Is this Hilbert-related transform bounded?

Question

I have trouble deciding whether the transform $$T(f)(x)=\int_{\lvert x-t\rvert>1}\frac{f(t)}{\lvert x-t\rvert}\mathrm{d}t\text,\qquad f\in L^2(\mathbb{R})$$ is bounded. In fact, whether its values actually lie in $L^2(\mathbb{R})$. It seems related to the Hilbert transform. In other words: does putting modulus in the denominator of the Hilbert transform preserves boundedness, if one cuts out a neighbourhood of zero? Can anyone help?