The characterization of the Triebel-Lizorkin spaces for \(p=\infty\)
Let \(-\infty<\alpha<\infty\), \(0<q\leq\infty\), and \(w\) be a non-trivial weight function in the Muckenhoupt class \(A_\infty\) on \(\mathbb R^n\). Choose a Schwartz function \(\psi\) such that \(\roman{supp} \hat\psi\subset \{1/2\leq|\xi|\leq 2\}\) and \(\sum_{j=-\infty}^{\infty}\hat\psi(2^{-j}\xi)=1\) for \(\xi\neq 0\). For each \(j\in \mathbb Z\), let \(\psi_j(x)=\psi_{2^{-j}}(x) =2^{jn}\psi(2^{j}x)\). One defines the weighted version of the Triebel-Lizorkin space \(\dot F_{\infty,q}^\alpha\) introduced by \textit{M. Frazier} and \textit{B. Jawerth} [J. Funct. Anal. 93, No. 1, 34-170 (1990; Zbl 0716.46031)] as follows: For \(f\in \mathcal S'\), one lets \[ \|f\|_{\dot F_{\infty,q}^{\alpha,w}}= \sup_{Q}\Biggl\{w(Q)^{-1}\int_Q\sum_{j=-\log_2 \ell(Q)}^{\infty}(2^{j\alpha} |\psi_j*f(x)|)^qw(x) dx) \Biggr\}^{1/q}, \] with the interpretation that when \(q=\infty\), \[ \|f\|_{\dot F_{\infty,\infty}^{\alpha,w}}= \sup_{Q}w(Q)^{-1}\int_Q\sum_{j=-\log_2 \ell(Q)}^{\infty}2^{j\alpha} |\psi_j*f(x)|w(x) dx, \] where the supremum is taken over all dyadic cubes \(Q\), and \(\ell(Q)\) denotes the side-length of the cube \(Q\). The authors define \(\dot F_{\infty,q}^{\alpha,w}=\{f\in \mathcal S'/\{\text{polynomials}\};\;\|f\|_{\dot F_{\infty,q}^{\alpha,w}}<\infty \}\). When \(q=\infty\), this definition is different from those by Frazier and Jawerth where they set \(\dot F_{\infty,\infty}^\alpha = \dot B_{\infty,\infty}^\alpha\)(Besov space) by definition. The authors give a characterization of these spaces by means of a ``generalized'' Littlewood-Paley function which is based on a kernel satisfying ``minimal'' moments and Tauberian conditions. And they give that \(\dot F_{\infty,\infty}^{\alpha,w}= \dot B_{\infty,\infty}^\alpha =\dot F_{\infty,\infty}^\alpha \). For the case \(\dot B_{p,q}^{\alpha,w}\) \((0<p, q\leq \infty)\) and \(\dot F_{p,q}^{\alpha,w}\) \((0<p<\infty, 0<q\leq \infty)\), similar characterizations are given by \textit{H.-Q. Bui, M. Paluszyński} and \textit{M. H. Taibleson} [Stud. Math. 119, No. 3, 219-246 (1996; Zbl 0861.42009); J. Fourier Anal. Appl. 3, Spec. Iss., 837-846 (1997; Zbl 0897.42010)].
scientific article; zbMATH DE number 1539364
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