When studying Fourier analysis, I have come across inequalities of the form $$ |\hat{K}_j(\xi)|\leq \min (|2^j\xi|^{-a},|2^j\xi|) $$ where we have the dilation operator $K_j(x)=2^{-jn}K(2^{-j}x),~j\in\mathbb{Z}$ and $K\in L^1(\mathbb{R}^n)$ (ie, $K$ is an integrable function over $\mathbb{R}^n$) has compact support. Also, $$ |I(t)|\leq \min (1,|t|^{-1}) $$ where $I(t)$ is the integral $$ I(t)=\int^2_1 e^{-2\pi irt}\frac{dr}{r}. $$ Is there a name for this type of inequality? Is there a more general form of this inequality? How do we go about proving this statement?
Please accept my apologies in advance if this is an easy UG inequality that I should know.
Thanks in advance for your help and comments.
As far as I know, there is no special name for these inequalities. Let's prove the last one. On the one hand $$ |I(t)|\le\int_1^2\frac{dr}r=\log2<1. $$ On the other, integration by parts gives $$ I(t)=\frac{1}{2\,\pi\,i\,t}\Bigl(-\frac{e^{-2\pi irt}}{r}\,\Bigr|_1^2-\int^2_1 e^{-2\pi irt}\,\frac{dr}{r^2}\Bigr) $$ and $$ |I(t)|\le\frac{1}{|t|}. $$