I am consulting some papers (references below) about the *Carleson's problem* for the pointwise convergence of the Schrödinger group
\begin{equation}
S_t=e^{i t \Delta}.
\end{equation}
In this context it is important to obtain $L^p$-estimates for the *maximal function*
\begin{equation}
S^\star f(x)=\sup_{t\in(0,1)}\lvert \left(S_tf\right)(x)\rvert,
\end{equation}
where $f\in H^s(\mathbb{R})$. Precisely, we want to find conditions on $s\ge 0$ such that the following inequality holds for some $p\in [1, \infty)$ and all $f\in H^s(\mathbb{R})$:
\begin{equation}\tag{1}
\lVert S^\star f\rVert_{L^p(\left[-1, 1\right])}\le C \lVert f\rVert_{H^s(\mathbb{R})}.
\end{equation}

Question. Both referenced papers claim without further explanation that (1) follows from \begin{equation}\tag{2} \left\lVert \left(S_{t(x)}f\right)(x)\right\rVert_{L^p([-1,1], dx)}\le C \lVert f\rVert_{H^s(\mathbb{R})}, \qquad \forall t\colon \mathbb{R}\to\mathbb{R}\ \text{measurable}.\end{equation} Can you prove the implication $(2)\Rightarrow (1)$ in full detail?

I guess that we should begin by assuming that $f$ is Schwartz, so that $S_tf(x)$ is a continuous function of $(x, t)$. In this case we can use selection theorems (cfr. this answer by Ilya on MathSE) to write
$$S^\star f(x)=\left\lvert S_{t(x)}f(x)\right \rvert,$$
for a Borel-measurable function $t=t(x)$, and so obtain (1) from (2). This answers the question *when $f$ is Schwartz*.

I am not sure on how to eliminate this restriction. Can you help me?

References:

- B. Dahlberg, C. Kenig, "A note on the almost everywhere behaviour of solutions to the Schrödinger equation", Harmonic Analysis (Minneapolis, Minn. 1981), Lecture Notes in Math., vol. 908, Springer, Berlin, 1982, pp.205-209;
- K. Rogers, A. Vargas, L. Vega, "Pointwise convergence of solutions to the nonelliptic Schrödinger equation", Indiana University Mathematics Journal, Vol. 55 No. 6 (2006) pp. 1893-1906.