Define a convolution type operator $T_m$ by
$$T_m(f) = p.v.\int_\mathbb{R}f(x-y)\frac{\log^m|y|}{y}dy.$$ Here $m\ge0$ is an integer.
Consider $f \in H^s (s > 0)$  which is the usual Sobolev space. We know that if $m = 0$, $T$ is the Hilbert transform and is a bounded operator on $H^s$. What can we say about the case $m\ge 1$? Can we get the same conclusion?

**Edit: a further question**

From the answer of Christian we know that $T_m$ cannot be a bounded operator on the Sobolev space $H^s$. It seems there is only slight loss of regularity. Can we show that if $f \in H^s$, then for any $\epsilon > 0$ and $m \ge 1$,
$T_m(f) \in H^{s-\epsilon}$?