Necessary Conditions for Fredholmness of Singular Integral Operators with Shifts and Slowly Oscillating Data

Abstract

Abstract. Suppose α is an orientation-preserving diffeomorphism (shift) of R + = (0, ∞) onto itself with the only fixed points 0 and ∞. In [6] we found sufficient conditions for the Fredholmness of the singular integral operator with shiftacting on L p (R + ) with 1 < p < ∞, where P ± = (I ± S)/2, S is the Cauchy singular integral operator, and W α f = f • α is the shift operator, under the assumptions that the coefficients a, b, c, d and the derivative α ′ of the shift are bounded and continuous on R + and may admit discontinuities of slowly oscillating type at 0 and ∞. Now we prove that those conditions are also necessary.

Sufficient Conditions for Fredholmness of Singular Integral Operators with Shifts and Slowly Oscillating Data

“…Finally, we note that conditions (i) and (ii) of Theorem 1.2 are also necessary for the Fredholmness of the operator N . This statement will be proved in the forthcoming paper [17].…”

“…Finally, we note that conditions (i) and (ii) of Theorem 1.2 are also necessary for the Fredholmness of the operator N . This statement will be proved in the forthcoming paper [17]. Conversely, if {t n } ⊂ R + is a sequence such that t n → s as n → ∞, then there exists a functional ξ ∈ M s (SO(R + )) such that (2.1) holds.…”

On a Weighted Singular Integral Operator with Shifts and Slowly Oscillating Data

“…This paper is in some sense a continuation of our papers [7,8,9], where singular integral operators with shifts were studied under the mild assumptions that the coefficients belong to SO(R + ) and the shifts belong to SOS(R + ). In [7,8] we found a Fredholm criterion for the singular integral operator N = (aI − bW α )P + p + (cI − dW α )P − p with coefficients a, b, c, d ∈ SO(R + ) and a shift α ∈ SOS(R + ). However, a formula for the calculation of the index of the operator N is still missing.…”