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  "body": "\n## Statement\n\nLet $(X, \\mu)$ and $(Y, \\nu)$ be [[σ-finite|WEBBT18D4X]] measure spaces and let $1 \\le p_0 \u003c p_1 \\le \\infty$.\nLet $T$ map $L^{p_0}(\\mu) + L^{p_1}(\\mu)$ into the measurable functions on $Y$.\nAssume that $T$ is sublinear: for all $f, g$ and scalars $c$, almost everywhere,\n$$\\abs{T(f + g)} \\le \\abs{T f} + \\abs{T g}, \\qquad \\abs{T(c f)} = \\abs{c} \\abs{T f}.$$\nAssume that $T$ is of weak type $(p_0, p_0)$: there is $A_0$ with\n$$\\nu(\\set{y : \\abs{T f(y)} \u003e t}) \\le \\left( \\frac{A_0 \\norm{f}_{p_0}}{t} \\right)^{p_0} \\quad \\text{for all } f \\in L^{p_0}(\\mu) \\text{ and } t \u003e 0.$$\nAssume that $T$ is of weak type $(p_1, p_1)$ in the same sense with a constant $A_1$, where for $p_1 = \\infty$ this means $\\norm{T f}_\\infty \\le A_1 \\norm{f}_\\infty$.\nThen for every $p$ with $p_0 \u003c p \u003c p_1$ there is a constant $A$, depending only on $p$, $p_0$, $p_1$, $A_0$ and $A_1$, with\n$$\\norm{T f}_p \\le A \\norm{f}_p \\quad \\text{for all } f \\in L^p(\\mu).$$\n\n## Remarks\n\nThe weak type condition says that $T$ maps $L^{p_i}$ boundedly into the [[weak Lp space|H4C888M5AZ]] $L^{p_i,\\infty}$.\nThe proof writes $\\norm{T f}_p^p = p \\int_0^\\infty t^{p-1} \\nu(\\set{\\abs{T f} \u003e t}) \\, dt$ by the [[layer cake representation|H78RXJC1K1]].\nFor each $t$, it splits $f$ into the part where $\\abs{f} \u003e c t$, which lies in $L^{p_0}$, and the rest, which lies in $L^{p_1}$.\nThe two weak-type bounds and Fubini's theorem then give the bound.\n\nExample: the [[Hardy–Littlewood maximal function|VX1A56P5KQ]] operator $M$ on $\\R^n$ is of weak type $(1, 1)$, and $\\norm{M f}_\\infty \\le \\norm{f}_\\infty$.\nThe theorem gives $\\norm{M f}_p \\le A_p \\norm{f}_p$ for $1 \u003c p \u003c \\infty$.\n\nThe endpoints are not included in the conclusion.\nThe 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.\n\n## Explanations\n\n### L1 human\n\nSome important operators on functions are not bounded on $L^1$, but satisfy a weaker estimate there that controls the measure of the set where the output is large.\nThe theorem says that two such weak estimates, at two exponents, give a genuine $L^p$ bound for every exponent strictly between them.\nIt works even for operators that are not linear, such as maximal functions.\n\n### L2 human\n\nIn 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.\nMarcinkiewicz interpolation then gives all the $L^p$ bounds in between.\nCompared with the [[Riesz–Thorin theorem|AGRKRG1TMJ]], it needs less (weak type, sublinearity) and gives less (an unspecified constant).\nThere 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 $p \\le q$.\n",
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