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  "body": "\n## Summary\n\nA locally integrable function has bounded mean oscillation if its average deviation from its own mean, over every cube, is bounded by one constant.\nThe space BMO of such functions contains $L^\\infty$ and also some unbounded functions such as $\\log \\abs{x}$.\nIt is the dual of the real Hardy space $H^1$, by Fefferman's theorem.\nIt is the natural replacement for $L^\\infty$ in the theory of singular integrals.\n\n## Explanations\n\n### L1 human\n\nTake any cube, compute the average of $f$ on it, and measure how far $f$ is from that average on the cube, on average.\nIf this number stays below a fixed bound for all cubes, large and small, then $f$ is in BMO.\nA bounded function is always in BMO, because it can never be far from its averages.\nThe function $\\log \\abs{x}$ is unbounded, but rescaling only adds a constant to it, so its oscillation looks the same at every scale and stays bounded.\nThe function $f(x) = x$ on $\\R$ is not in BMO: on $[0, L]$ its mean deviation is $L/4$, which grows without bound.\n",
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