue spaces $$L^p({\mathbb R},w)$$ with $$p\in(1,\infty)$$ and Muckenhoupt weights $$w$$ for some classes of piecewise quasicontinuous functions $$a\in PQC$$ and $$b\in PQC_{p,w}$$ on the real line $${\mathbb R}$$. Then we study two $$C^*$$-algebras $$Z_1$$ and $$Z_2$$ generated by the operators $$aW^0(b)$$, where $$a,b$$ are piecewise quasicontinuous functions admitting slowly oscillating discontinuities at $$\infty$$ or, respectively, quasicontinuous functions on $${\mathbb R}$$ admitting piecewise slowly oscillating discontinuities at $$\infty$$. We describe the maximal ideal spaces and the Gelfand transforms for the commutative quotient $$C^*$$-algebras $$Z_i^\pi=Z_i/{\mathcal K}$$ $$(i=1,2)$$ where $${\mathcal K}$$ is the ideal of compact operators on the space $$L^2({\mathbb R})$$, and establish the Fredholm criteria for the operators $$A\in Z_i$$.

### MSC:

 47B47 Commutators, derivations, elementary operators, etc. 45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) 46J10 Banach algebras of continuous functions, function algebras 47A53 (Semi-) Fredholm operators; index theories 47G10 Integral operators
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