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  "body": "\n## Statement\n\nLet $n \\ge 1$, let $1 \u003c p \\le 2$, and let $q$ be the conjugate exponent, $\\frac{1}{p} + \\frac{1}{q} = 1$.\nPut $A_p = \\left( p^{1/p} / q^{1/q} \\right)^{1/2}$.\nLet $\\hat f(\\xi) = \\int_{\\R^n} f(x) e^{-2\\pi i x \\cdot \\xi}\\, dx$ be the [[Fourier transform|DPVM34ADZV]].\nThen for every $f \\in L^1(\\R^n) \\cap L^p(\\R^n)$,\n$$\\norm{\\hat f}_{L^q(\\R^n)} \\le A_p^n\\, \\norm{f}_{L^p(\\R^n)}.$$\nThe constant $A_p^n$ is the best possible: equality holds for $f(x) = e^{-\\pi \\abs{x}^2}$.\n\n## Remarks\n\nThe inequality sharpens the [[Hausdorff–Young inequality|HXNJH6W23S]], whose constant is $1$; for $1 \u003c p \u003c 2$ one has $A_p \u003c 1$.\nK. I. Babenko proved it when $q$ is an even integer, and W. Beckner proved it for all $p$ in the range.\nBeckner's proof goes through a sharp hypercontractive estimate for the Hermite semigroup and a central limit argument.\n\nThe equality case can be checked directly with the [[Fourier transform of the Gaussian|ZNEHMR2CCP]]: for $f(x) = e^{-\\pi \\abs{x}^2}$ one has $\\hat f = f$ and $\\norm{f}_r = r^{-n/(2r)}$ for every $r \u003e 0$.\nSo $\\norm{\\hat f}_q / \\norm{f}_p = \\left( p^{1/p} / q^{1/q} \\right)^{n/2} = A_p^n$.\n\nDifferentiating the inequality in $p$ at $p = 2$, where both sides are equal by Plancherel, gives the entropic uncertainty principle.\n\n## Explanations\n\n### L1 human\n\nThe Hausdorff–Young inequality says that the Fourier transform does not increase a certain size of a function.\nThe Babenko–Beckner inequality finds the exact best factor by which it shrinks that size.\nThe bell curve is the function for which the factor is reached.\n\n### L2 human\n\nThis is a typical sharp inequality in analysis, with Gaussians as extremal functions.\nThe same happens for the logarithmic Sobolev inequality and the hypercontractivity of the Ornstein–Uhlenbeck semigroup, to which it is closely related.\nIts derivative at $p = 2$ gives the Hirschman–Beckner entropic uncertainty principle, which in turn implies the Heisenberg inequality.\n",
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