An atomic decomposition of variable Besov and Triebel-Lizorkin spaces

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FUNCTION SPACES WITH VARIABLE EXPONENTS – AN INTRODUCTION – (2014, unknown): “Remark that much was done by Xu [219, 220, 221, 222] (2008, 2008, 2009, 2012) when q(·) = q is a constant.”
Spaces of Variable Smoothness and Integrability: Characterizations by Local Means and Ball Means of Differences (2011, methodology): “Taking s ∈ R and q ∈ (0,∞] as constants we derive the spaces F s p(·),q(Rn) and B p(·),q(Rn) studied by Xu in [61] and [62].”
Besov spaces with variable smoothness and integrability (2010, background): “There was taken a step further by Xu [44, 45, 46], who considered Besov B p(·), q and Triebel–Lizorkin F α p(·), q spaces with variable p, but fixed q and α.”
A new framework for generalized Besov-type and Triebel-Lizorkin-type spaces (2012, unknown): “Another direction is pursued via replacing the space Lp(Rn) by the Morrey space Mu(R); see [48, 52, 53], or generalizations thereof [43, 80, 82, 89, 95, 96, 98, 99, 104].”
Triebel-Lizorkin-Type Spaces with Variable Exponents (2015, background): “Later, Xu [73, 74, 75] studied Besov spaces B p(·),q(R ) and Triebel-Lizorkin spaces F s p(·),q(R ) with the variable exponent p(·) but invariable exponents q and s.”