I'm starting to work with evolution PDEs and I'm still a little bit unfamiliar with Bochner spaces. I will ask my concrete question, but probably I might be missing some nuances that I will have to learn along the way.
Consider the following paper by Ionescu and Kenig in which they prove a certain well-posedness result for the initial value problem for the Schrödinger equation $$(i\partial_t+\Delta)u=Vu \quad \textrm{in }\mathbb{R}\times\mathbb{R}^n,$$ $$u(0,\centerdot)=f\quad \textrm{in }\mathbb{R}^n.$$ Here, $V$ is a potential with certain degree of regularity. The thing is, Theorem 1 in the paper says the solution $u(t,x)$ lays in the space $B= C([0,T],L^2(\mathbb{R}^n))\cap L_t^2L_x^{p_n}$, where $p_n$ is the endpoint index for the $H^1$-Hardy-Littlewood-Sobolev embedding, i.e. $1/p_n=1/2-1/n$, and the map $f\mapsto u$ is bounded from $L^2(\mathbb{R}^n)$ to $B$.
This boundedness implies also (which is actually very classical) that the map $f\mapsto u(T,\centerdot)$ is bounded from $L^2(\mathbb{R}^n)$ to $L^2(\mathbb{R}^n)$. In this case, this map is just the composition of the map $f\mapsto u$ with restriction to $t=T$.
I would like to say that $f\mapsto u(T,\centerdot)$ is also bounded from $L^2(\mathbb{R}^n)$ to $L^{p_n}(\mathbb{R}^n)$. Unfortunately the restriction to $t=T$ does not need to even make in principle, since here we are time-wise in $L^2([0,T])$. However, such restriction does exist as a function, by the paragraph above, so we are left to prove boundedness. Do you have any ideas on how to approach it?