Jump multipliers may lie outside the Mihlin criterion

Assume Countable Choice for the library Fourier and $L^p$ conventions. In one dimension the Mihlin convention of Mihlin smoothness convention above half the dimension uses $q=\lfloor 1\mathrm{/}2\rfloor +1=1$: a symbol must agree almost everywhere with a $C^1$ function $m_0$ on $\mathbb{R}\setminus \{0\}$ satisfying $\mathrm{\mid }{\mathrm{\partial }}^{\alpha }{m}_0(\xi )\mathrm{\mid }\le C_{\alpha }\mathrm{\mid }\xi {\mathrm{\mid }}^{-\mathrm{\mid }\alpha \mathrm{\mid }}$ for $\mathrm{\mid }\alpha \mathrm{\mid }\le 1$ and $\xi \mathrm{\ne }0$. In this convention:

Unshifted signum satisfies the criterion. The function $m_0(\xi )=\mathrm{sgn}\xi$ is constant on each of the two components $(-\mathrm{\infty },0)$ and $(0,\mathrm{\infty })$ of the punctured line, hence $C^1$ there with all derivatives zero; the bounds hold with $C_0=1$ and $C_1=0$. Its only jump is at the excluded frequency $0$. (This corrects a claim that the unshifted signum symbol fails the punctured-domain condition.)

Shifted signum fails the criterion. The symbol $m(\xi )=\mathrm{sgn}(\xi -1)$ equals $-1$ on $(0,1)$ and $+1$ on $(1,\mathrm{\infty })$. A function $m_0$ continuous on $\mathbb{R}\setminus \{0\}$ that agreed with $m$ almost everywhere would have to equal $-1$ everywhere on $(0,1)$ and $+1$ everywhere on $(1,\mathrm{\infty })$, since it is continuous and the two open intervals have full measure in themselves; continuity at the nonzero point $\xi =1$ would then fail. So the jump at the nonzero frequency $1$ is not excused by the punctured domain, and $m$ is not a Mihlin symbol here.

Recorded $L^p$ boundedness. Write $h(\xi )=-i\mathrm{sgn}\xi$; the Hilbert transform on $\mathbb{R}$ is the Fourier multiplier with symbol $h$ (Williams §3.1). Then $m(\xi )=\mathrm{sgn}(\xi -1)=ih(\xi -1)\mathrm{.}$ The shifted symbol is a unimodular constant times the frequency translate of the Hilbert multiplier, and frequency translation preserves the $L^p$ multiplier norm (Grafakos Proposition 2.5.14; modulation of the operator is an $L^p$ isometry). So the $L^p(\mathbb{R})$ boundedness of $\mathrm{sgn}(\xi -1)$ for $1<p<\mathrm{\infty }$, hence its membership in this library's $M^p$, is exactly the $L^p$ boundedness of the Hilbert transform, a recorded result assigned to the later singular-integral material and not proved or used on this page.

Thus the Mihlin condition is sufficient but not necessary: a multiplier with a jump at a nonzero frequency can still be bounded on every $L^p(\mathbb{R})$, $1<p<\mathrm{\infty }$. This remark is a recorded orientation leaf; the only locally checked content is the elementary smoothness comparison and the constant modulation identity above. Countable Choice is inherited from the cited conventions (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)).