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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $f$ be a function of [[bounded mean oscillation|QJYTXFT8Y4]] on $\\R^n$.\nFor a cube $Q$ with sides parallel to the axes, let $\\abs{Q}$ be its [[Lebesgue measure|2Z2QVWZ262]] and $f_Q = \\frac{1}{\\abs{Q}} \\int_Q f$.\nFor $\\delta \u003e 0$ let\n$$\\omega_f(\\delta) = \\sup_{\\abs{Q} \\le \\delta} \\frac{1}{\\abs{Q}} \\int_Q \\abs{f(x) - f_Q} \\, dx.$$\nThe function $f$ has *vanishing mean oscillation* if $\\omega_f(\\delta) \\to 0$ as $\\delta \\to 0$.\nThe space of such functions is denoted $\\mathrm{VMO}(\\R^n)$.\n\n## Remarks\n\nVMO is a closed subspace of BMO, and Sarason showed that it is the closure in BMO of the uniformly continuous functions in BMO.\nSome authors use a smaller space on $\\R^n$, also called VMO: the closure in BMO of the continuous functions with compact support.\nThe dual space of this smaller space is the real Hardy space $H^1$.\n\nExample: every uniformly continuous $f$ in BMO is in VMO.\nIf $\\abs{f(x) - f(y)} \\le \\varepsilon$ whenever $\\abs{x - y} \\le \\eta$, then on a cube $Q$ of diameter at most $\\eta$ one has $\\abs{f(x) - f_Q} \\le \\varepsilon$ for $x \\in Q$, so $\\omega_f(\\delta) \\le \\varepsilon$ for $\\delta$ small.\n\nExample with a singularity: if $n \\ge 2$ and $\\nabla f \\in L^n(\\R^n)$, then the [[Poincaré inequality|JS2CCR3RNS]] on cubes gives $\\frac{1}{\\abs{Q}} \\int_Q \\abs{f - f_Q} \\le C \\left( \\int_Q \\abs{\\nabla f}^n \\right)^{1/n}$, which tends to $0$ uniformly as $\\abs{Q} \\to 0$.\nSo such $f$ are in VMO, although they can be unbounded.\n\nNon-example: $f(x) = \\log \\abs{x}$ on $\\R$ is in BMO but not in VMO.\nFor $r \u003e 0$, $f(r x) = f(x) + \\log r$, so the mean oscillation of $f$ on $[-r, r]$ equals its mean oscillation on $[-1, 1]$, which is a fixed positive number.\nHence $\\omega_f(\\delta)$ does not tend to $0$.\n",
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