Weak-type estimate

Summary

A weak-type estimate bounds the measure of the set where $\abs{Tf}$ is larger than $\lambda$, instead of bounding a norm of $Tf$. An operator $T$ is of weak type $(p, q)$ if this measure is at most $(C\norm{f}_p/\lambda)^q$. Equivalently, $T$ maps $L^p$ boundedly into the weak Lp space $L^{q,\infty}$. Such estimates are the natural substitute for $L^1$ bounds, which often fail, for maximal functions and singular integrals. By the Marcinkiewicz interpolation theorem, two weak-type estimates give strong $L^p$ bounds for all exponents in between.