 Dyadic energy decay

If $E(R/2)\le \theta E(R)$ and $E$ is nondecreasing, then $E(2^{-k}R)\le {\theta }^{k}E(R)$. Between dyadic radii, monotonicity gives $E(r)\le 2^{\mu }(r/R{)}^{\mu }E(R)$ for any $0<\mu \le -{\log }_2\theta$. This transfers a discrete contraction into a uniform power bound.