I am currently reading On Uniqueness Properties of Solutions of Schrödinger Equations and a having trouble understanding a claim made on page 1819.

Context from the paper: let $g\in C^\infty_0(\mathbb{R}^{n+1})$ have its support contained in $$\{(\vec{x},t):\big|\vec{x}/R+\phi(t)e_1 \big|\ge 1\},$$ where $e_1=(1,0, \ldots, 0)$, $R>0$, $\phi\in C^\infty([0,1])$, and $f = e^{\alpha|\vec{x}/R+\phi(t)e_1|^2}g$. We are looking to lower bound the following expression: $$\frac{16\alpha^3}{R^4}\int\big|\vec{x}/R+\phi(t)e_1 \big|^2|f|^2\,dx\,dt+\frac{8\alpha}{R^2}\int|\nabla f|^2\,dx\,dt\tag{1}$$ $$+2\alpha\int\left[\left(\frac{x_1}{R}+\phi\right)\phi''+\phi'^2\right]\,dx\,dt - \frac{8\alpha i}{R}\int\phi'f^\dagger\partial_{x_1}f\,dx\,dt.$$ The authors state "...using the hypothesis on the support on $g$ and the Cauchy–Schwarz inequality, the absolute value of the last two terms in (1) can be bounded by a fraction of the first two terms of (1), when $\alpha ≥ cR^2$ for some large $c$ depending on $\|\phi'\|_\infty+\|\phi''\|_\infty$." How is this done explicitly?