Worked oscillation decay and its Hölder modulus

Given: An open set $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$, a function $u:\mathrm{\Omega }\to \mathbb{R}$ with finite oscillation on every compactly contained ball satisfying ${\mathrm{osc}}_{B_r(x)}u\le \theta {\mathrm{osc}}_{B_{2r}(x)}u$ whenever $B_{2r}(x)⋐\mathrm{\Omega }$, and the two values $\theta =\frac{1}{2}$ and $\theta =2^{-1\mathrm{/}3}$.

Oscillation-to-modulus conversion: under the stated finite-oscillation hypothesis, put ${\alpha }_0=\log (1\mathrm{/}\theta )\mathrm{/}\log 2$ and $\alpha =\min \{{\alpha }_0,1\mathrm{/}2\}$; then $\mathrm{\mid }u(x)-u(y)\mathrm{\mid }\le 4^{\alpha }(\mathrm{\mid }x-y\mathrm{\mid }\mathrm{/}R{)}^{\alpha }{\mathrm{osc}}_{B_R(x_0)}u$ for all $x,y\in B_{R\mathrm{/}2}(x_0)$, with $4^{{\alpha }^{\mathrm{\prime }}}$ for every $0<{\alpha }^{\mathrm{\prime }}<\alpha$ (Geometric oscillation decay implies a Hölder modulus).

Real powers with positive base satisfy $2^{-a}=1\mathrm{/}{2}^a$, $\log (2^a)=a\log 2$ and $\log (1\mathrm{/}\theta )=-\log \theta$ for $\theta \in (0,1)$ (Real powers for positive bases, with the zero-base positive-exponent convention); the Hölder seminorm on balls is defined as in Local Hölder and scaled C-two-alpha norms on balls.

The De Giorgi oscillation reduction produces a ratio $\theta \in (0,1)$ on balls satisfying its doubled-ball condition. Reserving this interior margin permits dyadic iteration; the resulting power-law conversion is the one illustrated in [F1] (De Giorgi oscillation reduction: one half-level set is small).