Local well-posedness for fourth order Benjamin-Ono type equations

We continue to study the local well-posedness for higher order Benjamin-Ono type equations, especially fourth order equations. The proof is based on the energy methods with correction terms. Although one of correction terms can eliminate the highest order derivative loss in the energy inequality, it may yield a lower order derivative loss than the worst term. In order to cancel this derivative loss, we define correction terms inductively.