Dyadic energy decay
= Dyadic energy decay
{title2=$E(r)\leq C(r/R)^\mu E(R)$}
If $E(R/2)\leq\theta E(R)$ and $E$ is nondecreasing, then $E(2^{-k}R)\leq\theta^kE(R)$. Between dyadic radii, monotonicity gives $E(r)\leq2^\mu(r/R)^\mu E(R)$ for any $0<\mu\leq-\log_2\theta$. This transfers a discrete contraction into a uniform power bound.