This is an announcement for the paper "Sharp constants related to the triangle inequality in Lorentz spaces" by Sorina Barza, Viktor Kolyada, and Javier Soria.

Abstract: We study the Lorentz spaces $L^{p,s}(R,\mu)$ in the range $1<p<s\le \infty$, for which the standard functional $$ ||f||_{p,s}=\left(\int_0^\infty (t^{1/p}f^*(t))^s\frac{dt}{t}\right)^{1/s} $$ 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: $$ ||f||_{(p,s)}=\inf\bigg{\sum_{k}||f_k||_{p,s}\bigg}, $$ where the infimum is taken over all finite representations $f=\sum_{k}f_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 $p,s>1$.

Archive classification: math.FA math.CA

Mathematics Subject Classification: 46E30, 46B25

Remarks: 24 pages

The source file(s), Norms-Constants.tex: 47398 bytes

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