Boundary chart lifting and compact-uniform Fourier projections

The local graph charts of the smooth boundary form an open cover. Shrinking them gives graph-chart neighborhoods whose coordinate images have compact closure inside Euclidean balls; compactness selects a finite subcover. Rescale each coordinate ball affinely into $(1\mathrm{/}4,3\mathrm{/}4{)}^{d-1}$. On each compact chart closure, the map and inverse have bounded derivatives, so the restricted coordinate maps are bi-Lipschitz. Apply [F1] to the finite cover and combine partition functions sharing one chart to obtain nonnegative smooth ${\theta }_j$ compactly supported in $U_j$, with ${\sum }_j{\theta }_j=1$. Since ${\sum }_j{\theta }_j^2\ge J^{-1}$, ${\psi }_j={\theta }_j\mathrm{/}({\sum }_i{\theta }_i^2{)}^{1\mathrm{/}2}$ is smooth, has the same support, and satisfies ${\sum }_j{\psi }_j^2=1$.

For each pair set ${\mathrm{\Phi }}_{jk}(x,y)=({\varphi }_j(x),{\varphi }_k(y))$ and ${\chi }_{jk}(x,y)={\psi }_j(x){\psi }_k(y)$. Define $(Gh{)}_{jk}(z)=({\chi }_{jk}h)({\mathrm{\Phi }}_{jk}^{-1}z)$ on ${\mathrm{\Phi }}_{jk}(U_j\times U_k)$ and zero elsewhere, and define $(Lv)(x,y)={\sum }_{j,k}{\chi }_{jk}(x,y)v_{jk}({\mathrm{\Phi }}_{jk}(x,y))$ on the chart pairs, zero outside. These are finite sums. The chart Jacobians and surface-density factors are bounded above and below on compact supports by bi-Lipschitz regularity and [F2]–[F3], with Fubini [F5] for the product chart factors. Thus the pullbacks, zero extensions and smooth multiplications are bounded on $L^2$, so $G$ and $L$ are bounded linear maps.

Substitution in the finite sum gives $(LGh)(x,y)={\sum }_{j,k}{\chi }_{jk}(x,y{)}^{2}h(x,y)=({\sum }_j{\psi }_j(x{)}^2)({\sum }_k{\psi }_k(y{)}^2)h(x,y)=h(x,y)$ almost everywhere. Thus $LG=I$ as maps on $L^2(M)$.

For smooth $h$, the product and chain rules show that first derivatives of $(Gh{)}_{jk}$ are bounded chart coefficients times $h$ and its first tangential derivatives. The Jacobian bounds in step 2.1 give $\mathrm{\parallel }Gh{\mathrm{\parallel }}_{\oplus H^1}\le C\mathrm{\parallel }h{\mathrm{\parallel }}_{H^1(M)}$ and $\mathrm{\parallel }Lv{\mathrm{\parallel }}_{H^1(M)}\le C\mathrm{\parallel }v{\mathrm{\parallel }}_{\oplus H^1}$. Since the weights vanish near patch edges, zero extension remains smooth. Density in the completion definition extends these maps boundedly to $H^1$.

By [F4], $P_N$ is a contraction on the finite direct sum and converges strongly to the identity. The image $G(C)$ is compact by step 2.1. For $\epsilon >0$, take a finite $\epsilon \mathrm{/}3$-net $v_1,\dots ,v_s$ for $G(C)$ and then take $N$ large enough that $\mathrm{\parallel }{P}_N{v}_i-v_i\mathrm{\parallel }<\epsilon \mathrm{/}3$ for all $i$. For any $v\in G(C)$ choose $v_i$ with $\mathrm{\parallel }v-v_i\mathrm{\parallel }<\epsilon \mathrm{/}3$; contraction gives $\mathrm{\parallel }{P}_{N}v-v\mathrm{\parallel }\le \mathrm{\parallel }{P}_N(v-v_i)\mathrm{\parallel }+\mathrm{\parallel }{P}_N{v}_i-v_i\mathrm{\parallel }+\mathrm{\parallel }{v}_i-v\mathrm{\parallel }<\epsilon$. This proves compact-uniform convergence.

For $w\in H^1({\mathbb{T}}^n)$, weak integration by parts against a character gives $\widehat{{\mathrm{\partial }}_{\mathrm{\ell }}w}(k)=2\pi i{k}_{\mathrm{\ell }}\widehat{w}(k)$. Parseval [F4] yields $\mathrm{\parallel }(I-P_N)w{\mathrm{\parallel }}_2^2={\sum }_{\mathrm{\parallel }k{\mathrm{\parallel }}_{\mathrm{\infty }}>N}\mathrm{\mid }\widehat{w}(k){\mathrm{\mid }}^2\le (2\pi N{)}^{-2}{\sum }_k\mathrm{\mid }2\pi k{\mathrm{\mid }}^2\mathrm{\mid }\widehat{w}(k){\mathrm{\mid }}^2\le (2\pi N{)}^{-2}\mathrm{\parallel }\mathrm{\nabla }w{\mathrm{\parallel }}_2^2$ for $N\ge 1$. Apply this to each component of $Gh$ and use step 3.2. For $N=0$, $\mathrm{\parallel }Gh{\mathrm{\parallel }}_2\le \mathrm{\parallel }Gh{\mathrm{\parallel }}_{H^1}\le C\mathrm{\parallel }h{\mathrm{\parallel }}_{H^1}$; enlarging $C_1$ proves the stated tail estimate.