Pseudodifferential Operators with Compound Slowly Oscillating Symbols

Abstract

Let V (ℝ) denote the Banach algebra of absolutely continuous functions of bounded total variation on ℝ. We study an algebra $$ \mathfrak{B} $$ of pseudodifferential operators of zero order with compound slowly oscillating V (ℝ)-valued symbols (x, y) ↦ a(x, y, ·) of limited smoothness with respect to x, y ∈ ℝ. Sufficient conditions for the boundedness and compactness of pseudodifferential operators with compound symbols on Lebesgue spaces L p(ℝ) are obtained. A symbol calculus for the algebra $$ \mathfrak{B} $$ is constructed on the basis of an appropriate approximation of symbols by infinitely differentiable ones and by use of the techniques of oscillatory integrals. A Fredholm criterion and an index formula for pseudodifferential operators A ∈ $$ \mathfrak{B} $$ are obtained. These results are carried over to Mellin pseudodifferential operators with compound slowly oscillating V (ℝ)-valued symbols. Finally, we construct a Fredholm theory of generalized singular integral operators on weighted Lebesgue spaces L p with slowly oscillating Muckenhoupt weights over slowly oscillating Carleson curves.