Definition of the space \(H^ 1\) with the help of the maximal function (Q1070176)

The paper deals with the problem of \textit{G. Weiss} [Proc. Symp. Pure Math. 35, 189-200 (1979; Zbl 0417.30025)] concerning maximal functions. Let \(H_{\phi}\) be the space of integrable functions, for which \(M_{\phi}f\) is integrable (here \(M_{\phi}f\) is the maximal function). It is proved that for even and monotonic (when \(x>0)\) functions \(\phi\), either \(H_{\phi}\) contains \(H^ 1\), or \(H_{\phi}\) is trivial. Some of the properties of the spaces \(H_{\phi}\) are discussed as well.