Criteria for n ( d ) ‐normality of weighted singular integral operators with shifts and slowly oscillating data

Let α , β be orientation‐preserving homeomorphisms of [ 0 , ∞ ] onto itself, which have only two fixed points at 0 and ∞ , and whose restrictions toR + = ( 0 , ∞ )are diffeomorphisms, and letU α , U βbe the corresponding isometric shift operators on the spaceL p ( R + ) , 1 < p < ∞ , given byU μ f = ( μ ′ ) 1 / p( f ∘ μ )for μ ∈ { α , β } . We prove criteria for the n ‐normality and d ‐normality on the spaceL p ( R + )of singular integral operators of the formA + P γ + + A − P γ − , whereA + = ∑ k ∈ Za k U α k ,A − = ∑ k ∈ Zb k U β kare operators in the Wiener algebras of functional operators with shifts and the operatorsP γ ± = ( I ± S γ ) / 2 are associated to the weighted Cauchy singular integral operator( S γ f ) ( t ) = 1 π i∫ R +t τ γf ( τ ) τ − td τ ,where γ ∈ C satisfies the condition 0 < 1 / p + ℜ γ < 1 . We assume that the coefficientsa k , b kfor k ∈ Z and the derivatives of the shiftsα ′ , β ′are bounded continuous functions on R + which may have slowly oscillating discontinuities at 0 and ∞ .