Microlocal regularity of nonlinear PDE in quasi-homogeneous Fourier Lebesgue spaces

We study the continuity in weighted Fourier Lebesgue spaces for a class of pseudodifferential operators, whose symbol has finite Fourier Lebesgue regularity with respect to $x$ and satisfies a quasi-homogeneous decay of derivatives with respect to the $xi$ variable. Applications to Fourier Lebesgue microlocal regularity of linear and nonlinear partial differential equations are given.