Non-smooth atomic decomposition of 2-microlocal spaces and application to pointwise multipliers

We provide non-smooth atomic characterizations for 2-microlocal Besov and Triebel-Lizorkin spaces with variable integrability Bp(⋅),q(⋅)w(Rn) and Fp(⋅),q(⋅)w(Rn). These spaces cover the classical Besov and Triebel-Lizorkin spaces as well as spaces of variable smoothness and integrability. As an application, we state a pointwise multipliers result for these spaces.