Parabolic maximal functions and potentials of distributions in H p
In their joint paper [Harmonic analysis, Conf. in Honor A. Zygmund, Vol. 2, 394-415 (1983; Zbl 0504.35040)], \textit{A. B. Gatto, J. R. Jiménez} and \textit{C. Segovia} have identified the space of solutions of the equation \(\Delta mF=f\) for f in the Hardy space H \(p({\mathbb{R}}\) n) as a topological vector space \({\mathcal H} p_{q,m}\) (provided \(n(2m+n/q)^{- 1}<p\leq 1)\), which under the operator \(\Delta\) m is topologically isomorphic to H p. Moreover they have given an atomic decomposition of these spaces \({\mathcal H} p_{q,m}.\) In the present paper the author shows that the definition and atomic decomposition of \({\mathcal H} p_{q,m}\) extend to the case of non-isotropic dilations on \({\mathbb{R}}^ n,\) and that the analogue of the result above for \(\Delta\) holds true for more general elliptic operators with symbols of the form \(\xi_ 1^{2k_ 1}+...+\xi_ n^{2k_ n},\quad k_ 1,...,k_ n\in {\mathbb{N}}.\)
scientific article; zbMATH DE number 3873948