In this study we consider the Cauchy problem for the nonlinear Schrödinger equations with data which belong to (Formula presented.)(Formula presented.) In particular, we discuss analytic smoothing effect with data which satisfy exponentially decaying condition at spatial infinity in (Formula presented.)(Formula presented.) We construct solutions in the function space of analytic vectors for the Galilei generator and the analytic Hardy space with the phase modulation operator based on (Formula presented.).
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