  Given two intervals $I, J \subset \mathbb{R}$, we ask whether it is possible
to reconstruct a real-valued function $f \in L^2(I)$ from knowing its Hilbert
transform $Hf$ on $J$. When neither interval is fully contained in the other,
this problem has a unique answer (the nullspace is trivial) but is severely
ill-posed. We isolate the difficulty and show that by restricting $f$ to
functions with controlled total variation, reconstruction becomes stable. In
particular, for functions $f \in H^1(I)$, we show that $$ \|Hf\|_{L^2(J)} \geq
c_1 \exp{\left(-c_2 \frac{\|f_x\|_{L^2(I)}}{\|f\|_{L^2(I)}}\right)} \| f
\|_{L^2(I)} ,$$ for some constants $c_1, c_2 > 0$ depending only on $I, J$.
This inequality is sharp, but we conjecture that $\|f_x\|_{L^2(I)}$ can be
replaced by $\|f_x\|_{L^1(I)}$.
