Pseudodifferential operators with compound non-regular symbols

The boundedness and compactness of Fourier pseudodifferential operators with compound symbols in subclasses of $$L^\infty\left(\mathbb{R}^2, L^{1}\left(\mathbb{R}\right)\right)$$ is studied on weighted Lebesgue spaces $$L^p\left(\mathbb{R}, w\right)$$ with $$p\;\in\;\left(1,\;\infty\right)$$ and Muckenhoupt weights $$w\;\in\;A_p\left(\mathbb{R}\right)$$ 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 $$L^\infty\left(\mathbb{R}^2_{+}, L^{1}\left(\mathbb{R}\right)\right),$$ which act on the spaces $$L^p\left(\mathbb{R}_{+}, d\mu\right),$$ where $$d\mu \left(t\right)\;=\;dt/t\; \mathrm{for}\;t\in \mathbb{R}_{+}.$$ 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 $$L^p\left(\Gamma, w\right)$$ .