Fefferman--Stein-type estimates and fractional NLS in Fourier Sobolev spaces

In this paper, we prove a sharp Fefferman–Stein-type estimate for the fractional Schrödinger equation, which can be regarded as a generalized Strichartz estimate for data in the Fourier Lebesgue space $\widehat{L^p}$. Then, as an application of the Fefferman–Stein inequality and its off-diagonal generalization, we prove large data local well-posedness and small data global well-posedness results for the one dimensional fractional nonlinear Schrödinger equation with pure power nonlinearities in the homonegeneous and inhomogeneous Fourier–Sobolev spaces $\widehat{{\dot{H}}_p^s},\widehat{H_p^s}$. Solutions are established in $L_x^r(\mathbb{R};L_t^q(I))$ spaces in order to overcome the difficulty of a loss of derivatives in the standard Strichartz estimates.