Pseudodifferential operators with compound non-regular symbols

Let V(R) denote the Banach algebra of absolutely continuous functions of bounded total variation on R. We study an algebra B of pseudodifferential operators of zero order with compound slowly oscillating V(R)-valued symbols (x, y) a(x, y, center dot) that satisfy a Lipschitz condition with respect to the spatial variables x, y is an element of R. Sufficient conditions for the boundedness and compactness of pseudodifferential operators with compound symbols on the Lebesgue spaces L-p (1(8), for p = 2 and 1 < p < infinity, are obtained. A Fredholm criterion and an index formula for pseudodifferential operators A is an element of B are presented. 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.