Abstract of a paper by Sorina Barza, Viktor Kolyada, and Javier Soria
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 The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0709.0647 or http://arXiv.org/abs/0709.0647 or by email in unzipped form by transmitting an empty message with subject line uget 0709.0647 or in gzipped form by using subject line get 0709.0647 to: math@arXiv.org.
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Dale Alspach