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  "body": "\n## Summary\n\nThe real Hardy space $H^1(\\R^n)$ is the space of integrable functions $f$ on $\\R^n$ whose Riesz transforms $R_1 f, \\dots, R_n f$ are also integrable.\nIt is a proper subspace of $L^1(\\R^n)$, and its elements have integral zero.\nIt is the right replacement for $L^1$ in harmonic analysis.\nSingular integrals such as the Hilbert and Riesz transforms map $H^1$ to $L^1$, but not $L^1$ to $L^1$.\nIts dual space is the space [[BMO|QJYTXFT8Y4]] of functions of bounded mean oscillation, and its elements are sums of simple building blocks called atoms.\nFor $n = 1$ it is the space of real parts of boundary values of functions in the holomorphic [[Hardy space|PNNNFJVF6W]] $H^1$ of the upper half-plane.\n\n## Explanations\n\n### L1 human\n\nAn integrable function can still behave badly under singular integrals, like the Hilbert transform, which can turn it into a non-integrable function.\nThe real Hardy space keeps exactly the integrable functions for which this does not happen.\nSuch functions must have some cancellation: their total integral is zero.\nA typical element on $\\R$ is $\\mathbf{1}_{[0, 1]} - \\mathbf{1}_{[-1, 0]}$, a positive bump next to a negative bump of the same size.\n",
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