Data oscillation reduction under three-generation refinement

For each parent $K\in \mathcal{T}$, let $K^{\mathrm{\prime }}$ range over its descendants in ${\mathcal{T}}_{\ast }$, including $K^{\mathrm{\prime }}=K$ if it is unrefined. Nestedness of the piecewise-constant spaces gives ${\sum }_{K^{\mathrm{\prime }}\subset K}\mathrm{\parallel }f-{\mathrm{\Pi }}_{K^{\mathrm{\prime }}}^{0}f{\mathrm{\parallel }}_{L^2(K^{\mathrm{\prime }})}^2\le \mathrm{\parallel }f-{\mathrm{\Pi }}_K^{0}f{\mathrm{\parallel }}_{L^2(K)}^2\mathrm{.}$ Indeed, the childwise projection space contains the parent constant space, and the $L^2$ orthogonal projection minimizes squared error. If $K\in M_O$, the completed call supplies three generations, so [F1] gives $h_{K^{\mathrm{\prime }}}^2\le h_K^2\mathrm{/}4$ and the sum of descendant oscillations is at most one quarter of the parent oscillation. For every other parent, containment gives $h_{K^{\mathrm{\prime }}}\le h_K$, so its descendant oscillations sum to at most its original contribution. An unrefined cell retains its contribution exactly.

Summing step 1.1 over refined parents gives ${\mathrm{osc}}_{{\mathcal{T}}_{\ast }}^2\le {\mathrm{osc}}_{\mathcal{T}}^2-\frac{3}{4}{\sum }_{K\in M_O}{\mathrm{osc}}_{\mathcal{T}}(K{)}^2\mathrm{.}$ Apply the bulk inequality [F2] to obtain the stated ${\gamma }_O$. If all oscillations vanish, both sides are zero and the same inequality holds.