Sharp constants related to the triangle inequality in Lorentz spaces

Barza, Sorina; Kolyada, Viktor; Soria, Javier

doi:10.1090/S0002-9947-09-04739-4

Sharp constants related to the triangle inequality in Lorentz spaces
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by Sorina Barza, Viktor Kolyada and Javier Soria PDF

Trans. Amer. Math. Soc. 361 (2009), 5555-5574 Request permission

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 of $p,s>1$.

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