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  "body": "\n## Summary\n\nThe Riesz transforms $R_1, \\ldots, R_n$ are the $n$ [[singular integral operators|HTMCEARGDD]] on $\\R^n$ with Fourier multipliers $-i\\xi_j/\\abs{\\xi}$.\nThey are the higher-dimensional analogue of the [[Hilbert transform|NSCEBXM7ZM]], which is the case $n = 1$.\nThey express second derivatives through the Laplacian, by $\\partial_j\\partial_k = -R_j R_k \\Delta$, so their $L^p$ bounds give $L^p$ estimates for elliptic equations.\nThey are bounded on $L^p(\\R^n)$ for $1 \u003c p \u003c \\infty$ and commute with translations and dilations.\n\n## Explanations\n\n### L1 human\n\nThe Hilbert transform on the line keeps the size of each frequency and turns its phase by a quarter turn.\nIn several variables there is one such operator for each coordinate direction.\nThe $j$-th Riesz transform weights the frequency $\\xi$ by the $j$-th component of its direction $\\xi/\\abs{\\xi}$, and turns the phase by a quarter turn.\nTogether they turn a function into a vector field, much as the gradient does, but without changing the size of frequencies.\nFor example, the sum $R_1^2 + \\cdots + R_n^2$ is minus the identity, because the squared components of a unit vector add up to $1$.\n",
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