Uniqueness of $L^{(1,\infty)}$-Hamiltonians and almost-every-moment   Lagrangian disjunction
  This paper is withdrawn by the author due to a critical error in the proof of the main theorem, Theorem 3.10. (More specifically the identity (5.18) is incorrect in that the $T^*S^1(2)$-factor is missing in the right hand side of the equality.) Therefore the uniquenss question for $L^{(1,\infty)}$-case is still open. 
Yong-Geun Oh