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  "body": "\n## Statement\n\nLet $\\mathbb{T} = \\R / 2\\pi\\Z$, identified with $[-\\pi, \\pi)$ with [[Lebesgue measure|2Z2QVWZ262]] $\\abs{\\cdot}$.\nThere is an absolute constant $C$ with the following property.\nFor every integrable function $f$ on $\\mathbb{T}$, with [[conjugate function|A1H48DDW49]] $\\tilde f$, and every $\\lambda \u003e 0$,\n$$\\abs{\\set{\\theta \\in \\mathbb{T} : \\abs{\\tilde f(\\theta)} \u003e \\lambda}} \\le \\frac{C}{\\lambda} \\int_{\\mathbb{T}} \\abs{f(\\theta)} \\, d\\theta.$$\nThat is, $f \\mapsto \\tilde f$ is of weak type $(1, 1)$.\n\n## Remarks\n\nA real-variable proof uses the [[Calderón–Zygmund lemma|K1YKXVPVDG]] at height $\\lambda$ to write $f = g + b$.\nThe good part $g$ is bounded by $2\\lambda$ and is handled by the $L^2$ bound and Chebyshev's inequality.\nThe bad part $b$ has mean zero on each interval $Q_k$.\nSo the smoothness of the kernel $\\cot(t/2)$ away from $0$ makes $\\tilde b$ small off the doubled intervals.\nThese have total length at most $C\\lambda^{-1}\\norm{f}_1$.\n\nThe strong bound $\\norm{\\tilde f}_{L^1} \\le C\\norm{f}_{L^1}$ fails.\nFor example, if $f$ is the Poisson kernel $P_r$ and $r \\to 1^-$, then $\\norm{P_r}_{L^1}$ stays bounded while $\\tilde P_r$, the conjugate Poisson kernel, has $L^1$ norm comparable to $\\log\\frac{1}{1 - r}$.\n\nSo $\\tilde f$ is finite almost everywhere for $f \\in L^1$, and it lies in weak $L^1$, but it need not be integrable.\n\n## Explanations\n\n### L1 human\n\nFor an integrable function, the conjugate function can fail to be integrable, so one cannot bound its size in the $L^1$ norm.\nKolmogorov's estimate says that it is still not large on a big set: the set where it exceeds $\\lambda$ has measure at most a constant times $\\norm{f}_1/\\lambda$.\nThis is the same bound that Chebyshev's inequality would give if $\\tilde f$ were integrable.\n\n### L2 human\n\nTogether with the $L^2$ bound from Parseval's identity, the Marcinkiewicz interpolation theorem turns this estimate into the $L^p$ bounds of the [[M. Riesz theorem|PJFH51Z48M]] for $1 \u003c p \\le 2$, and duality covers $2 \\le p \u003c \\infty$.\nIt is the model for the weak type $(1, 1)$ bounds of all [[Calderón–Zygmund operators|3SCXCW4AH3]].\n",
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