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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 $e_1=(1,0, \ldots, 0)$, $R>0$, and $\phi\in C^\infty([0,1])$. Let $f = e^{\alpha|\vec{x}/R+\phi(t)e_1|^2}g$. 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?

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    $\begingroup$ You are missing a factor of $|f|^2$ in the third integrand. The third integrand is pointwise dominated by the first up to factors depending on $\phi'$ and $\phi''$; the fourth integrand is pointwise dominated by $|f| |\nabla f|$ up to similar factors, and then Cauchy-Schwarz (and the lower bound on $|\vec x/R + \phi(t) e_1|$, and Young's inequality $ab \leq \frac{1}{2\lambda} a^2 + \frac{\lambda}{2} b^2$ for a suitable $\lambda>0$) takes you home. $\endgroup$
    – Terry Tao
    Commented Dec 20, 2021 at 7:44
  • $\begingroup$ Could you add more details as to how to lower bound the third integral?. We can say that it is$$ \ge -2|\alpha|\int \left(|x/R+\phi| \|\phi''\|_\infty +\|\phi'^2\|_\infty \right)|f|^2,$$ but we have no lower bound for $-|x/R+\phi|$. $\endgroup$
    – Dispersion
    Commented Dec 23, 2021 at 0:37
  • $\begingroup$ $|\vec x/R + \phi(t)e_1|$ is lower bounded by $1$ on the support of $f$, so the expression in parentheses can be bounded by a constant multiple of $|\vec x/R + \phi(t)e_1|^2$ on this support. $\endgroup$
    – Terry Tao
    Commented Dec 23, 2021 at 3:52

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