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  "body": "\n## Summary\n\nA Carleson measure is a positive measure on the [[unit disc|8G5ABXRK85]] with a size bound near the boundary.\nIts mass in each box touching the boundary is at most a constant times the length of the box.\nCarleson showed that these are exactly the measures $\\mu$ for which the [[Hardy space|PNNNFJVF6W]] $H^p$ embeds continuously into $L^p(\\mu)$.\nThey appear in the [[corona theorem|WMGA92P2G4]], in interpolation problems and in the duality between $H^1$ and [[BMO|QJYTXFT8Y4]].\n\n## Explanations\n\n### L1 human\n\nPicture a small arc $I$ on the circle and the \"box\" in the disc that sits just inside it, of depth equal to the length of $I$.\nA measure is Carleson if each such box carries mass at most a constant times the length of $I$.\nArea measure on the disc is Carleson, since a box of side $\\ell$ has area about $\\ell^2$, which is at most a constant times $\\ell$.\nA measure that piles too much mass close to one boundary point is not Carleson.\nThe condition is exactly what is needed to integrate $\\abs{f}^p$ against $\\mu$ when $f$ is in $H^p$.\n",
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