Let $\{\phi(n)\}_{n\in\mathbb Z}$ be a sequence of complex numbers with the following properties:

$\phi(0)=0$ and $|\phi(n)|\leq \frac{C_1}{|n|}$ for all $n\neq 0$ and $C_1>0$ is independent of $n.$

$|\phi(n+1)-\phi(n)|\leq \frac{C_2}{n^2}$ for all $n\neq 0$ and $C_2>0$ is independent of $n.$

$\sum_{-N}^N\phi(n)$ converges as $N\to\infty.$

Consider $$ K(x)=\sum_{n\in\mathbb Z}\phi(n)\chi_{\left[n-\frac{1}{2},n+\frac{1}{2}\right)}(x), $$ which exists as a function in $L_p(\mathbb R).$ Can anyone prove that $$ \lim_{\epsilon\to \infty}\int\limits_{\frac{1}{\epsilon}<|x|<\epsilon}K(x)\,\mathrm{d}x $$ exists? Also is the following true? $|K(x)-K(x-y)|\leq C_3\frac{|y|}{|x|^2}$ for $|x|>2|y|.$ We also have by http://matwbn.icm.edu.pl/ksiazki/cm/cm66/cm66211.pdf that $|K(x)|\leq C_4|x|^{-1}$ for $x\neq 0.$ That is $K$ is standard Calderon-Zygmund kernel.

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