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Algebras of Pseudodifferential Operators with Discontinuous Oscillating Symbols

Commun. Math. Anal. Conf.
Conference 3 (2011), 108 - 130

Algebras of Pseudodifferential Operators with Discontinuous Oscillating Symbols

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Abstract

Non-closed algebras Ad;g of pseudodifferential operators with slowly oscillating Lipschitz symbols in LSO d;g(R£R) with d; g 2 (1=2;1] and the minimalC¤-algebra A containing all Ad;g are studied on the Lebesgue space L2(R). Applying results on the boundedness and compactness of pseudodifferential operators A 2 A, a commutative algebra of their Fredholm symbols is described. A Fredholm criterion and an index formula for the operators A 2 A are obtained. Then we study the Fredholmness for the C¤-algebra B generated by the operators A 2 A with multiplicatively slowly oscillating Lipschitz symbols and by the operators of multiplication aI and convolution operators W0(b) with piecewise continuous functions a;b : R!C. The algebra of Fredholm symbols for the operators A 2 B is not commutative. A Fredholm criterion for the operators A 2B is established.