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Characterization of the Besov-Lipschitz and Triebel-Lizorkin spaces. The case $$q<1$$. (English) Zbl 0897.42010
Let $$\mathcal S$$ be the space of Schwartz functions in $$\mathbb R^n$$. Let $$w$$ be a Muckenhoupt’s $$A_\infty$$ weight on $$\mathbb R^n$$, and $$\dot F_{p,q}^{\alpha, w}$$ and $$\dot B_{p,q}^{\alpha, w}$$ be the weighted homogeneous Triebel-Lizorkin and Besov spaces, respectively. Suppose $$\mu\in \mathcal S$$ satisfies that for each $$\xi \neq 0$$ there exists a $$t>0$$ such that $$\hat\nu(t\xi)\neq 0$$. Then the authors give the following: Let $$-\infty<\alpha<\infty$$, and $$0<p, q\leq \infty$$ (in the case of $$\dot F_{p,q}^{\alpha, w}$$, $$p<\infty$$). Then there exits a constant $$C>0$$ independent of $$f\in \mathcal S'$$ such that \begin{aligned}\| f\| _{\dot F_{p,q}^{\alpha, w}}&\leq C\left\| \Biggl(\int_0^\infty | t^{-\alpha}(\nu_t\ast f)| ^q t^{-1}dt \Biggr)^{1/q}\right\| _{p,w},\\ \text{and} \| f\|_{\dot B_{p,q}^{\alpha, w}}&\leq C\Biggl(\int_0^\infty (t^{-\alpha}\| \nu_t\ast f\| _{p,w})^q t^{-1}dt\Biggr)^{1/q}, \end{aligned} where $$\nu_t(x)=t^{-n}\nu(x/t)$$. This result improves their results in Stud. Math. 119, No. 3, 219-246 (1996; Zbl 0861.42009) (the cases $$q\geq 1$$ for $$\dot F_{p,q}^{\alpha, w}$$ and $$p, q\geq 1$$ for $$\dot B_{p,q}^{\alpha, w}$$). Inhomogeneous cases are also treated.
Reviewer: K.Yabuta (Nara)

##### MSC:
 42B25 Maximal functions, Littlewood-Paley theory 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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##### References:
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