Pseudodifferential operators with compound non-regular symbols

Abstract

The boundedness and compactness of Fourier pseudodifferential operators with compound symbols in subclasses of L∞(ℝ2, L1(ℝ) is studied on weighted Lebesgue spaces Lp(ℝ, w) with p ϵ (1, ∞) and Muckenhoupt weights w ϵ Ap(ℝ) by applying the techniques of oscillatory integrals. The boundedness and compactness conditions are also obtained for Mellin pseudodifferential operators with compound symbols in subclasses of L∞(ℝ2+, L1(ℝ) which act on the spaces Lp(ℝ+, dμ) where dμ(t) = dt/t for t ϵ ℝ+. The latter results allow one to reduce the smoothness of slowly oscillating Carleson curves Γ and slowly oscillating Muckenhoupt weights w in the Fredholm study of singular integral operators with shifts on weighted Lebesgue spaces Lp(Γ, w).