I am reading the paper in the title, and try to understand the inequality (4.14). The last case works with standard techniques using the Mean Value inequality. The first case can be done by localization and using the Schwartz tails of $\Psi$. But I can't get the middle case to work. I.e I'm trying to show:
$$\int _{|x|>2|y'|} \int _{{2^{1-l}}<|z|<2^{1+l}} |\Psi (x-y'-z)-\Psi(x-z)| |h_k(z)| dz dx \leq C||h_k||_1$$ for $1<|y'|<2^{2l}, \Psi$ a Schwartz function and $h_k\in L^1$
Thanks in advance!
This is very similar in spirit to Young's inequality with $p=q=r=1$. Define $f(x) = \Psi(x-y^\prime) - \Psi(x)$ and bound above by the integral over $\Bbb{R}^{2d}$:
$$ I\leq \int_{\Bbb{R}^d}\int_{\Bbb{R}^d} \vert f(x-z)\vert \vert h_k(z)\vert dzdx \tag{1} $$ Then apply Tonelli's theorem and shift invariance of the integral:
$$ (1) = \int_{\Bbb{R}^d}\vert h_k(z)\vert \int_{\Bbb{R}^d}\vert f(x-z)\vert dx dz = \|f\|_1\|h_k\|_1 $$ We know that $\|f\|_1<\infty$ because $\Psi$ is Schwartz:
$$ \|f\|_1 \leq 2\|\Psi\|_1<\infty $$ Thus $I\leq C\|h_k\|_1$, in particular with $C=2\|\Psi\|_1$.