Chart independence of the fractional boundary norm

The distance hypothesis. By the bi-Lipschitz hypothesis choose $L,M>0$ such that $M^{-1}\mathrm{\mid }x-y\mathrm{\mid }\le \mathrm{\mid }\mathrm{\Psi }(x)-\mathrm{\Psi }(y)\mathrm{\mid }\le L\mathrm{\mid }x-y\mathrm{\mid }$ for all $x,y\in V$. These inequalities are assumed for arbitrary open sets; no convexity of $V$ or $V^{\mathrm{\prime }}$ is inferred. On a sufficiently small ball around a point where $D\mathrm{\Psi }$ is invertible, such inequalities follow by integrating $D\mathrm{\Psi }-D\mathrm{\Psi }(x_0)$ on segments and using continuity to make that difference smaller than half the least stretching of $D\mathrm{\Psi }(x_0)$.

Multiplication by a bounded Lipschitz function is bounded on $W^{s,p}$. Let $a$ be bounded and Lipschitz on ${\mathbb{R}}^d$ and $v$ measurable. Then $\mathrm{\mid }a(x)v(x)-a(y)v(y){\mathrm{\mid }}^p\le 2^{p-1}(\mathrm{\mid }a(x){\mathrm{\mid }}^p\mathrm{\mid }v(x)-v(y){\mathrm{\mid }}^p+\mathrm{\mid }a(x)-a(y){\mathrm{\mid }}^p\mathrm{\mid }v(y){\mathrm{\mid }}^p)$, and with $\mathrm{\parallel }a{\mathrm{\parallel }}_{\mathrm{\infty }}$ and $[a{]}_{\mathrm{L}\mathrm{i}\mathrm{p}}$ the two bounds, integrating against $\mathrm{\mid }x-y{\mathrm{\mid }}^{-d-sp}dxdy$ gives $[av{]}_{s,p}^p\le 2^{p-1}(\mathrm{\parallel }a{\mathrm{\parallel }}_{\mathrm{\infty }}^p[v{]}_{s,p}^p+[a{]}_{\mathrm{L}\mathrm{i}\mathrm{p}}^p{\int }_{\mathrm{\mid }h\mathrm{\mid }\le 1}\mathrm{\mid }h{\mathrm{\mid }}^{p-d-sp}dh\int \mathrm{\mid }v{\mathrm{\mid }}^{p}d\xi +2^p\mathrm{\parallel }a{\mathrm{\parallel }}_{\mathrm{\infty }}^p{\int }_{\mathrm{\mid }h\mathrm{\mid }>1}\mathrm{\mid }h{\mathrm{\mid }}^{-d-sp}dh\int \mathrm{\mid }v{\mathrm{\mid }}^p)$ by translating the second term in $x$ for fixed $y$. Both constants are finite because $p-sp>0$ for $s<1$ and $d+sp>d$; hence $[av{]}_{s,p}\le C(a,d,p,s)(\mathrm{\parallel }v{\mathrm{\parallel }}_{L^p}+[v{]}_{s,p})$, and $\mathrm{\parallel }av{\mathrm{\parallel }}_{L^p}\le \mathrm{\parallel }a{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\parallel }v{\mathrm{\parallel }}_{L^p}$, so $\mathrm{\parallel }av{\mathrm{\parallel }}_{W^{s,p}}\le C^{\mathrm{\prime }}(a,d,p,s)\mathrm{\parallel }v{\mathrm{\parallel }}_{W^{s,p}}$.

