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In this paper, we establish a standard $L^p$-theory of solutions to onedimensional 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 subcriticalnonlinearities, 2. Small data global well-posedness for criticalnonlinearities, 3. Large data global well-posedness if the subcriticalnonlinearity is given by $|u|^{\alpha-1}u$.

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