The details about Lorentz space are given in Wikipedia page. It is easy to prove that $L^{p,q}$ is contained in $L^{p,r}$ whenever $q<r$. I'm trying to construct an example which shows that the inclusion is proper. I tried examples like $f=\frac{1}{x^{1/p}}\chi_{(0,1)}$. But they are not working. Can I get a precise example for the same?
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