I'm trying to understand scattering for the classical nonlinear Schrödinger equation and for that i'm studying a scattering criterion on Tao's paper. At Lema 3.1 he states that $$\left\|e^{it\Delta}f\right\|_{L^4_tL^{\infty}_x}\lesssim\left\|\nabla f\right\|_{L^2},$$ where $\left\|g\right\|_{L^4_tL^{\infty}_x}:=\left\|\left\|g(x,t)\right\|_{L^{\infty}_x}\right\|_{L^4_t}$. Problem is for the proof he uses Lorentz and Besov spaces, which i'm not familiar with.
Is there a way to prove this inequality by using just basics about Sobolev spaces and interpolation inequalities? I've studied the Riesz-Thorin and the Marcinciewicz theorems, Hardy-Littlewood-Sobolev, the Strichartz estimates and some Sobolev embeddings. Thanks in advance.
