Zero-mean L2 decomposition over a finite cover

Root the finite spanning tree. Set $g_i={\mathbf{1}}_{E_i}q$, $m_i={\int }_D{g}_i$, and let $M_i$ be the sum of $m_j$ over the subtree rooted at $i$. For a nonroot vertex $i$ with parent-edge bump ${\psi }_i$, set $q_i=g_i+{\sum }_{j\text{ child of }i}{M}_j{\psi }_j-M_i{\psi }_i$; at the root omit the final term. Every summand is supported in $U_i$ by [F1].

Since each bump integrates to one, a nonroot patch has integral $m_i+{\sum }_j{M}_j-M_i=0$ by the subtree identity $M_i=m_i+{\sum }_j{M}_j$. At the root, the integral is $m_{\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{t}}+{\sum }_j{M}_j=M_{\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{t}}={\sum }_i{m}_i={\int }_{D}q=0$.

In the sum of all $q_i$, every tree-edge term $-M_i{\psi }_i$ at a child cancels the $+M_i{\psi }_i$ term at its parent, leaving ${\sum }_i{g}_i=q$ almost everywhere because the $E_i$ partition $D$. All coefficients and operations are linear in $q$.

For every subtree, $\mathrm{\mid }{M}_i\mathrm{\mid }=\mathrm{\mid }{\int }_{{\cup }_{j\text{ in subtree }i}{E}_j}q\mathrm{\mid }\le \mathrm{\parallel }q{\mathrm{\parallel }}_1\le \mathrm{\mid }D{\mathrm{\mid }}^{1\mathrm{/}2}\mathrm{\parallel }q{\mathrm{\parallel }}_2$ by [F2], and $\mathrm{\parallel }{g}_i{\mathrm{\parallel }}_2\le \mathrm{\parallel }q{\mathrm{\parallel }}_2$. The triangle inequality applied to the definition in step 1.1 gives $\mathrm{\parallel }{q}_i{\mathrm{\parallel }}_2\le \mathrm{\parallel }q{\mathrm{\parallel }}_2+\mathrm{\mid }D{\mathrm{\mid }}^{1\mathrm{/}2}\mathrm{\parallel }q{\mathrm{\parallel }}_2{\sum }_{e\text{ incident to }i}\mathrm{\parallel }{\psi }_e{\mathrm{\parallel }}_2=D_i\mathrm{\parallel }q{\mathrm{\parallel }}_2$.