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

Abstract

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. The spaces $B^{s(\cdot)}_{p(\cdot),q(\cdot)}(\mathbb{R}^{n})$ and $F^{s(\cdot)}_{p(\cdot),q(\cdot)}(\mathbb{R}^{n})$ have been introduced in Almeida and Hästö (J. Funct. Anal. 258(5):1628–2655, 2010) and Diening et al. (J. Funct. Anal. 256(6):1731–1768, 2009) by Fourier analytical tools, as the decomposition of unity. Surprisingly, our main result states that these spaces also allow a characterization in the time-domain with the help of classical ball means of differences. To that end, we first prove a local means characterization for $B^{s(\cdot)}_{p(\cdot),q(\cdot)}(\mathbb{R}^{n})$ with the help of the so-called Peetre maximal functions. Our results do also hold for 2-microlocal function spaces $B^{\boldsymbol{w}}_{{p(\cdot)},{q(\cdot)}}(\mathbb{R}^{n})$ and $F^{\boldsymbol{w}}_{{p(\cdot)},{q(\cdot)}}(\mathbb{R}^{n})$ which are a slight generalization of generalized smoothness spaces and spaces of variable smoothness.