The Mihlin–Hörmander Fourier multiplier theorem

The dyadic pieces and the kernel. The hypotheses of [F1] are satisfied, so with $\zeta$, $m_j$, $K_j$ as there the weighted bounds hold; [F2] then supplies the tempered distribution $W={\mathcal{F}}^{-1}(u_m)$, the almost-everywhere convergent series $k={\sum }_j{K}_j$ on ${\mathbb{R}}^n\setminus \{0\}$ coinciding with $W$ off the origin, and the annular and Hörmander bounds $\le C_{n}A$ for $k$.

On Schwartz inputs, [F4] gives $\mathcal{F}(W\ast f)=u_{m\widehat{f}}$, so $W\ast f=T_{m}f$. Suppose $x\mathrm{\notin }\mathrm{supp}f$ and put $\psi (y)=f(x-y)$, which vanishes near zero. Cut $\psi$ off at infinity using the fixed cutoff ${\chi }_R$ of [F1]. Then ${\chi }_R\psi \to \psi$ in Schwartz space, and [F2] gives $⟨W,{\chi }_R\psi ⟩=\int k{\chi }_R\psi$. The annular bound makes $\int \mathrm{\mid }k\psi \mathrm{\mid }$ finite: on $2^{j}d\le \mathrm{\mid }y\mathrm{\mid }\le 2^{j+1}d$, its contribution is at most $C_{n}A{\sup }_{\mathrm{\mid }y\mathrm{\mid }\ge 2^{j}d}\mathrm{\mid }\psi (y)\mathrm{\mid }$, summable for $j\ge 0$ by Schwartz decay, where $d>0$ is smaller than the distance from zero to the support of $\psi$. Dominated convergence [F6] gives $T_{m}f(x)=\int k(y)f(x-y)dy$. Only annular size and local integrability are used.

For compactly supported $f\in L^2$, its mollifications $f_{\epsilon }=f\ast {\phi }_{\epsilon }$ are smooth with uniformly bounded compact support and converge to $f$ in $L^1$ and $L^2$ by [F5]. Fix a compact $E$ disjoint from $\mathrm{supp}f$; for sufficiently small $\epsilon$ all these supports lie in a fixed compact set $S$ disjoint from $E$. The compact difference set $E-S$ avoids zero. Tonelli and [F2] give ${\int }_E{\int }_S\mathrm{\mid }k(x-y)\mathrm{\mid }\mathrm{\mid }{f}_{\epsilon }(y)-f(y)\mathrm{\mid }dydx\le \mathrm{\parallel }{f}_{\epsilon }-f{\mathrm{\parallel }}_1{\int }_{E-S}\mathrm{\mid }k(z)\mathrm{\mid }dz\to 0$. The same estimate with $f$ proves absolute convergence of its kernel integral almost everywhere on $E$. Step 1.2 identifies $T_m{f}_{\epsilon }$ there with the kernel integral of $f_{\epsilon }$. By [F3], $T_m{f}_{\epsilon }\to T_{m}f$ in $L^2$, hence in $L^1(E)$ by Cauchy–Schwarz. Uniqueness of the $L^1(E)$ limit identifies $T_{m}f$ with the kernel integral of $f$. A countable compact exhaustion proves the off-support representation on $(\mathrm{supp}f{)}^c$.

The kernel $k$ has annular and Hörmander constants at most $C_{n}A$, and $T_m$ has $L^2$ norm $B=\mathrm{\parallel }m{\mathrm{\parallel }}_{\mathrm{\infty }}$. Step 2.1 proves its off-support representation, so $T_m$ is a Calderón–Zygmund operator. Apply the dimension-only estimate proved in the quantitative argument of Calderón–Zygmund operators are bounded on Lp to get $\mathrm{\parallel }{T}_{m}g{\mathrm{\parallel }}_p\le C_n\max (p,(p-1{)}^{-1})(A+\mathrm{\parallel }m{\mathrm{\parallel }}_{\mathrm{\infty }})\mathrm{\parallel }g{\mathrm{\parallel }}_p$. The constant has no unrecorded dependence on $p$.

Consequently $m$ is an $L^p$ Fourier multiplier in the sense of the multiplier definition: [F3] gives $\mathcal{S}\subseteq D_m$, for $f\in \mathcal{S}$ the distribution $T_{m}f$ is the regular distribution of the $L^2$ class ${\mathcal{F}}_2^{-1}(m{\mathcal{F}}_{2}f)$, which is the class assigned to $f$ by the $L^p$ extension of step 3.1 (the two extensions of the Schwartz-core action agree on the dense subspace $\mathcal{S}$), and step 3.1 is exactly the required norm bound. Hence $\mathrm{\parallel }m{\mathrm{\parallel }}_{M_p}\le C_n\max (p,(p-1{)}^{-1})(A+\mathrm{\parallel }m{\mathrm{\parallel }}_{\mathrm{\infty }})$ for every $1<p<\mathrm{\infty }$, which is the assertion.