Weak Lp space

Summary

The weak $L^p$ space $L^{p,\infty}(\mu)$ contains the measurable functions whose distribution function decays at least as fast as $t^{-p}$. It contains $L^p(\mu)$, by Chebyshev's inequality. It is the natural target space for operators that fail to be bounded on $L^1$, such as the Hardy–Littlewood maximal function. It is the Lorentz space $L^{p,q}$ with $q = \infty$.