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

在本文中，我们提供了具有可变指数\（B ^ {\ varvec {w}，\ phi} _ {p（\ cdot），q （\ cdot）}（{\ mathbb {R}} ^ n）\）和\（F ^ {\ varvec {w}，\ phi} _ {p（\ cdot），q（\ cdot）}（{\ mathbb {R}} ^ n）\）。总的来说，这里最重要的工具是通过最大函数和局部手段对空间进行刻画，这也是我们现在要介绍的。这些空间是由Wu等人最近引入的。不仅涵盖变量2-microlocal Besov和Triebel–Lizorkin空间\（B ^ {\ varvec {w}} _ {p（\ cdot），q（\ cdot）}（{\ mathbb {R}} ^ n） \）和\（F ^ {\ varvec {w}} _ {p（\ cdot），q（\ cdot}}（{\ mathbb {R}} ^ n）\），但还有更经典的平滑度Morrey空间\（B ^ {s，\ tau} _ {p，q}（{\ mathbb {R}} ^ n）\）和\（F ^ {s，\ tau} _ {p，q}（{\ mathbb {R}} ^ n）\）。然后，我们针对该标度陈述一个逐点乘数断言。