Non-smooth atomic decomposition of variable 2-microlocal Besov-type and Triebel–Lizorkin-type spaces

Abstract In this paper we provide non-smooth atomic decompositions of 2-microlocal Besov-type and Triebel–Lizorkin-type spaces with variable exponents $$B^{\varvec{w}, \phi }_{p(\cdot ),q(\cdot )}({\mathbb {R}}^n)$$ B p ( · ) , q w ϕ R n $$F^{\varvec{w}, F . Of big importance in general, an essential tool here, are the characterizations via maximal functions local means, that also present. These were recently introduced by Wu et al. cover not only Besov Triebel–Lizorkin $$B^{\varvec{w}}_{p(\cdot $$F^{\varvec{w}}_{p(\cdot , but more classical smoothness Morrey $$B^{s, \tau }_{p,q}({\mathbb s τ $$F^{s,\tau Afterwards, state a pointwise multipliers assertion for scale.