P0 averages approximate smooth densities in H negative one half

We first construct a surface approximation at a length $0<ϵ\le 1$, using only the fixed atlas. For each zero-extended localized pullback $g_j=({\chi }_{j}v)\circ {\mathrm{\Phi }}_j$ in the chart-space definition, let $g_{j,ϵ}$ have Fourier transform $1_{\{\mathrm{\mid }\xi \mathrm{\mid }\le {ϵ}^{-1}\}}{\widehat{g}}_j$. The weighted characterization and differentiation in [F4] give $\mathrm{\parallel }{g}_j-g_{j,ϵ}{\mathrm{\parallel }}_2\le C{ϵ}^{1\mathrm{/}2}q(g_j)$ and $\mathrm{\parallel }{g}_{j,ϵ}{\mathrm{\parallel }}_{H^1}\le C{ϵ}^{-1\mathrm{/}2}q(g_j)$ by comparing the weights on the discarded and retained frequencies. Fix Lipschitz chart cutoffs ${\beta }_j$ equal to one on the fixed chart support of ${\chi }_j$ and compactly supported inside the charts. Transport ${\beta }_j{g}_{j,ϵ}$ back and sum to obtain $v_{ϵ}$. Bounded graph area densities and the finite partition give $\mathrm{\parallel }v-v_{ϵ}{\mathrm{\parallel }}_{L^2(\mathrm{\Gamma })}\le C{ϵ}^{1\mathrm{/}2}\mathrm{\parallel }v{\mathrm{\parallel }}_{H^{1\mathrm{/}2}(\mathrm{\Gamma })}$. Each band-limited function is smooth, and the compactly supported chart products transported to the surface are Lipschitz, including across coplanar face edges. Their restrictions thus belong to $H^1(T)$ on every triangle, even when it crosses such an edge. On each planar face the graph maps are affine with uniformly bounded derivative and inverse derivative; the product and chain rules give $\mathrm{\parallel }{\mathrm{\nabla }}_{\mathrm{\Gamma }}{v}_{ϵ}{\mathrm{\parallel }}_2\le C{ϵ}^{-1\mathrm{/}2}\mathrm{\parallel }v{\mathrm{\parallel }}_{H^{1\mathrm{/}2}(\mathrm{\Gamma })}$. These constants depend only on the fixed atlas and cutoffs. The averages form an $L^2$ orthogonal projection, so $\mathrm{\parallel }(I-{\mathrm{\Pi }}_h)z{\mathrm{\parallel }}_2\le \mathrm{\parallel }z{\mathrm{\parallel }}_2$. Applying the shape-independent convex Poincare inequality [F2] on every triangle gives $\mathrm{\parallel }(I-{\mathrm{\Pi }}_h)v_{ϵ}{\mathrm{\parallel }}_2\le Ch\mathrm{\parallel }{\mathrm{\nabla }}_{\mathrm{\Gamma }}{v}_{ϵ}{\mathrm{\parallel }}_2$. For $h\le 1$, take $ϵ=h$ and combine these estimates to obtain $\mathrm{\parallel }(I-{\mathrm{\Pi }}_h)v{\mathrm{\parallel }}_2\le C{h}^{1\mathrm{/}2}\mathrm{\parallel }v{\mathrm{\parallel }}_{H^{1\mathrm{/}2}}$. For $h>1$, use projection contraction and the fixed $H^{1\mathrm{/}2}\to L^2$ inclusion from [F1]. This proves the first estimate with no dependence on any triangle aspect ratio or family shape-regularity bound.

Under the face alignment hypothesis, facewise $C^1$ $\lambda$ with square-integrable derivatives restricts to $H^1(T)$ on each triangle. More generally assume directly that $\lambda {\mathrm{\mid }}_T\in H^1(T)$. Apply [F2] on the relative interior of each planar triangle, which is convex, to obtain $\mathrm{\parallel }\lambda -{\mathrm{\Pi }}_h\lambda {\mathrm{\parallel }}_{L^2(T)}\le C{h}_T\mathrm{\parallel }{\mathrm{\nabla }}_{\mathrm{\Gamma }}\lambda {\mathrm{\parallel }}_{L^2(T)}$. Squaring and summing yields $\mathrm{\parallel }\lambda -{\mathrm{\Pi }}_h\lambda {\mathrm{\parallel }}_{L^2(\mathrm{\Gamma })}\le Ch\mathrm{\parallel }{\mathrm{\nabla }}_{\mathrm{\Gamma }}\lambda {\mathrm{\parallel }}_{L^2(\mathrm{\Gamma })}$.

For any $\lambda \in L^2(\mathrm{\Gamma })$ put $e=\lambda -{\mathrm{\Pi }}_h\lambda$. For any $w\in H^{1\mathrm{/}2}(\mathrm{\Gamma })$, [F3] gives ${\int }_{T}eds=0$, hence ${\int }_{\mathrm{\Gamma }}ewds={\sum }_T{\int }_{T}e(w-w_T)ds\mathrm{.}$ Cauchy–Schwarz over the finite element sum and step 1.1 give $\mathrm{\mid }⟨e,w⟩\mathrm{\mid }\le \mathrm{\parallel }e{\mathrm{\parallel }}_{L^2}\mathrm{\parallel }w-{\mathrm{\Pi }}_{h}w{\mathrm{\parallel }}_{L^2}\le C{h}^{1\mathrm{/}2}\mathrm{\parallel }e{\mathrm{\parallel }}_{L^2}\mathrm{\parallel }w{\mathrm{\parallel }}_{H^{1\mathrm{/}2}}\mathrm{.}$ Taking the supremum over $\mathrm{\parallel }w{\mathrm{\parallel }}_{H^{1\mathrm{/}2}}\le 1$ proves the general negative-norm bound. When the hypotheses of step 1.2 hold, its additional factor $h\mathrm{\parallel }{\mathrm{\nabla }}_{\mathrm{\Gamma }}\lambda {\mathrm{\parallel }}_{L^2}$ proves the stated $h^{3\mathrm{/}2}$ bound.