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  "body": "\n## Statement\n\nLet $m \\colon \\R \\to \\C$ be a bounded function, with $\\abs{m(\\xi)} \\le A$ for all $\\xi \\in \\R$.\nFor each integer $j$, let $I_j = [2^j, 2^{j+1}]$ and $I_j' = [-2^{j+1}, -2^j]$.\nAssume that $m$ has bounded variation on each $I_j$ and each $I_j'$, with total variation at most $A$ on each of these intervals.\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 $C_p$, depending only on $p$, with\n$$\\norm{(m \\hat f)^\\vee}_{L^p(\\R)} \\le C_p A \\norm{f}_{L^p(\\R)}$$\nfor every Schwartz function $f$.\n\n## Remarks\n\nThe proof uses Littlewood–Paley theory.\nThe operators that cut the frequencies to the dyadic pieces $I_j$ and $I_j'$ split $f$ into pieces whose square function has $L^p$ norm comparable to that of $f$.\nOn each dyadic piece, a function of bounded variation is an average of indicators of intervals.\nThe multiplier of an interval is built from Hilbert transforms, which are bounded on $L^p$.\nA vector-valued bound for these interval operators then gives the theorem.\n\nExample: $m(\\xi) = \\abs{\\xi}^{it}$ for a real $t \\ne 0$ has modulus $1$, and on $[2^j, 2^{j+1}]$ its variation is $\\int_{2^j}^{2^{j+1}} \\abs{t}/\\xi \\, d\\xi = \\abs{t} \\log 2$.\nSo the imaginary powers of $\\abs{\\xi}$ are $L^p$ multipliers for $1 \u003c p \u003c \\infty$.\n\nThe conclusion fails at $p = 1$ and $p = \\infty$: the multiplier $m = \\mathbf{1}_{[0, \\infty)}$ satisfies the hypotheses but gives an operator built from the Hilbert transform, which is unbounded on $L^1$.\nThere is also a version on $\\R^n$, in which the condition is imposed on the dyadic rectangles, using mixed derivatives of $m$.\n\n## Explanations\n\n### L1 human\n\nCut the frequency line into blocks $[2^j, 2^{j+1}]$ that double in length, and the same on the negative side.\nThe theorem says that a bounded multiplier that does not wiggle much within each block preserves $L^p$ size for $1 \u003c p \u003c \\infty$.\nIt may jump between blocks, and it may have infinitely many jumps in total.\n\n### L2 human\n\nThis criterion needs no smoothness: only bounded variation on each dyadic block.\nIt is weaker in hypotheses than the [[Hörmander–Mikhlin multiplier theorem|F5V7CXW30E]] in one variable, where $\\abs{m'(\\xi)} \\le A/\\abs{\\xi}$ implies a variation of at most $A \\log 2$ on each block.\nIts proof is a model use of Littlewood–Paley theory together with the $L^p$ boundedness of the Hilbert transform.\n",
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