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Zbl 1183.46028
Barza, Sorina; Kolyada, Viktor; Soria, Javier
Sharp constants related to the triangle inequality in Lorentz spaces.
(English)
[J] Trans. Am. Math. Soc. 361, No. 10, 5555-5574 (2009). ISSN 0002-9947; ISSN 1088-6850/e

Summary: We study the Lorentz spaces $L^{p,s}(R,\mu)$ in the range $1<p<s\le\infty$, for which the standard functional $$\|f\|_{p,s}=\left(\int^\infty_0(t^{1/p}f^*(t))^s\,\frac{dt}{t}\right)^{1/s}$$ is only a quasi-norm. We find the optimal constant in the triangle inequality for this quasi-norm, which leads us to consider the following decomposition norm: $$\|f\|_{p,s}=\inf\left\{\sum_k\|f\|_{p,s}\right\},$$ where the infimum is taken over all finite representations $f=\sum_kf_k$. We also prove that the decomposition norm and the dual norm $$\|f\|_{p,s}'=\sup\left\{\int_R fg\,d\mu:\|g\|_{p',s'}=1\right\}$$ agree for all values of $p,s>1$.
MSC 2000:
*46E30 Spaces of measurable functions
46B25 Classical Banach spaces in the general theory of normed spaces

Keywords: equivalent norms; level function; Lorentz spaces; sharp constants; quasi-norm; decomposition norm

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