Marcinkiewicz interpolation theorem

Remarks

The weak type condition says that $T$ maps $L^{p_i}$ boundedly into the weak Lp space $L^{p_i,\infty}$. The proof writes $\norm{T f}_p^p = p \int_0^\infty t^{p-1} \nu(\set{\abs{T f} > t}) \, dt$ by the layer cake representation. For each $t$, it splits $f$ into the part where $\abs{f} > c t$, which lies in $L^{p_0}$, and the rest, which lies in $L^{p_1}$. The two weak-type bounds and Fubini's theorem then give the bound.

Example: the Hardy–Littlewood maximal function operator $M$ on $\R^n$ is of weak type $(1, 1)$, and $\norm{M f}_\infty \le \norm{f}_\infty$. The theorem gives $\norm{M f}_p \le A_p \norm{f}_p$ for $1 < p < \infty$.

The endpoints are not included in the conclusion. The maximal operator is not bounded on $L^1(\R^n)$: for $f$ the indicator function of the unit ball, $M f(x)$ is of order $\abs{x}^{-n}$ for large $\abs{x}$, which is not integrable.

Going deeper

For a student

In harmonic analysis one usually proves a weak $(1, 1)$ bound by a covering or Calderón–Zygmund argument and an $L^2$ or $L^\infty$ bound by other means. Marcinkiewicz interpolation then gives all the $L^p$ bounds in between. Compared with the Riesz–Thorin theorem, it needs less (weak type, sublinearity) and gives less (an unspecified constant). There is also an off-diagonal version, from weak type $(p_i, q_i)$ to strong type $(p, q)$, which requires $p_0 \le q_0$, $p_1 \le q_1$, $q_0 \ne q_1$, and gives the conclusion for $p \le q$.