Besov-type spaces with variable smoothness and integrability
From a so-called admissible system \(\{\varphi_j\}_{j\in\mathbb{N}_0}\) of functions on \(\mathbb{R}^n\), the authors define the Besov-type space \(B^{s(\cdot),\phi}_{p(\cdot),q(\cdot)}(\mathbb{R}^n)\) of variable smoothness and integrability as the collection of all tempered distributions \(f\) on \(\mathbb{R}^n\) such that \[ \|f\|_{B^{s(\cdot),\phi}_{p(\cdot),q(\cdot)}(\mathbb{R}^n)} := \sup_{Q} \frac{1}{\phi(Q)} \, \left\| \left\{2^{js(\cdot)} |\varphi_j\ast f|\right\}_{j\geq (j_Q\vee 0)}\right\|_{\ell^{q(\cdot)}(L^{p(\cdot)}(Q))} \,<\infty, \] where the supremum is taken over all dyadic cubes in \(\mathbb{R}^n\). Here the variable exponents \(s(\cdot),\;p(\cdot),\;q(\cdot)\) are assumed to be regular (e.g., \(\log\)-Hölder continuous), \(\ell^{q(\cdot)}(L^{p(\cdot)})\) are mixed Lebesgue sequence spaces and \(\phi\) satisfies some compatibility and doubling conditions. In particular, the scale \(B^{s(\cdot),\phi}_{p(\cdot),q(\cdot)}(\mathbb{R}^n)\) includes the Besov spaces \(B^{s(\cdot)}_{p(\cdot),q(\cdot)}(\mathbb{R}^n)\) of variable smoothness and integrability (when \(\phi(Q)=1\) for all cubes \(Q\)) introduced in [\textit{A. Almeida} and \textit{P. Hästö}, J. Funct. Anal. 258, No. 5, 1628--1655 (2010; Zbl 1194.46045)], and the Besov-type spaces \(B^{s,\tau}_{p,q}(\mathbb{R}^n)\) (in the case \(\phi(Q)=|Q|^\tau, \; t\in(0,\infty)\), and constant \(s,\,p,\,q\)) introduced in [\textit{W. Yuan} et al., Morrey and Campanato meet Besov, Lizorkin and Triebel. Lecture Notes in Mathematics 2005. Berlin: Springer (2010; Zbl 1207.46002)]. The main results of the paper provide characterizations in terms of Peetre maximal functions, \(\varphi\)-transforms and smooth atoms. Moreover, the authors also obtain a Sobolev type embedding and a trace result for these variable exponent Besov-type spaces.
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