Well-posedness and decay estimates for 1D nonlinear Schr\"odinger equations with Cauchy data in $L^p$

In this paper, we establish a standard $L^p$-theory of solutions to one dimensional nonlinear Schr"odinger equations with the power like nonlinearity. More precisely, we extend the following three well-known results in the $L^2$ space into $L^p$ setting: 1. Large data local well-posedness for subcritical nonlinearities, 2. Small data global well-posedness for critical nonlinearities, 3. Large data global well-posedness if the subcritical nonlinearity is given by $\mathrm{\mid }u{\mathrm{\mid }}^{\alpha -1}u$.