In a paper I'm reading, they state without proof that if $\phi\in C_c^\infty(R^2)$, then
$|\phi(x_1,x_2)|\le \frac{1}{2}\int_{-\infty}^{\infty}|\ \phi_{x_1}(t,x_2)|dt$
I understand one should use the fundamental theorem of calculus but I cannot justify the constant.
Here, $x_2$ is only used as a constant, and so we can consider a function of one variable, where the analogous statement is $|f(x)|\leq \frac{1}{2}\int_{-\infty}^{\infty} |f'(t)|dt$
Because $f$ is compactly supported, $f(x)=f'(x)=0$ for all sufficiently large values of $|x|$, and so by the fundamental theorem of calculus, $f(x)=\int_{-\infty}^x f'(t)dt$. Similarly, $-f(x)=\int_{x}^{\infty} f'(t)dt$. Combining the two, we have
$ 2|f(x)|=\left|\int_{-\infty}^x f'(t)dt \right|+\left|\int_{x}^{\infty} f'(t)dt \right|\leq \int_{-\infty}^x \left|f'(t) \right|dt +\int_{x}^{\infty} \left|f'(t) \right|dt = \int_{-\infty}^{\infty} \left|f'(t) \right|dt. $