Atomic, molecular and wavelet decomposition of 2-microlocal Besov and Triebel-Lizorkin spaces with variable integrability

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Triebel-Lizorkin-Type Spaces with Variable Exponents (2015, background): “Subsequently, Vyb́ıral [72] established Sobolev and Jawerth embeddings of these spaces and, moreover, Kempka [34] characterized F s(·) p(·),q(·)(R n) by local means, and Kempka and Vyb́ıral [36] obtained the equivalent characterization via ball means of differences.”
Besov-Type Spaces with Variable Smoothness and Integrability (2015, methodology): “More generally, 2-microlocal Besov and Triebel-Lizorkin spaces with variable, B w(·) p(·),q(·)(R ) and F w(·) p(·),q(·)(R ), were introduced by Kempka [27, 28] and provided a unified approach that cover the classical Besov and Triebel-Lizorkin spaces as well as versions of variable smoothness and in”
Generalized coorbit space theory and inhomogeneous function spaces of Besov-Lizorkin-Triebel type (2010, methodology): “ales of inhomogeneous 2-microlocal Besov and Lizorkin-Triebel spaces Bw p,q(R d,v) and Fw p,q(R d,v) on Rd with Muckenhoupt weights. These spaces represent a symbiosis of the spaces studied by Kempka [33, 34, 35], Bui [5, 6], Bui et al. [7, 8], and Haroske et al. [31]. The scales Bw p,q(R d,v) and F”