Diffeomorphism invariance (i). Let $u$ be measurable on $V^{\mathrm{\prime }}$ and apply the bi-Lipschitz upper bound of step 1.1: $\mathrm{\mid }x-y{\mathrm{\mid }}^{-d-sp}\le L^{d+sp}\mathrm{\mid }\mathrm{\Psi }(x)-\mathrm{\Psi }(y){\mathrm{\mid }}^{-d-sp}$, so $[u\circ \mathrm{\Psi }{]}_{s,p}^p\le L^{d+sp}{\int }_V{\int }_V\mathrm{\mid }u(\mathrm{\Psi }x)-u(\mathrm{\Psi }y){\mathrm{\mid }}^p\mathrm{\mid }\mathrm{\Psi }(x)-\mathrm{\Psi }(y){\mathrm{\mid }}^{-d-sp}dxdy$. The map $\mathrm{\Theta }:=\mathrm{\Psi }\times \mathrm{\Psi }:V\times V\to V^{\mathrm{\prime }}\times V^{\mathrm{\prime }}$ is a $C^1$ diffeomorphism of open subsets of ${\mathbb{R}}^{2d}$ with $\mathrm{\mid }\det D\mathrm{\Theta }(x,y)\mathrm{\mid }=\mathrm{\mid }\det D\mathrm{\Psi }(x)\mathrm{\mid }\mathrm{\mid }\det D\mathrm{\Psi }(y)\mathrm{\mid }$ and $\det D{\mathrm{\Theta }}^{-1}(\xi ,\eta )=\det D{\mathrm{\Psi }}^{-1}(\xi )\det D{\mathrm{\Psi }}^{-1}(\eta )$; the change-of-variables theorem [F2] applied to the nonnegative measurable integrand $f(\xi ,\eta ):=\mathrm{\mid }u(\xi )-u(\eta ){\mathrm{\mid }}^p\mathrm{\mid }\xi -\eta {\mathrm{\mid }}^{-d-sp}$ yields ${\int }_{V\times V}f(\mathrm{\Theta }(x,y))dxdy={\int }_{V^{\mathrm{\prime }}\times V^{\mathrm{\prime }}}f(\xi ,\eta )\mathrm{\mid }\det D{\mathrm{\Psi }}^{-1}(\xi )\mathrm{\mid }\mathrm{\mid }\det D{\mathrm{\Psi }}^{-1}(\eta )\mathrm{\mid }d\xi d\eta \le M^{2d}{\int }_{V^{\mathrm{\prime }}\times V^{\mathrm{\prime }}}f$, where $M$ also bounds $\mathrm{\mid }\det D{\mathrm{\Psi }}^{-1}\mathrm{\mid }\le M^d$ after enlarging the constant. For the $L^p$ term, [F2] applied in the form ${\int }_V\mathrm{\mid }u(\mathrm{\Psi }x){\mathrm{\mid }}^{p}dx\le M^d{\int }_{V^{\mathrm{\prime }}}\mathrm{\mid }u{\mathrm{\mid }}^p$ gives $\mathrm{\parallel }u\circ \mathrm{\Psi }{\mathrm{\parallel }}_{L^p(V)}\le M^{d\mathrm{/}p}\mathrm{\parallel }u{\mathrm{\parallel }}_{L^p(V^{\mathrm{\prime }})}$. Adding the two bounds, $\mathrm{\parallel }u\circ \mathrm{\Psi }{\mathrm{\parallel }}_{W^{s,p}(V)}\le C(\mathrm{\Psi },s,p)\mathrm{\parallel }u{\mathrm{\parallel }}_{W^{s,p}(V^{\mathrm{\prime }})}$; exchanging $\mathrm{\Psi }$ and ${\mathrm{\Psi }}^{-1}$ gives the symmetric inequality.

Chart independence (ii). Let two atlases and partitions be as in [F5]. For each pair $j,k$, the support of ${\chi }_j{\chi }_k^{\mathrm{\prime }}$ on the boundary is compact inside the chart overlap. Cover it by finitely many small coordinate balls whose slightly larger closures remain in that overlap. Step 1.1 makes each transition bi-Lipschitz on those larger balls; its Jacobians and inverse Jacobians are bounded there by [F4]. Choose a finite smooth coordinate partition equal to one near this compact support. On each piece, the identity $({\chi }_j{\chi }_k^{\mathrm{\prime }}g)\circ {\mathrm{\Psi }}_j^{-1}=(({\chi }_k^{\mathrm{\prime }}g)\circ ({\mathrm{\Psi }}_k^{\mathrm{\prime }}{)}^{-1})\circ ({\mathrm{\Psi }}_k^{\mathrm{\prime }}\circ {\mathrm{\Psi }}_j^{-1})({\chi }_j\circ {\mathrm{\Psi }}_j^{-1})$ and steps 1.2 and 2.1 control the norm restricted to the ball. All multiplier factors are bounded Lipschitz there; multiplying by the coordinate cutoff extends them by zero to bounded Lipschitz functions on ${\mathbb{R}}^d$. The localised function is supported a positive distance $\delta$ from the ball's complement, so the extra cross term in its zero-extension seminorm is at most $C{\delta }^{-sp}\mathrm{\parallel }v{\mathrm{\parallel }}_p^p$, obtained by integrating $\mathrm{\mid }h{\mathrm{\mid }}^{-d-sp}$ over $\mathrm{\mid }h\mathrm{\mid }\ge \delta$. Thus its whole-space norm is bounded by the second atlas norm. Sum over the finite pieces and use ${\sum }_k{\chi }_k^{\mathrm{\prime }}=1$ to obtain the first atlas norm bounded by the second. Exchange the atlases for the reverse inequality.