Abstract
Recently it was shown by Z. Guo and the author independently that the Cauchy problem for the nonperiodic Korteweg-de Vries equation is locally well-posed in the Sobolev space of the critical regularity. Their proofs were based on the iteration argument in the Besov endpoint of the Bourgain space with some additional modification in low frequency, and it was conjectured that the Bourgain space with the Besov-type modification only does not restore the bilinear estimate which is essential to the iteration. In the present article we give the answer to this conjecture by constructing an exact counterexample to the bilinear estimate.
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