Example for proper inclusion in Lorentz space

Question

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?

Answer 1

Try $f=\sum_{j=1}^\infty j^{-q}2^{-j/p}\chi_{(0,2^{-j})}$, or equivalently $f(x)=x^{-1/p}|\log x|^{-q}\chi_{(0,\frac12)}$.