Define a convolution type operator $T$ by
$$T(f) = \int_\mathbb{R}f(x-y)\frac{\log^m(y)}{y}dy.$$ 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?