  We study the Cauhcy problem for space-time fractional nonlinear Schr\"odinger
equation with a general nonlinearity. We prove the local well-posedness of it
in fractional Sobolev spaces based on the decay estimates and H\"older type
estimates. Due to the lack of the semigroup structure of the solution
operators, we deduce the decay estimates and H\"older type estimates via the
asymptotic expansion of the Mittag-Leffler functions and Bessel functions. In
particular, these results also show the dispersion of the solutions.
