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The semilinear space-time fractional Schr\"odinger equation is considered.First, we give the explicit form for the fundamental solutions by using the Fox$H$-functions in order to to establish some $L^s$ decay estimates. After that,we give some space-time estimates for the mild solutions from which the localwell-posedness is derived on some proper Banach space.

Local well-posedness of semilinear space-time fractional Schr\"odinger equation