Ill-posedness for one-dimensional wave maps at the critical regularity

We show that the wave map equation in R¹⁺¹ is in general ill posed in the critical space Ḣ^1/2, and the Besov space Ḃ^1/2,1₂. The problem is attributed to the bad behavior of the one-dimensional bilinear expression D^-1(fDg) in these spaces.