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  "body": "\n## Statement\n\nLet $n \\ge 1$, let $H^1(\\R^n)$ be the [[real Hardy space|75WGJ014GR]], and let $\\mathrm{BMO}(\\R^n)$ be the space of functions of [[bounded mean oscillation|QJYTXFT8Y4]], modulo constants.\nLet $H^1_0$ be the dense subspace of $H^1(\\R^n)$ of finite linear combinations of atoms.\n\n1. For every $b \\in \\mathrm{BMO}(\\R^n)$ the linear functional $\\ell_b(f) = \\int_{\\R^n} f b \\, dx$ on $H^1_0$ extends to a bounded linear functional on $H^1(\\R^n)$, with $\\norm{\\ell_b} \\le C \\norm{b}_{\\mathrm{BMO}}$.\n2. Every bounded linear functional $\\ell$ on $H^1(\\R^n)$ is of the form $\\ell_b$ for a function $b \\in \\mathrm{BMO}(\\R^n)$, unique up to an additive constant, with $\\norm{b}_{\\mathrm{BMO}} \\le C \\norm{\\ell}$.\n\nHere $C$ depends only on $n$.\nSo the [[continuous dual space|4A82S5FVRT]] of $H^1(\\R^n)$ is $\\mathrm{BMO}(\\R^n)$.\n\n## Remarks\n\nThe theorem is due to Charles Fefferman, and a full account appeared in the work of Fefferman and Stein on $H^p$ spaces.\n\nWith the [[atomic decomposition|AQ15060AAA]], part 1 is short.\nFor an atom $a$ on a ball $B$, the cancellation $\\int a = 0$ gives $\\abs{\\int a b} = \\abs{\\int_B a (b - b_B)} \\le \\frac{1}{\\abs{B}} \\int_B \\abs{b - b_B} \\le C \\norm{b}_{\\mathrm{BMO}}$, and the bound passes to sums of atoms.\nFor part 2, the definition of $H^1$ through the [[Riesz transforms|FRNDFJ9WQ5]] embeds $H^1$ isometrically into a closed subspace of $L^1 \\times \\cdots \\times L^1$ ($n + 1$ factors).\nBy the Hahn–Banach theorem, $\\ell$ is given by bounded functions $b_0, \\dots, b_n$, so $b = b_0 - \\sum_j R_j b_j$, and each $R_j$ maps $L^\\infty$ into BMO.\n\nThe pairing $\\int f b$ need not converge absolutely for general $f \\in H^1$ and $b \\in \\mathrm{BMO}$, which is why it is first defined on $H^1_0$.\nAn analogous duality on the circle identifies the dual of the real $H^1$ of the circle with BMO of the circle.\n\n## Explanations\n\n### L1 human\n\nEvery continuous linear way of measuring functions in the Hardy space $H^1$ is given by integrating against a function of bounded mean oscillation, and every BMO function gives one.\nThis is like the familiar fact that the dual of $L^1$ is $L^\\infty$, with $L^1$ replaced by the smaller space $H^1$ and $L^\\infty$ by the larger space BMO.\n\n### L2 human\n\nThe theorem explains why BMO appears so often as the endpoint of $L^p$ theory at $p = \\infty$.\nSince Calderón–Zygmund operators map $H^1$ into $L^1$, their adjoints map $L^\\infty$ into BMO by duality.\nIt is also the basis for interpolation between $H^1$ and $L^p$, or between $L^p$ and BMO.\nCombined with the Carleson measure characterization of BMO, it links the real-variable theory with harmonic extensions to the half-space.\n",
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