On well-posedness for nonlinear Schrِdinger equations with power nonlinearity in fractional order Sobolev spaces

We study the well-posedness for the nonlinear Schrödinger equation (NLS) i ∂ t u + 1 2 Δ u = λ | u | p − 1 u in R 1 + n , where p > 1 , λ ∈ C , and prove that (NLS) is locally well-posed in H s if 2 < s < 4 and s / 2 < p < 1 + 4 / ( n − 2 s ) + . To obtain a good lower bound for p , we systematically use Strichartz type estimates in fractional order Besov spaces for the time variable.