Definition $1$: A distribution $K$ is regular if and only if $K$ agree with a $C^{\infty}$ function away from the origin .
Definition 2 : $K \hat{}$ denote the fourier transform of the distribution $K$
$C^{(n)}$-normalized bump function : Function $\varphi \in C^{\infty}$ supported in the unit ball and $\sup_x |\partial_x^{\alpha}\varphi(x)|\le1$ for all $|\alpha|\le n$ .
Cancelation property : For some fixed $n\ge1$ , there is an $A$ so that $\sup_{0 \lt r} |K(\varphi(rx)|\le A$ for all $C^{(n)}$-normalized bump function $\varphi$ .
Theorem : If $K$ is a tempered distribution , and $m=K \hat{}$ is a $C^{\infty}$ function away from the origin that satisfied $|\partial_x^{\alpha} m(x)|\le c_{\alpha} |x|^{-\alpha}$ . Then $K$ is also a regular distribution .
In Stein's functional analysis Page$_{135}$ , it said the following three definition of Calderon-Zygmund distributions are equivalent .
$(1)$ $K$ is regular and satisfies the differential inequalities $|\partial_x^{\alpha} k(x)|\le c_{\alpha}|x|^{-d-|\alpha|}$ for all $\alpha$ together with the cencelation property .
$(2)$ $K$ is tempered , and $m=K \hat{}$ is a function that is $C^{\infty}$ away from the origin that satisfies $|\partial_x^{\alpha}m(x)|\le c_{\alpha}'|x|^{-|\alpha|}$ for all $\alpha$ .
$(3)$ $K$ is a regular distribution that satisfies the differential inequalities $|\partial_x^{\alpha} k(x)|\le c_{\alpha}|x|^{-d-|\alpha|}$ and $K \hat{}$ is a bound function .
If the word "tempered" in $(2)$ was replaced by "regular" , then I can show three definition above are equivalent . However , in stein's proof ,it seems that he forget to prove this .
My attempt : First consider the case when $m$ has compact support . In this case , we can let $$k(x)=\int m(x) e^{2\pi i nx}\,dx$$ Note that both $\varphi \hat{}$ and $m$ are in $L^1$ , then we have $$K(\varphi \hat{})=K \hat{}(\varphi)=\int m(x)\varphi(x) \,dx=\int k(x)\varphi \hat{} (x) \,dx$$ For all $\varphi \in S(R^d)$ , and suppored outside origin . Then we find $K_0(\varphi)=\int k(x)\varphi(x)$ is regular and $K-K_0$ is supported at origin . Since $|\partial_x^{\alpha}m(x)|\le c_{\alpha}'|x|^{-|\alpha|}$ for all $\alpha$ , we have $K_0=K$ . However , when $m$ is not compact , $\int m(x) e^{2\pi i nx}$ might make no sense . For example , $K=\delta $ , then $m=1$ satisfied condition $(2)$ but $\int e^{2\pi inx}$ diverges while $\delta$ agrees with function $0$ .