REMARK ON WELL-POSEDNESS AND ILL-POSEDNESS FOR THE KDV EQUATION

We consider the Cauchy problem for the KdV equation with low regularity initial data given in the space H-s,H-a(R), which is defined by the norm parallel to phi parallel to H-s,H-a = parallel to <xi >(s-a)vertical bar xi vertical bar(a)(phi) over cap parallel to(L xi 2). We obtain the local well-posedness in H-s,H-a with s >= max{-3/4, -a - 3/2}, -3/2 < a <= 0 and (s, a) not equal (-3/4, -3/4). The proof is based on Kishimoto's work [12] which proved the sharp well-posedness in the Sobolev space H-3/4(R). Moreover we prove ill-posedness when s < max{-3/4, -a - 3/2}, a <= -3/2 or a > 0.