Dyadic pieces are uniformly Mihlin multipliers and uniformly Lp-bounded

Uniform Mihlin constants. Fix a multi-index $\alpha$ with $\mathrm{\mid }\alpha \mathrm{\mid }\le q$. For $j\ge 1$ and $\xi \mathrm{\ne }0$, [F1] and [F2] give $\mathrm{\mid }{\mathrm{\partial }}^{\alpha }{\phi }_j(\xi )\mathrm{\mid }\le 2^{\mathrm{\mid }\alpha \mathrm{\mid }}{C}_{\alpha }\mathrm{\mid }\xi {\mathrm{\mid }}^{-\mathrm{\mid }\alpha \mathrm{\mid }}$, and for $j=0$ the same bound holds with constant $2^{\mathrm{\mid }\alpha \mathrm{\mid }}{C}_{\alpha }$: for $\mathrm{\mid }\xi \mathrm{\mid }<2$ one has $2^{\mathrm{\mid }\alpha \mathrm{\mid }}\mathrm{\mid }\xi {\mathrm{\mid }}^{-\mathrm{\mid }\alpha \mathrm{\mid }}\ge 1$ so the bound follows from $\mathrm{\mid }{\mathrm{\partial }}^{\alpha }{\phi }_0\mathrm{\mid }\le C_{\alpha }$, while for $\mathrm{\mid }\xi \mathrm{\mid }\ge 2$ all derivatives of ${\phi }_0$ vanish by [F2]. Hence ${\phi }_j$ is a Mihlin symbol with constants $A_{\alpha }:=2^{\mathrm{\mid }\alpha \mathrm{\mid }}{C}_{\alpha }$ for every $j\ge 0$. For the companions, $\mathrm{\mid }{\mathrm{\partial }}^{\alpha }{\tilde{\phi }}_j\mathrm{\mid }\le \mathrm{\mid }{\mathrm{\partial }}^{\alpha }{\phi }_{j-1}\mathrm{\mid }+\mathrm{\mid }{\mathrm{\partial }}^{\alpha }{\phi }_j\mathrm{\mid }+\mathrm{\mid }{\mathrm{\partial }}^{\alpha }{\phi }_{j+1}\mathrm{\mid }\le 3A_{\alpha }\mathrm{\mid }\xi {\mathrm{\mid }}^{-\mathrm{\mid }\alpha \mathrm{\mid }}$ for $\xi \mathrm{\ne }0$ (each summand satisfying the same bound, with ${\phi }_{-1}=0$ for $j=0$), so the constants $A_{\alpha }^{\mathrm{\prime }}:=3A_{\alpha }$ are uniform in $j$ as well.

Uniform $L^p$ multiplier bounds. By step 1.1 the symbols ${\phi }_j,{\tilde{\phi }}_j$ are Mihlin symbols with constants bounded by the $j$-independent numbers $A:={\max }_{\mathrm{\mid }\alpha \mathrm{\mid }\le q}{A}_{\alpha }$ and $A^{\mathrm{\prime }}={\max }_{\mathrm{\mid }\alpha \mathrm{\mid }\le q}{A}_{\alpha }^{\mathrm{\prime }}$, and $\mathrm{\mid }m\mathrm{\mid }\le \mathrm{\parallel }m{\mathrm{\parallel }}_{\mathrm{\infty }}\le 1$ for both. [F3] therefore makes each of them an $L^p$ Fourier multiplier with $\mathrm{\parallel }{\phi }_j{\mathrm{\parallel }}_{M_p}\le C_n\max (p,(p-1{)}^{-1})(A+1)$ and $\mathrm{\parallel }{\tilde{\phi }}_j{\mathrm{\parallel }}_{M_p}\le C_n\max (p,(p-1{)}^{-1})(A^{\mathrm{\prime }}+1)$, uniformly in $j$, and the corresponding operators $T_{{\phi }_j},T_{{\tilde{\phi }}_j}$ act boundedly on $L^p$ by the definition of the multiplier norm.

The extension is the convolution. Fix $j\ge 0$. On $\mathcal{S}$ the operator $T_{{\phi }_j}$ agrees with the convolution representative $f\mapsto f\ast K_j$ by [F4], and the convolution operator is bounded on $L^p$ with norm at most $C$ by Young's inequality [F4]; since $\mathcal{S}$ is dense in $L^p$ ($p$ finite) and the bounded extension of $T_{{\phi }_j}$ is unique, the $L^p$ multiplier operator equals the convolution operator, so for every $f\in L^p$ the class ${\mathrm{\Delta }}_{j}f$ has the honest representative $f\ast K_j$ and $\mathrm{\parallel }{\mathrm{\Delta }}_{j}f{\mathrm{\parallel }}_p\le C\mathrm{\parallel }f{\mathrm{\parallel }}_p$; the same argument applies to ${\tilde{K}}_j$.

Conclusion. Steps 1.1 and 2.1 give the uniform Mihlin property and the uniform multiplier bounds, and step 3.1 identifies the extensions with the convolution representatives; the asserted inequalities follow with $C_{n,\psi }$ a constant depending only on $n,\psi$ (absorbing $C_n(A+1)$ and the $L^1$ bound, and enlarging it for the companions).