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2-microlocal Besov and Triebel-Lizorkin spaces of variable integrability. (English) Zbl 1166.42011

Let \(\alpha\geq 0\), \(\alpha_1,\, \alpha_2\in \mathbb{R}\) and \(\alpha_1 \leq \alpha_2.\) A sequence of non-negative measurable functions \(w=\{w_j\}_{j=0}^{\infty}\) belongs to the class \({\mathcal W}^\alpha_{\alpha_1,\alpha_2}\) if and only if (i) There exists a positive constant \(C\) such that \(0<w_j(x)\leq Cw_j(y) (1+2^j|x-y|)^\alpha\) for all \(j\in \mathbb{N}\cup \{0\}\) and \(x,y\in \mathbb{R}^n;\) (ii) For all \(j\in \mathbb{N}\cup \{0\}\) and \(x\in \mathbb{R}^n,\) \(2^{\alpha_1}w_j(x)\leq w_{j+1}(x) \leq 2^{\alpha_2}w_j(x).\) Let \(\varphi_0 \in \mathcal{S}(\mathbb{R}^n)\) with \(\varphi_0(x)=1\) for \(|x|\leq 1\) and \({\text{supp}}\, {\varphi_0} \subseteq \{x\in \mathbb{R}^n:\, |x|\leq2 \}\). For all \(j\in \mathbb{N},\) let \(\varphi_j(x)=\varphi_0(2^{-j}x)\;- \;\varphi_0(2^{-j+1}x) \) for all \(x \in \mathbb{R}^n.\) Denote by \(\mathcal{P}(\mathbb{R}^n)\) the class of all measurable \(p:\, \mathbb{R}^n\to(0,\infty]\) such that \(p^->0\), where \(p^-={\text{essinf}}_{x\in \mathbb{R}^n}p(x)\). Let \(w\in \mathcal{W}_{\alpha_1,\alpha_2}^\alpha\), \(p\in \mathcal{P}(\mathbb{R}^n)\) with \(0<p^-\leq p^+={\text{esssup}}_{x\in\mathbb{R}^n}p(x)\leq \infty,\) and \(0<q \leq \infty.\) The space \(B_{p(\cdot),q}^w(\mathbb{R}^n)\) is defined as \(B_{p(\cdot),q}^w(\mathbb{R}^n)= \{ f\in\mathcal{S}^\prime(\mathbb{R}^n):\, \|f\,|B_{p(\cdot),q}^w(\mathbb{R}^n)\|_\varphi < \infty \},\) where \(\| f\,|B_{p(\cdot),q}^w(\mathbb{R}^n) \|_\varphi= \{\sum_{j=0}^{ \infty } \| w_j(\varphi_j \widehat{f})^{\vee} \,|L_{p(\cdot)}(\mathbb{R}^n)\|^q\}^{1/q}.\) If \(p^+ < \infty,\) the space \(F_{p(\cdot),q}^w(\mathbb{R}^n)\) is defined by \(F_{p(\cdot),q}^w(\mathbb{R}^n)= \{ f\in\mathcal{S}^\prime(\mathbb{R}^n):\, \| f\,|F_{p(\cdot),q}^w(\mathbb{R}^n) \|_\varphi < \infty \},\) where \(\| f|F_{p(\cdot),q}^w(\mathbb{R}^n) \|_\varphi= \| \{\sum_{j=0}^{ \infty }|w_j(\varphi_j \widehat{f} )^{\vee}|^q\}^{1/q} \mid L_{p(\cdot)}(\mathbb{R}^n) \|\). In this paper, the author presents the local means characterization of these spaces. Moreover, the author also obtains a similar characterization of the space \(F_{p(\cdot),q(\cdot)}^w(\mathbb{R}^n).\)

MSC:

42B35 Function spaces arising in harmonic analysis
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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