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  "body": "\n## Definition\n\nLet $n \\ge 1$ and $j \\in \\set{1, \\ldots, n}$.\nLet $c_n = \\Gamma\\big(\\frac{n+1}{2}\\big) / \\pi^{(n+1)/2}$, where $\\Gamma$ is the gamma function.\nFor a [[Schwartz function|CG9JEATW3Q]] $f$ on $\\R^n$, the *$j$-th Riesz transform* of $f$ is the [[principal value|0SYR10HD4R]]\n$$R_j f(x) = c_n \\lim_{\\varepsilon \\to 0^+} \\int_{\\abs{y} \u003e \\varepsilon} \\frac{y_j}{\\abs{y}^{n+1}} f(x - y) \\, dy.$$\n\n## Remarks\n\nWith $\\hat f(\\xi) = \\int f(x) e^{-2\\pi i x \\cdot \\xi} \\, dx$, the [[Fourier transform|DPVM34ADZV]] of $R_j f$ is\n$$\\widehat{R_j f}(\\xi) = -i \\frac{\\xi_j}{\\abs{\\xi}} \\hat f(\\xi).$$\nSo $\\sum_{j=1}^n R_j^2 = -I$ on $L^2(\\R^n)$, and each $R_j$ is bounded on $L^2$ with norm $1$.\nEach $R_j$ extends to a bounded operator on $L^p(\\R^n)$ for $1 \u003c p \u003c \\infty$.\n\nExample: for $n = 1$, $c_1 = \\Gamma(1)/\\pi = 1/\\pi$ and the kernel is $\\frac{1}{\\pi}\\frac{y}{\\abs{y}^2} = \\frac{1}{\\pi y}$.\nSo $R_1$ is the [[Hilbert transform|NSCEBXM7ZM]], with multiplier $-i\\sgn\\xi$.\n\nNon-example: the kernel $\\abs{y}^{-n}$, with the same homogeneity but without the odd factor $y_j/\\abs{y}$, does not define a principal value operator.\nIt has no cancellation, and $\\int_{\\varepsilon \u003c \\abs{y} \u003c 1} \\abs{y}^{-n} \\, dy$ grows like $\\log(1/\\varepsilon)$.\n\nThe kernel $c_n y_j/\\abs{y}^{n+1}$ is odd, homogeneous of degree $-n$, and smooth away from $0$, which makes $R_j$ a [[singular integral operator|HTMCEARGDD]].\n",
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    "kind": "definition",
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    "name": "Riesz transform",
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        "url": "https://en.wikipedia.org/wiki/Riesz_transform"
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    "tag": "W3TAPJEX29",
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  "msc": [
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  "name": "Riesz transform",
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