Local well-posedness for the dispersion generalized periodic KdV equation

In this paper, we set up the local well-posedness of the initial value problem for the dispersion generalized periodic KdV equation: ∂tu+∂xα|Dx|u=∂xu2, u(0)=φ for α>2, and φ∈Hs(T). And we show that the is a lower endpoint to obtain the bilinear estimates (1.2) and (1.3) which are the crucial steps to obtain the local well-posedness by Picard iteration. The case α=2 was studied in Kenig et al. (1996) [10].