I am reading the following paper of Calderon and Zygmund http://matwbn.icm.edu.pl/ksiazki/sm/sm14/sm14122.pdf On page 3 (252) they provide an estimate (2.3) which is following $$ |K(x-x_{\nu}) - K(-x_{\nu})| \leq \frac{1}{|x_{\nu}|^k} \omega\left(\frac{A}{|x_{\nu}|}\right) $$ where they have said earlier that A is a constant which depends at most on $\Omega$ but I am unable to get the following estimate, infact going through usual way to estimate i.e. writing $$ |K(x-x_{\nu}) - K(-x_{\nu})| \leq |\frac{\Omega\left(\frac{x-x_{\nu}}{|x-x_{\nu}|}\right) - \Omega\left(\frac{-x_{\nu}}{|x_{\nu}|}\right)}{|x-x_{\nu}|^k} | + |\Omega\left(-\frac{x_{\nu}}{|x_{\nu}|}\right)\left(\frac{1}{|x_{\nu}|^k} - \frac{1}{|x-x_{\nu}|^k} \right)| $$ I can make the first term of above form (with $A$ also depending on $x$) but the second term I am not able to, what I am getting is something of form $\frac{c|x|}{|x-x_{\nu}|^{k+1}}$. Which can for $|x_{\nu}| > 2 |x|$ can be made in form $\frac{c|x|}{|x_{\nu}|^k}$
These estimates is required to show that terms in left of 2.3 forms an absolutely convergent series. I think with what estimates I have got we can still show that, atleast for any fixed $x$, but I am not sure how to get these stronger estimates
The paper above also refer to page 95 of following paper, but there are also I am not sure how they do these estimates https://projecteuclid.org/journals/acta-mathematica/volume-88/issue-none/On-the-existence-of-certain-singular-integrals/10.1007/BF02392130.full There they have got $I_k$ using calderon zygmund decompostion of some $L^p$ function, here I dont have that to start with, so I thought of taking grid in $\mathbb{Z}^k$ But still the second part in above inequality I cannot estimate. Am I missing something here?
Thanks in advance for help.
In second reference they have shown in footnote that we may without loss of generality assume that $\omega(t) \geq t$, and with this we can get such bounds.