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  "body": "\n## Summary\n\nA weak-type estimate bounds the measure of the set where $\\abs{Tf}$ is larger than $\\lambda$, instead of bounding a norm of $Tf$.\nAn operator $T$ is of weak type $(p, q)$ if this measure is at most $(C\\norm{f}_p/\\lambda)^q$.\nEquivalently, $T$ maps $L^p$ boundedly into the [[weak Lp space|H4C888M5AZ]] $L^{q,\\infty}$.\nSuch estimates are the natural substitute for $L^1$ bounds, which often fail, for maximal functions and singular integrals.\nBy the [[Marcinkiewicz interpolation theorem|R1PHMVBJ9J]], two weak-type estimates give strong $L^p$ bounds for all exponents in between.\n\n## Explanations\n\n### L1 human\n\nBounding $\\int \\abs{Tf}$ asks for control of the total size of $Tf$.\nA weak-type bound asks for less: it only controls how much room the large values of $Tf$ take up.\nFor example, the function $1/x$ on $(0, \\infty)$ has infinite integral, but the set where it exceeds $\\lambda$ is $(0, 1/\\lambda)$, of length exactly $1/\\lambda$.\nThat is the typical shape of a weak $L^1$ function, and the Hilbert transform and the maximal function produce such tails.\n",
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