Looking at $(t,x)\in[0,1]\times\mathbb{R}^2$, I came across the statement (for sufficiently smooth) real-valued $f$ that $ \|f(t,x)\|_{L^\infty_tL^2_x} \lesssim \|f\|_{L^2_tL^2_x}^{1/2}\|\partial_t f\|_{L^2_tL^2_x}^{1/2} + \|f\|_{L^2_tL^2_x}, $ which is a "simple calculus identity."
I do not know how to prove this. I tried to let $t\in[0,1]$ and then bound $ \int f(t,x)^2dx$ by the above by writing $f(t,x)$ as $\int_0^t\partial_s f(s,x)ds+f(0,x)$ and similar tricks, but could not get it to work out. I also tried to use the Sobolev embedding of $W^{1,2}(\mathbb{R})$ into $L^\infty(\mathbb{R})$, but could not get that to work (nor is that "simple calculus" in my opinion). Any help would be greatly appreciated, thanks!
(In case it is not standard, the mixed norm notation is $ \|f(t,x)\|_{L^q_tL^p_x} = \left(\int\|f(t,\cdot)\|^q_{L^p}dt\right)^{1/q}. $ )
(For reference, the original statement was found in the paper "Sharp Trace Theorems for Null Hypersurfaces on Einstein Metrics with Finite Curvature Flux" by S. Klainerman and I. Rodnianski, Geom. funct. anal. Vol. 16 (2006) 164-229.)