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We study the spaces \(B^{s(\cdot)}_{p(\cdot),q(\cdot)}(\mathbb {R}^{n})\) and \(F^{s(\cdot)}_{p(\cdot),q(\cdot)}(\mathbb{R}^{n})\) of Besov and Triebel-Lizorkin type as introduced recently in Almeida and Hästö (J. Funct. Anal. 258(5):1628–2655, 2010) and Diening et al. (J. Funct. Anal. 256(6):1731–1768, 2009). Both scales cover many classical spaces with fixed exponents as well as function spaces of variable smoothness and function spaces of variable integrability.

Spaces of Variable Smoothness and Integrability: Characterizations by Local Means and Ball Means of Differences