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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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