The chart and chordal surface H1/2 norms are equivalent

In a graph chart write $\mathrm{\Phi }(p)=(p,\psi (p))$. The map $\mathrm{\Phi }$ is bi-Lipschitz: $\mathrm{\mid }p-q\mathrm{\mid }\le \mathrm{\mid }\mathrm{\Phi }(p)-\mathrm{\Phi }(q)\mathrm{\mid }\le (1+\mathrm{Lip}(\psi {)}^2{)}^{1\mathrm{/}2}\mathrm{\mid }p-q\mathrm{\mid }$. Its surface Jacobian $J_{\mathrm{\Phi }}=(1+\mathrm{\mid }\mathrm{\nabla }\psi {\mathrm{\mid }}^2{)}^{1\mathrm{/}2}$ is bounded above and below almost everywhere. Therefore the $L^2$ norm and the double-integral seminorm of a function supported a positive distance inside this chart are comparable to those of its pullback; pairs with one point outside the chart contribute at most a constant times the $L^2$ norm because the support is separated from the chart edge. By [F1], each localized chart Bessel norm is thus equivalent to the localized surface chordal norm.

Write $g_j={\chi }_{j}g$, so $g={\sum }_{j=1}^J{g}_j$. The inequality $\mathrm{\mid }g(x)-g(y){\mathrm{\mid }}^2\le J{\sum }_j\mathrm{\mid }{g}_j(x)-g_j(y){\mathrm{\mid }}^2$ shows that the global chordal norm is bounded by the sum of localized chordal norms from step 1.1. Conversely, $\mathrm{\mid }{g}_j(x)-g_j(y)\mathrm{\mid }\le \mathrm{\mid }{\chi }_j(x)\mathrm{\mid }\mathrm{\mid }g(x)-g(y)\mathrm{\mid }+\mathrm{\mid }g(y)\mathrm{\mid }\mathrm{\mid }{\chi }_j(x)-{\chi }_j(y)\mathrm{\mid }\mathrm{.}$ Each ${\chi }_j$ is Lipschitz. The second term contributes at most $C\mathrm{\parallel }g{\mathrm{\parallel }}_{L^2}^2$ to the squared seminorm, since ${\int }_{\mathrm{\Gamma }}\mathrm{\mid }x-y{\mathrm{\mid }}^{-1}d{s}_x$ is uniformly bounded in $y$: in each chart the surface area of a radius-$r$ disk is at most $C{r}^2$, and summing dyadic annuli gives ${\int }_{\mathrm{\mid }x-y\mathrm{\mid }<1}\mathrm{\mid }x-y{\mathrm{\mid }}^{-1}d{s}_x\le C{\sum }_{m\ge 0}{2}^{-m}<\mathrm{\infty }$; the part $\mathrm{\mid }x-y\mathrm{\mid }\ge 1$ is bounded by the finite surface area. Hence the sum of localized chordal norms is bounded by the global chordal norm. Combining both comparisons with step 1.1 proves the two-sided equivalence for smooth $g$. Equivalent norms have the same completion, which proves the stated extension.