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  "body": "\n## Statement\n\nLet $n \\ge 1$, let $f \\in L^1(\\R^n)$ be an integrable function, and let $\\alpha \u003e 0$.\nThen there are disjoint sets $F$ and $U$ with $\\R^n = F \\cup U$ such that:\n\n1. $\\abs{f(x)} \\le \\alpha$ for almost every $x \\in F$;\n2. $U$ is a countable union of closed cubes $Q_k$ with sides parallel to the axes and pairwise disjoint interiors;\n3. for every $k$,\n$$\\alpha \u003c \\frac{1}{\\abs{Q_k}}\\int_{Q_k} \\abs{f(x)} \\, dx \\le 2^n\\alpha,$$\nwhere $\\abs{Q_k}$ is the [[Lebesgue measure|2Z2QVWZ262]] of $Q_k$.\n\n## Remarks\n\nThe proof is a stopping-time argument on dyadic cubes.\nCut $\\R^n$ into a grid of cubes so large that the average of $\\abs{f}$ on each is at most $\\alpha$.\nDivide each cube into $2^n$ subcubes by halving its sides, keep the subcubes on which the average exceeds $\\alpha$, and continue dividing the others.\nA kept cube has a parent with average at most $\\alpha$, which gives the bound $2^n\\alpha$, and the Lebesgue differentiation theorem gives $\\abs{f} \\le \\alpha$ almost everywhere on the points never kept.\n\nClause 3 gives $\\abs{U} = \\sum_k \\abs{Q_k} \\le \\alpha^{-1}\\norm{f}_{L^1}$.\nSetting $g = f$ on $F$ and $g$ equal to the average of $f$ on each $Q_k$, and $b = f - g$, one obtains the Calderón–Zygmund decomposition $f = g + b$.\nHere $\\abs{g} \\le 2^n\\alpha$ almost everywhere, and $b$ has mean zero on each cube $Q_k$ and vanishes off $U$.\n\nExample: for $n = 1$, $f = \\mathbf{1}_{[0, 1]}$ and $\\alpha = 1/2$, one may take $U = [0, 1]$ and $F = \\R \\setminus [0, 1]$; the average of $\\abs{f}$ on $[0, 1]$ is $1$, which lies in $(1/2, 1]$.\n\n## Explanations\n\n### L1 human\n\nThe lemma splits space into a part where $f$ is small, at most $\\alpha$, and a collection of cubes where $f$ is large on average, but not too large.\nThe large part occupies little volume, at most $\\norm{f}_{L^1}/\\alpha$.\nIt is the basic tool for studying functions that are only integrable, without any better bound.\n\n### L2 human\n\nThe lemma, through the decomposition $f = g + b$, is the key step in proving that [[singular integral operators|HTMCEARGDD]] are of weak type $(1, 1)$.\nThe good part $g$ is bounded and integrable, so it lies in $L^2$, where the operator is bounded.\nThe bad part $b$ has mean zero on each cube, and the smoothness of the kernel controls the operator on it away from the cubes.\nThe same stopping-time idea gives the weak type $(1, 1)$ bound for the [[Hardy–Littlewood maximal function|VX1A56P5KQ]] and the John–Nirenberg inequality.\n",
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