## 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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