Hole-filling argument ($E(R)\le \frac{C}{1+C}E(2R)$)

An estimate $E(R)\le C(E(2R)-E(R))$ bounds the energy in a ball by the energy in the surrounding annulus. Moving the smaller-ball energy to the left gives the strict contraction $E(R)\le \theta E(2R)$ with $\theta =C/(1+C)<1$. Iteration turns this contraction into decay at small scales.

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.