The Littlewood-Paley operator is sometimes also improperly known as a projection since $$||P_kP_kf||_{L^p}\approx ||P_kf||_{L^p}$$
I wish to prove this fact so this would mean I'd need to show that there exists a constant $C>0$ such that$$\frac{1}{C}||P_kP_kf||_{L^p} \stackrel{(1)}{\le} ||P_kf||_{L^p} \stackrel{(2)}{\le} C ||P_kP_kf||_{L^p}$$
I am able to show the first inequality since the annulus test function (let's call it $\chi_k$) is – by definition – less than $1$ in the whole of $\mathbb{R}^n$. However, I am not able to establish the second inequality. I've tried invoking the definition of $\chi_k$ again but this time, I am unable to pin down the exact constant here since $\chi_{k\pm1}\chi_k$ not necessarily less than $(\chi_k)^2$. How should I go about proving this?