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  "body": "\n## Statement\n\nLet $1 \u003c p \u003c \\infty$, and let $H$ be the [[Hilbert transform|NSCEBXM7ZM]] on $\\R$.\nThen there is a constant $C_p$, depending only on $p$, such that\n$$\\norm{Hf}_{L^p(\\R)} \\le C_p \\norm{f}_{L^p(\\R)} \\quad \\text{for all } f \\in L^p(\\R).$$\nThe same holds for the [[conjugate function|A1H48DDW49]] on the circle: $\\norm{\\tilde f}_{L^p} \\le C_p \\norm{f}_{L^p}$ for all $f \\in L^p(\\R/2\\pi\\Z)$.\n\n## Remarks\n\nMarcel Riesz's proof for $p = 2k$ an even integer uses complex analysis.\nIf $f$ is real and $F = f + iHf$ extends holomorphically to the upper half-plane, then $\\int F^{2k} = 0$, and taking real parts bounds $\\norm{Hf}_{2k}$ by $\\norm{f}_{2k}$.\nInterpolation then gives $2 \\le p \u003c \\infty$, and duality gives $1 \u003c p \\le 2$, since $H$ is skew-adjoint.\nA real-variable proof combines the bound on $L^2$ with a weak type $(1, 1)$ bound, through the Marcinkiewicz interpolation theorem.\nThe best constant is $C_p = \\tan\\frac{\\pi}{2p}$ for $1 \u003c p \\le 2$ and $C_p = \\cot\\frac{\\pi}{2p}$ for $2 \\le p \u003c \\infty$, a result of Pichorides.\n\nExample: for $p = 2$, $H$ is an isometry, since its multiplier $-i\\sgn\\xi$ has modulus $1$, and $C_2 = \\tan(\\pi/4) = 1$.\n\nThe theorem fails for $p = 1$ and $p = \\infty$.\nFor $f = \\mathbf{1}_{[0, 1]}$, one has $Hf(x) = \\frac{1}{\\pi} \\log\\abs{\\frac{x}{x - 1}}$.\nThis is unbounded near $x = 0$ and $x = 1$, and it behaves like $\\frac{1}{\\pi x}$ at infinity, so it is not integrable.\n\n## Explanations\n\n### L1 human\n\nThe Hilbert transform turns each cosine wave into a sine wave, and so turns the real part of a holomorphic function into its imaginary part.\nThis theorem says it does not make functions much bigger, measured in any $L^p$ norm with $1 \u003c p \u003c \\infty$.\nAt the two ends, $p = 1$ and $p = \\infty$, a jump in the function creates a logarithmic blow-up, and the bound fails.\n\n### L2 human\n\nThe theorem is equivalent to the convergence of Fourier series in $L^p(\\R/2\\pi\\Z)$ for $1 \u003c p \u003c \\infty$.\nIndeed, the partial sum operators are combinations of modulated conjugate function operators, so the bound for $\\tilde f$ gives uniform bounds for the partial sums.\nIt is the first case of the theory of singular integrals, which extends it to the [[Riesz transforms|FRNDFJ9WQ5]] and to [[Calderón–Zygmund operators|3SCXCW4AH3]].\n",
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