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  "body": "\n## Statement\n\nLet $n \\ge 1$.\nFor a ball $B$, let $\\abs{B}$ be its [[Lebesgue measure|2Z2QVWZ262]].\nAn *atom* is a measurable function $a$ on $\\R^n$ with a ball $B$ such that $a = 0$ outside $B$, $\\abs{a} \\le 1/\\abs{B}$, and $\\int_B a \\, dx = 0$.\nLet $H^1(\\R^n)$ be the [[real Hardy space|75WGJ014GR]].\nThen a measurable $f$ lies in $H^1(\\R^n)$ exactly when there are atoms $a_k$ and complex numbers $\\lambda_k$ with\n$$\\sum_{k} \\abs{\\lambda_k} \u003c \\infty \\quad \\text{and} \\quad f = \\sum_k \\lambda_k a_k,$$\nthe series converging in $L^1(\\R^n)$.\nMoreover there are constants $0 \u003c c \\le C$, depending only on $n$, such that\n$$c \\norm{f}_{H^1} \\le \\inf \\sum_k \\abs{\\lambda_k} \\le C \\norm{f}_{H^1},$$\nwhere the infimum is over all such representations of $f$.\n\n## Remarks\n\nEach atom has $\\norm{a}_{L^1} \\le 1$, and a computation with the Riesz transforms shows $\\norm{a}_{H^1} \\le C$; this gives the easy direction.\nThe hard direction starts from the maximal function characterization of $H^1$.\nFor each $k \\in \\Z$, the level set $\\set{f^* \u003e 2^k}$ of a suitable maximal function $f^*$ is decomposed into [[Whitney cubes|FBERT0KQXN]].\nA Calderón–Zygmund decomposition at height $2^k$ produces functions supported on these cubes.\nThe differences between consecutive levels split into pieces that live on slightly enlarged Whitney cubes $Q$, have integral zero and are bounded by $C 2^k$.\nEach such piece is a multiple $\\lambda$ of an atom with $\\abs{\\lambda} \\le C 2^k \\abs{Q}$, and summing over $k$ gives $\\sum \\abs{\\lambda} \\le C \\sum_k 2^k \\abs{\\set{f^* \u003e 2^k}} \\le C \\int f^* \\le C \\norm{f}_{H^1}$.\n\nThe theorem is due to Coifman for $n = 1$ and to Latter in all dimensions.\nIt reduces many questions about $H^1$ to atoms.\nFor example, let $T$ be a Calderón–Zygmund operator and $a$ an atom on a ball $B$.\nThen $Ta$ is controlled on the double ball $2B$ by the $L^2$ bound of $T$.\nOff $2B$ it is controlled by the cancellation of $a$ and the smoothness of the kernel.\nThis gives $\\norm{Ta}_{L^1} \\le C$ for all atoms, and hence $T$ is bounded from $H^1$ to $L^1$.\nCare is needed in such arguments, since for a general operator a uniform bound on atoms does not by itself imply boundedness on all of $H^1$.\nFor Calderón–Zygmund operators the passage from atoms to $H^1$ can be justified.\n\n## Explanations\n\n### L1 human\n\nEvery function in the Hardy space $H^1$ can be built from simple pieces, called atoms.\nAn atom lives on a ball, is not too large there, and has integral zero.\nThe theorem says that $H^1$ consists exactly of absolutely summable combinations of atoms.\nSo, to show that an operator behaves well on $H^1$, it is often enough to check it on a single atom.\n\n### L2 human\n\nThe atomic decomposition is the most used description of $H^1(\\R^n)$.\nIt gives a short proof of the duality between $H^1$ and BMO: pairing an atom $a$ on a ball $B$ with a BMO function $b$ gives $\\abs{\\int a b} = \\abs{\\int_B a (b - b_B)} \\le \\frac{1}{\\abs{B}}\\int_B \\abs{b - b_B}$.\nIt extends to the spaces $H^p(\\R^n)$ for $0 \u003c p \u003c 1$, where atoms must have vanishing moments up to an order depending on $p$.\nSimilar decompositions exist on spaces of homogeneous type, where the Fourier transform is not available.\n",
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