Consider any function (convolution kernel) $K(x):\mathbb{R}^d\to\mathbb{R}$. Suppose the Fourier transform of $K(x)$, denoted by $\hat{K}(\xi):\mathbb{R}^d\to\mathbb{R}$ satisfies the standard Mikhlin multiplier condition, \begin{equation*} \left|\nabla^n \hat{K}(\xi)\right|\le C |\xi|^{-n}, \end{equation*} for all $n=0,\dots,\frac{d}{2}+1$, and some $C>0$. Then $K(x)$ is also a Calderón–Zymund kernel. That is \begin{gather*} \left|K(x)\right|\le C'|x|^{-d}, \\ \left|\nabla K(x)\right|\le C' |x|^{-d-1}, \end{gather*} for some $C'>0$.
My question is a reference for this result, including also the relationship between $C$ and $C'$ (presumably $C'\le AC$ for some $A>0$).
EDIT: as kindly mentioned by Giorgio Metafune, the result is true if we replace $\frac{d}{2}+1$ by $d+2$, see for example the proof in Theorem 8.2, Classical and multilinear harmonic analysis. Vol. I, by Camil Muscalu and Wilhelm Schlag.
