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  "body": "\n## Definition\n\nLet $\\mathbb{D}$ be the open [[unit disc|8G5ABXRK85]] and $\\mathbb{T}$ the unit circle.\nFor an arc $I \\subseteq \\mathbb{T}$ of length $\\ell(I)$, the *Carleson box* over $I$ is\n$$S(I) = \\set{r \\zeta : \\zeta \\in I, \\ 1 - \\ell(I) \\le r \u003c 1}.$$\nA positive [[Borel measure|DYR1QFNGNJ]] $\\mu$ on $\\mathbb{D}$ is a *Carleson measure* if there is a constant $C$ such that\n$$\\mu(S(I)) \\le C \\, \\ell(I) \\quad \\text{for every arc } I \\subseteq \\mathbb{T}.$$\n\n## Remarks\n\nCarleson's embedding theorem: for $0 \u003c p \u003c \\infty$, $\\mu$ is a Carleson measure if and only if there is $C_p$ with $\\int_{\\mathbb{D}} \\abs{f}^p \\, d\\mu \\le C_p \\norm{f}_{H^p}^p$ for all $f$ in the [[Hardy space|PNNNFJVF6W]] $H^p$.\n\nExample: area measure $A$ on $\\mathbb{D}$.\nIn polar coordinates, $A(S(I)) \\le \\ell(I) \\int_{\\max(0, 1 - \\ell(I))}^1 r \\, dr \\le \\ell(I) \\min(\\ell(I), 1) \\le \\ell(I)$.\nSo $A$ is a Carleson measure with $C = 1$.\n\nNon-example: let $z_n = 1 - 2^{-n}$ and $\\mu = \\sum_{n \\ge 1} n^{-2} \\delta_{z_n}$, a finite measure.\nLet $I_k$ be the arc of length $2^{-k}$ centred at $1$.\nThen $S(I_k)$ contains $z_n$ for all $n \\ge k$, so $\\mu(S(I_k)) \\ge \\sum_{n \\ge k} n^{-2} \u003e 1/k$.\nThe ratio $\\mu(S(I_k)) / \\ell(I_k) \u003e 2^k / k$ is unbounded, so $\\mu$ is not a Carleson measure.\n",
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