Pseudodifferential operators with compound non-regular symbols

Abstract.

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 $L^p\left(\Gamma, w\right)$.

The author was partially supported by the PFCE project (México).