Random signed dyadic sums have uniform Mihlin and Lp multiplier bounds

The symbol is well defined and bounded. At each fixed $\xi$ at most three of the numbers ${\phi }_j(\xi )$ are nonzero by [F1], so the series $m(\xi )={\sum }_j{c}_j{\phi }_j(\xi )$ is actually a finite sum at every point and converges locally uniformly to a smooth function; and $\mathrm{\mid }m(\xi )\mathrm{\mid }\le {\sum }_j{\phi }_j(\xi )=1$ because $\mathrm{\mid }{c}_j\mathrm{\mid }\le 1$ and ${\phi }_j\ge 0$.

Uniform Mihlin constants. For $\alpha =0$ step 1.1 gives $\mathrm{\mid }m\mathrm{\mid }\le 1$, which is the required degree-zero bound with constant $1$. Fix $\alpha$ with $1\le \mathrm{\mid }\alpha \mathrm{\mid }\le q$ and $\xi \mathrm{\ne }0$, and put $k:=\mathrm{\mid }\alpha \mathrm{\mid }$. By [F1], $\mathrm{\mid }{\mathrm{\partial }}^{\alpha }m(\xi )\mathrm{\mid }\le {\sum }_{j\ge 0}\mathrm{\mid }{\mathrm{\partial }}^{\alpha }{\phi }_j(\xi )\mathrm{\mid }$. If the $j=0$ derivative is nonzero, then $\mathrm{\mid }\xi \mathrm{\mid }\le 2$, so $\mathrm{\mid }\xi {\mathrm{\mid }}^k\mathrm{\mid }{\mathrm{\partial }}^{\alpha }{\phi }_0(\xi )\mathrm{\mid }\le 2^k{C}_{\alpha }$. For $j\ge 1$, a nonzero derivative requires $2^{j-1}\le \mathrm{\mid }\xi \mathrm{\mid }\le 2^{j+1}$; at most three integers $j$ satisfy both inequalities. On each such annulus, [F1] gives $\mathrm{\mid }\xi {\mathrm{\mid }}^k\mathrm{\mid }{\mathrm{\partial }}^{\alpha }{\phi }_j(\xi )\mathrm{\mid }\le \mathrm{\mid }\xi {\mathrm{\mid }}^k{C}_{\alpha }{2}^{-jk}\le 2^k{C}_{\alpha }$. Therefore $\mathrm{\mid }\xi {\mathrm{\mid }}^k\mathrm{\mid }{\mathrm{\partial }}^{\alpha }m(\xi )\mathrm{\mid }\le 4\cdot 2^k{C}_{\alpha }$, which is the required homogeneous Mihlin estimate. Thus $m$ is a Mihlin symbol with constants $A_0:=1$ and $A_{\alpha }:=4\cdot 2^{\mathrm{\mid }\alpha \mathrm{\mid }}{C}_{\alpha }$ for $1\le \mathrm{\mid }\alpha \mathrm{\mid }\le q$, depending only on $n,\psi ,\alpha$, uniformly in the coefficients.

Uniform $L^p$ bounds. By step 2.1 the symbol $m$ is Mihlin with $A:={\max }_{\mathrm{\mid }\alpha \mathrm{\mid }\le q}{A}_{\alpha }$ and, by step 1.1, $\mathrm{\parallel }m{\mathrm{\parallel }}_{\mathrm{\infty }}\le 1$; [F2] therefore gives the $L^p$ multiplier bound $\mathrm{\parallel }{T}_{m}f{\mathrm{\parallel }}_p\le C_n\max (p,(p-1{)}^{-1})(A+1)\mathrm{\parallel }f{\mathrm{\parallel }}_p$ for every $f\in L^p$, with constants depending only on $n,\psi ,p$.

Truncations and signs. A finite truncation ${\sum }_{j<N}{\epsilon }_j{\phi }_j$ is the symbol $m$ for the sequence $c_j={\epsilon }_j{\mathbf{1}}_{j<N}$, which again satisfies $\mathrm{\mid }{c}_j\mathrm{\mid }\le 1$; the bound of step 3.1 therefore applies to all such truncations and to all sign sequences $c_j=\pm 1$, with the same constant $C_{n,\psi }$.