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A note on the Besov-Lipschitz and Triebel-Lizorkin spaces. (English) Zbl 0849.46021

Marcantognini, S. A. M. (ed.) et al., Harmonic analysis and operator theory. A conference in honor of Mischa Cotlar, January 3-8, 1994, Caracas, Venezuela. Proceedings. Providence, RI: American Mathematical Society. Contemp. Math. 189, 95-101 (1995).
The paper gives some inequalities concerning the Besov-Lipschitz, and Triebel-Lizorkin spaces norms from above and from below. More precisely, let \(\varphi (x)\in S\) be such that \(\text{supp } \widehat {\varphi} \subset \{{1 \over 2}\leq |\xi |\leq 2\}\), \(|\widehat {\varphi} (\xi) |\geq c> 0\), for \({3\over 5}\leq |\xi|\leq {5\over 3}\), \(\varphi_t (x)= t^{-n} \varphi ({x\over t})\), \(t> 0\). For \(\alpha\in \mathbb{R}\), \(1\leq p,q\leq \infty\) define \[ |f|_{\dot F_p^{\alpha,q}}= \Biggl|\Biggl( \int_0^\infty (t^{- \alpha} |\varphi_t * f(x) |)^q {dt \over t} \biggr)^{1 \over q} \Biggr|_p, \qquad |f|_{\dot B_p^{\alpha, q}}= \Biggl( \int_0^\infty (t^{- \alpha} |\varphi_t * f|_p)^q {dt \over t} \Biggr)^{1\over q}. \] Let \(\mu\), \(\nu\) be two finite measures. Suppose that \(\mu\) is either compactly supported, or absolutely continuous with density in \(S\) (in symbols \(\mu\in S\)), and satisfies the standard Tauberian condition: \(\forall \xi\in \mathbb{R}^n\), \(\xi\neq 0\), \(\exists t>0\) such that \(\widehat {u} (t\xi) \neq 0\). Suppose that \(\nu\in S\) and has \(m_0\) vanishing moments \(\int x^\gamma d\nu=0\), \(\forall \gamma\), \(|\gamma |\leq m_0\). The two given theorems say, respectively, that \[ \begin{aligned} {1\over c} \Biggl( \int^\infty_0 (s^{-\alpha} |\nu_s * f|_p )^q {ds\over s} \Biggr)^{1 \over q} &\leq |f|_{\dot B_p^{\alpha, q}} \leq C\Biggl( \int^\infty_0 (s^{- \alpha} |\mu_s *f |_p)^q {ds \over s} \Biggr)^{1\over q},\\ {1\over c} \Biggl|\Biggl( \int^\infty_0 (s^{- \alpha} |\nu_s * f|)^q {ds \over s} \Biggr)^{1\over q} \Biggr|_p &\leq |f|_{\dot F_p^{\alpha, q}} \leq C\Biggl|\Biggl( \int_0^\infty (s^{-\alpha} |\mu_s * f(x) |^q {ds \over s} \Biggr)^{1 \over q} \Biggr|_p. \end{aligned} \] When \(\mu= \nu\), then the theorems provide new characterizations of the Besov-Lipschitz spaces and the Triebel-Lizorkin spaces.
For the entire collection see [Zbl 0826.00021].

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
42B25 Maximal functions, Littlewood-Paley theory
26B35 Special properties of functions of several variables, Hölder conditions, etc.
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