On a theorem of Bui, Paluszyński, and Taibleson (Q2714722)

One of the important tasks in the theory of function spaces is to find different equivalent norms of given spaces. In the case of the Besov \(B^s_{p,q}(\mathbb{R}^n)\) and Lizorkin-Triebel \(F^s_{p,q}(\mathbb{R}^n)\) spaces a lot of norms are described by the convolutions with the appropriate functions, and the corresponding maximal operators. In 1996, \textit{H. Q. Bui}, \textit{M. Paluszyśnki} and \textit{M. H. Taibleson} [Stud. Math. 119, No. 3, 219-246 (1996; Zbl 0861.42009)] proved the theorem that asserts when a tempered distribution belongs to \(B^s_{p,q}(\mathbb{R}^n)\) or \(F^s_{p,q}(\mathbb{R}^n)\) in terms of convolutions with quite general functions and the corresponding Peetre maximal functions. In the paper the simpler version of the proof of the above theorem is given for the nonhomogeneous Besov and Lizorkin-Triebel spaces. Some corrections of the original proof are made. Moreover, the author deals with the spaces \(F^s_{\infty,q}(\mathbb{R}^n)\), which were not regarded in the above mention paper. The analog of the Bui-Paluszyński-Taibleson theorem is proved also in this case.NEWLINENEWLINEFor the entire collection see [Zbl 0952.00006].