Resumen de Lower bounds for the truncated Hilbert transform

Rima Alaifari, Lillian B. Pierce, Stefan Steinerberger

• Given two intervals I,J⊂R, we ask whether it is possible to reconstruct a real-valued function f∈L2(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 ff to functions with controlled total variation, reconstruction becomes stable. In particular, for functions f∈H1(I), we show that ∥Hf∥L2(J)≥c1exp(−c2∥fx∥L2(I)∥f∥L2(I))∥f∥L2(I), for some constants c1,c2>0 depending only on I,J. This inequality is sharp, but we conjecture that ∥fx∥L2(I) can be replaced by ∥fx∥L1(I).