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We consider the space-time fractional nonlinear Schrodinger equation. Wefirst give a decay estimates and Holder-type estimates of the evolutionoperators and estimates the nonlinearity in some Sobolev spacs and Besov spacesusing the harmonic tools. Then we give a priori estimates of the solution.After that we prove the local and global well-posedness of the solution and itsdispersion in some significant spaces.

On the solutions to the space-time fractional nonlinear Schrodinger equation and their dispersion