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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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Full Text: Euclid