Normability of gamma spaces. (Q2798069)

The author presents a complete characterization of parameters \(p\in(0,1)\) and a weight function \(w\) for which the classical Lorentz space \(\Gamma^p_w\) is normable. In fact, the result follows from the paper of \textit{A. Kamińska} and \textit{L. Maligranda} [Isr. J. Math. 140, 285--318 (2004; Zbl 1068.46019)], however the author presents here an alternative more direct proof. His method is based on suitable discretizations and weighted norm inequalities. Recall that \(\Gamma^p_w\) is a set of all \(\mu\)-measurable functions \(f\) on a measure space \((\mathcal R,\mu)\) (whose measure \(\mu\) is non-atomic, \(\sigma\)-finite and satisfies \(\mu(\mathcal R)=\infty\)) such that \(\|f\|_{\Gamma^p_w}=\big(\int_0^\infty f^{**}(t)^pw(t)\,\text{d}t\big)^{1/p}<\infty\). Here \(f^{**}(t)=(1/t)\int_0^tf^{*}(s)\text{d}s\), where \(f^*\) denotes the non-increasing rearrangement of the function \(f\) with respect to the measure \(\mu\), that is, \(f^*(t)=\inf\{s; \mu(\{|f|>s\})\leq t\}\).NEWLINENEWLINE The author proves that if \(p\in(0,1)\) and \(w\) is a weight (i.e. a positive measurable function) on \((0,\infty)\), then the functional \(\|f\|_{\Gamma^p_w}\) is equivalent to a norm on \(\Gamma^p_w\) if and only if both \(w(s)\) and \(w(s)s^{-p}\) are integrable functions on \((0,\infty)\). Moreover, it happens if and only if \(\Gamma^p_w=L^1+L^\infty\) (in the sense of equivalent norms).