Seeking an example of Schwartz function $f$ such that $ \int_{\bf R}\left|\frac{f(x-y)}{y}\right|\ dy=\infty$

Question

In the introduction section of Hilbert transform in Grafakos's Classical Fourier Analysis (3rd) (Section 5.1.1), it is said that

for Schwartz functions $f$, the integral $$ \int_{\bf R}\frac{f(x-y)}{y}\ dy\tag{1} $$ may not converge absolutely for any real number $x$.

This motivates the use of principal value integrals: $$ \lim_{\epsilon\to 0}\int_{|y|\geq\epsilon}\frac{f(x-y)}{y}\ dy. $$

Question: Could anyone give an example such that (1) does not converge absolutely?

Answer 1

If the Schwartz function $f$ is $0$ at $x,$ then the integral in question converges absolutely. Thus the Schwartz functions $f$ for which absolute convergence of the integral fails for every $x$ are precisely the Schwartz functions that are positive everywhere, or negative everywhere.