Let $E$ be a normed $\mathbb R$-vector space and $\lambda$ be a measure on $\mathcal B(E)$ with $$\int\min(1,\|x\|_E)\:\lambda({\rm d}x)<\infty.\tag1$$ Are we able to conclude $$\int\left|\underbrace{e^{{\rm i}\langle x,\:x'\rangle}-1-{\rm i}\langle x,x'\rangle1_{\overline B_1(0)}(x)}_{=:\:K(x,\:x')}\right|\:\lambda({\rm d}x)<\infty\tag2$$ for all $x'\in E'$?
$(1)$ is clearly equivalent to $$\int_{\overline B_1(0)}\|x\|_E^2\:\lambda({\rm d}x)<\infty\tag3$$ and $$\lambda\left({\overline B_1(0)}^c\right)<\infty\tag4.$$
Certainly. The integral over the ball is finite becasue of the standard inequality $|e^{it}-1-it|\leq \frac {t^{2}} 2$ valid for all real $t$ (and the inequality $|\langle x, x \rangle'|\leq \|x\| \|x'\|$) and the integral over the complement is finite because $|e^{it}-1|\leq 2$.