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Normability of gamma spaces. (English) Zbl 1374.46029
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 A. Kamińska and 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, $$f^*(t)=\inf\{s; \mu(\{|f|>s\})\leq t\}$$.
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).
Reviewer: Petr Gurka (Praha)

##### MSC:
 4.6e+31 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
##### Keywords:
Lorentz space; weight; normability
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