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  "body": "\n## Statement\n\nLet $n \\ge 1$.\nThere are constants $c_1, c_2 \u003e 0$, depending only on $n$, with the following property.\nLet $f$ be a function of [[bounded mean oscillation|QJYTXFT8Y4]] on $\\R^n$ that is not constant, with norm $\\norm{f}_{\\mathrm{BMO}}$.\nLet $Q \\subseteq \\R^n$ be a cube with sides parallel to the axes, let $\\abs{E}$ denote [[Lebesgue measure|2Z2QVWZ262]], and let $f_Q = \\frac{1}{\\abs{Q}}\\int_Q f$.\nThen for every $\\lambda \u003e 0$,\n$$\\abs{\\set{x \\in Q : \\abs{f(x) - f_Q} \u003e \\lambda}} \\le c_1 \\abs{Q} \\exp\\left( - \\frac{c_2 \\lambda}{\\norm{f}_{\\mathrm{BMO}}} \\right).$$\n\n## Remarks\n\nThe proof applies the [[Calderón–Zygmund lemma|K1YKXVPVDG]] to $f - f_Q$ on the cube $Q$, at a height of about $2\\norm{f}_{\\mathrm{BMO}}$.\nOn the selected subcubes $Q_k$ the mean $f_{Q_k}$ differs from $f_Q$ by at most a constant times $\\norm{f}_{\\mathrm{BMO}}$, and their total measure is at most half of $\\abs{Q}$.\nRepeating the argument inside each $Q_k$ shows that the set where $\\abs{f - f_Q}$ exceeds $C m \\norm{f}_{\\mathrm{BMO}}$ has measure at most $2^{-m}\\abs{Q}$, which is the exponential decay.\n\nConsequences: for $c \u003c c_2$ the function $\\exp(c \\abs{f - f_Q}/\\norm{f}_{\\mathrm{BMO}})$ has average over $Q$ bounded by a constant depending on $n$ and $c$.\nAlso, for every $1 \\le p \u003c \\infty$, the quantity $\\sup_Q \\left( \\frac{1}{\\abs{Q}} \\int_Q \\abs{f - f_Q}^p \\right)^{1/p}$ is a norm on BMO equivalent to $\\norm{f}_{\\mathrm{BMO}}$.\n\nExample: $f(x) = \\log \\abs{x}$ on $\\R$ and $Q = [-1, 1]$, where $f_Q = -1$.\nThe set $\\set{\\abs{f + 1} \u003e \\lambda}$ contains $\\set{\\abs{x} \u003c e^{-1 - \\lambda}}$, of measure $2 e^{-1 - \\lambda}$.\nSo the exponential decay in $\\lambda$ cannot be improved to a faster decay for all BMO functions.\n\n## Explanations\n\n### L1 human\n\nA BMO function can be unbounded, but only in a mild way.\nThe inequality says that on any cube, the set where $f$ is far from its average is exponentially small in the distance.\nSo BMO functions behave like logarithms: they can blow up, but only very slowly.\n\n### L2 human\n\nThe inequality shows that BMO functions are locally integrable to every power, and even exponentially integrable after subtracting means.\nIt is used to show that $\\log w$ is in BMO for every Muckenhoupt weight $w$, and conversely that $e^{\\delta b}$ is a Muckenhoupt weight for $b \\in$ BMO and $\\delta$ small.\nIt enters the proof of Fefferman's duality theorem and of the boundedness of commutators with BMO functions.\n",
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