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  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $k = \\floor{n/2} + 1$.\nLet $m \\colon \\R^n \\setminus \\set{0} \\to \\C$ be a function of class $C^k$.\nAssume that there is a constant $A$ such that\n$$\\abs{\\partial^\\alpha m(\\xi)} \\le A \\abs{\\xi}^{-\\abs{\\alpha}} \\quad \\text{for all } \\xi \\ne 0$$\nand all multi-indices $\\alpha$ with $\\abs{\\alpha} \\le k$.\nThen $m$ is an [[$L^p$ multiplier|RAGR38Q40B]] for every $1 \u003c p \u003c \\infty$.\nLet $\\hat f$ be the [[Fourier transform|DPVM34ADZV]] of $f$.\nThere is a constant $C$, depending only on $n$ and $p$, with\n$$\\norm{(m \\hat f)^\\vee}_{L^p(\\R^n)} \\le C A \\norm{f}_{L^p(\\R^n)}$$\nfor every Schwartz function $f$.\nMoreover, the operator $f \\mapsto (m \\hat f)^\\vee$ is of weak type $(1, 1)$.\n\n## Remarks\n\nThe condition with $\\alpha = 0$ says that $m$ is bounded, so the operator is bounded on $L^2$ by the Plancherel theorem.\nThe derivative conditions make the inverse Fourier transform $K$ of $m$ a kernel that satisfies the Hörmander condition $\\int_{\\abs{x} \u003e 2\\abs{y}} \\abs{K(x - y) - K(x)} \\, dx \\le CA$.\nThen the [[Calderón–Zygmund theorem|AE1HJ8D8GV]] gives the weak type $(1, 1)$ bound, interpolation gives $1 \u003c p \\le 2$, and duality gives $2 \\le p \u003c \\infty$.\nHörmander weakened the pointwise derivative bounds to bounds on the $L^2$ averages of the derivatives over dyadic shells.\n\nExample: in $\\R^n$, $m(\\xi) = \\xi_j / \\abs{\\xi}$ is smooth away from $0$ and homogeneous of degree $0$.\nSo $\\partial^\\alpha m$ is homogeneous of degree $-\\abs{\\alpha}$ and bounded on the unit sphere, which gives the hypothesis.\nHence the [[Riesz transforms|FRNDFJ9WQ5]] are bounded on $L^p(\\R^n)$ for $1 \u003c p \u003c \\infty$.\n\nSome smoothness is needed.\nThe indicator of a ball is bounded but not smooth.\nBy [[Fefferman's ball multiplier theorem|4A5MJY4XPV]], it is not an $L^p$ multiplier for $p \\ne 2$ when $n \\ge 2$.\n\n## Explanations\n\n### L1 human\n\nTake a multiplier that is smooth away from the origin, whose derivatives shrink at the right rate as the frequency grows.\nSuch a multiplier keeps $L^p$ size under control for every $p$ strictly between $1$ and $\\infty$.\nThe typical case is a function that depends only on the direction of $\\xi$, or that changes slowly on each dyadic scale.\nOne only needs to check about half as many derivatives as there are dimensions.\n\n### L2 human\n\nThis is the most used multiplier criterion in PDE.\nIt shows at once that the Riesz transforms, the operators $\\partial_i \\partial_j \\Delta^{-1}$, and the imaginary powers $(-\\Delta)^{it}$ are bounded on $L^p(\\R^n)$ for $1 \u003c p \u003c \\infty$.\nIt sits between the [[Marcinkiewicz multiplier theorem|375ZGE0V2W]], which uses dyadic rectangles, and the general Calderón–Zygmund theory, which supplies its proof.\n",
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