I am trying to do Exercise A.4 of Terence Tao's book Nonlinear dispersive equations: local and global analysis. The exercise is on p.342 in Appendix A. The problem statement is:
Let $f\in L^q(\mathbb{R}^d)$ for some $1<q<\infty$. Show that $$ \|f\|_{L^q}^q \sim_q \sum_{k\in\mathbb{Z}} 2^{k(1-q)} \sup_{E_k} \left|\int_{E_k} f(x)\,dx\right|^q $$ where $E_k$ ranges over bounded open sets of measure $2^k$.
The hint is to use the rearrangement $f^*(x) := \inf\{\alpha : |\{|f|>\alpha\}|<x\}$, show that $\sup_{E_k} \left|\int_{E_k} f(x)\,dx\right| \sim \int_0^{2^k} f^*(t)\,dt$, and then decompose the interval $[0,2^k]$ dyadically.
So far I have done the first part, but I have no idea how decomposing $[0,2^k]$ dyadically will help me. I do know, however, that $$ \int_{\mathbb{R}^d} |f(x)|^q\,dx = \int_0^\infty (f^*(t))^q\,dt, $$ so an estimate of the form $$ \left(\int_0^{2^k} f^*(t)\,dt\right)^q \sim_q 2^{k(q-1)}\int_{2^{k-1}}^{2^k} (f^*(t))^q\,dt $$ would suffice. However, it is easy to construct counterexamples where the above fails. What other approach might work here?