Which are the conditions for a Lorentz space $L^{p,q}$ to be o-c?

Question

Which are the conditions for a Lorentz space $L^{p,q}$ to be ord. continuous?

( A Banach function space is o-c $\equiv$ Increasing sequences of order-bounded positive functions converge in norm).

Thanks a lot for any help.

Answer 1

I think they are order continuous if $q < \infty$. If $0 \le f_1 \le f_2 \dots \le g$, then $\mu(g>t) \ge \mu(f - f_n > t) \to 0$ where $f = \sup f_n$. Since $$ \| f - f_n\|_{p,q}^q = \int_0^\infty [(f-f_n)^*(x)]^q dx^{p/q} = \int_0^\infty [\mu(f-f_n>t)]^{p/q} dt^q, $$ we see by dominated convergence that $f-f_n \to 0$ in $L^{p,q}$.

If $q = \infty$, then if the underlying measure space is $(0,1]$ or $(0,\infty)$, then define $$ f_n(x) = \cases{ x^{-1/p} & if $1/n <x \le 1$ \cr n^{1/p} & if $0 < x < 1/n$ \cr 0 & if $x \ge 1$ .\cr} $$ Then $f_n(x) \nearrow f(x) = x^{-1/p}I_{0<x<1}$, but $f_n$ does not converge to $f$ in the $L^{p,\infty}$ norm. A similar construction should work for any measure space for which $L^{p,\infty}$ is not finite dimensional.