For $q\in \mathbb{Z}$,denote $$\Delta_q^{'} y:=1_{2^q\le |\xi|\le 2^{q+1}}\,(D)u$$ Prove that the inquality $$ \lVert \Delta_q^{'}u\rVert_{L^p}\le C\lVert u\rVert_{L^p}$$ for some constant $C$ independent of $q$ is false in the case $p\ne 2$.
The defination of operator $\chi(D)$ is following. $$\chi (D)f:=\mathcal{F}^{-1}(\chi \mathcal{F(f)})$$ Hint:Try with the function $u=\chi$,$\chi$ is supported in the ball $\{\xi \in \mathbb{R}^N||\xi |\le \alpha\}$ ,and is a smooth function valued in $[0,1]$
My attempt:$$ \Delta_q^{'}u=(2\pi)^{-N}\int_{2^q\le |\xi|\le 2^{q+1}}e^{i\xi \cdot x}d\xi\int_{|y|\le \alpha}e^{-i\xi \cdot y}\chi(y)dy$$ In addition,we know that $\lVert\chi\rVert_p\le \infty$.My thought is to prove $\lVert \Delta'u\rVert_{L^p}$ is not bounded.
However,I don't know how to deal with it.