Lower bounds for the truncated Hilbert transform

Abstract: 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 $|f_x|_{L^1(I)}$.