Lorentz space

Remarks

Example: for $q = p$, $\norm{f}_{p,p}^p = \int_0^\infty f^*(t)^p \, dt = \int_X \abs{f}^p \, d\mu$, because $f^*$ and $\abs{f}$ have the same distribution function. So $L^{p,p}(\mu) = L^p(\mu)$ with equal norms.

Non-example: on $(0, \infty)$ with Lebesgue measure, $f(x) = x^{-1/p}$ is not in $L^{p,q}$ for $q < \infty$. Here $f^* = f$, so $t^{1/p} f^*(t) = 1$ and the integral $\int_0^\infty dt / t$ diverges. It does lie in $L^{p,\infty}$, with $\norm{f}_{p,\infty} = 1$.

The space $L^{p,\infty}$ is the weak $L^p$ space, since $\sup_t t^{1/p} f^*(t) = \sup_s s \, d_f(s)^{1/p}$. For fixed $p$, the spaces increase with $q$: $L^{p,q_1} \subseteq L^{p,q_2}$ when $q_1 < q_2$. In general $\norm{\cdot}_{p,q}$ is only a quasinorm.