Periodic averaging and rescaling bounds on the unit cube

Tile ${\mathbb{R}}^d$ by cells $C_k^{ϵ}=ϵ(k+Y)$. Only $O_d({ϵ}^{-d})$ cells meet the fixed compact support of $\phi$. On each such cell choose its lower corner $x_k=ϵk$. The mean-value bound gives $\mathrm{\mid }\phi (x)-\phi (x_k)\mathrm{\mid }\le \sqrt{d}ϵ\mathrm{\parallel }\mathrm{\nabla }\phi {\mathrm{\parallel }}_{\mathrm{\infty }}$. By [F3], ${\int }_{C_k^{ϵ}}\mathrm{\mid }g(x\mathrm{/}ϵ)\mathrm{\mid }dx={ϵ}^d\mathrm{\parallel }g{\mathrm{\parallel }}_{L^1(Y)}$. Therefore replacing $\phi$ by $\phi (x_k)$ on every intersecting cell changes the integral by at most $Cϵ\mathrm{\parallel }g{\mathrm{\parallel }}_1\mathrm{\parallel }\mathrm{\nabla }\phi {\mathrm{\parallel }}_{\mathrm{\infty }}$; for sufficiently small $ϵ$, every such cell is contained in $Q$ because the support is compact in $Q$. Extend $\phi$ by zero to a compactly supported $C^1$ function on ${\mathbb{R}}^d$. For all sufficiently small $ϵ$, the grid sum has only cells in a fixed bounded box, and $\mid {ϵ}^d{\sum }_k\phi (ϵk)-{\int }_{{\mathbb{R}}^d}\phi (x)dx\mid \le {\sum }_k{\int }_{C_k^{ϵ}}\mathrm{\mid }\phi (ϵk)-\phi (x)\mathrm{\mid }dx\le Cϵ\mathrm{\parallel }\mathrm{\nabla }\phi {\mathrm{\parallel }}_{\mathrm{\infty }},$ since the total volume of those cells is uniformly bounded. Thus the grid Riemann sums converge. By [F3], the change-of-variables identity first holds for $g^{+}$ and $g^{-}$ separately; subtracting their finite integrals gives ${\int }_{C_k^{ϵ}}g(x\mathrm{/}ϵ)dx={ϵ}^d{\int }_{Y}g$ for signed $g$. The cell-corner sum is therefore ${ϵ}^d{\sum }_k\phi (ϵk){\int }_{Y}g$, which converges to the stated mean times integral.

The same tiling bounds the $L^2$ rescaling: each cell has ${\int }_{C_k^{ϵ}}\mathrm{\mid }g(x\mathrm{/}ϵ){\mathrm{\mid }}^{2}dx={ϵ}^d\mathrm{\parallel }g{\mathrm{\parallel }}_2^2$ by [F3], and at most $C_d{ϵ}^{-d}$ cells meet $Q$ when $ϵ\le 1$. Summing gives the asserted uniform bound.

The averaging result applied to every $C_c^{\mathrm{\infty }}(Q)$ test gives convergence of pairings against that dense class. The uniform $L^2$ bound in step 1.2, together with $\mathrm{\mid }{\int }_{Y}g\mathrm{\mid }\le \mathrm{\parallel }g{\mathrm{\parallel }}_2$, lets one approximate an arbitrary $L^2(Q)$ test by [F2] and pass the pairing limit. This proves weak $L^2$ convergence.