Atlas-independent hypersurface Sobolev norms and $H^{3\mathrm{/}2}$ tangential derivatives

The coordinate functions $f_i$ are well-defined on almost-everywhere classes because the compatible charts are regular parametrizations, smooth coordinate changes preserve null sets, and the chart-independent surface measure has positive chart densities. At order zero their squared $L^2$ norms sum to a quantity comparable to $\mathrm{\parallel }f{\mathrm{\parallel }}_{L^2(S)}^2$: the upper bound follows from the finite bounded densities and cutoffs, while $f={\sum }_i{\chi }_{i}f$ gives the lower bound by the finite-sum inequality and change of variables. Thus the stated norm is definite for every $r\ge 0$ and controls $L^2(S)$.

For a second compatible atlas and subordinate partition $({\tilde{\chi }}_j,{\tilde{\kappa }}_j)$, write each localized new representative as the finite sum of the representations of ${\tilde{\chi }}_j{\chi }_{i}f$. Each summand is obtained from $f_i$ by an overlap transition and a smooth cutoff supported inside that overlap. Such overlap cutoffs exist after a finite refinement of the compact support intersections; extend them smoothly with compact margins inside the relevant chart. The localized-chart lemma bounds each summand in $H^r$. The finite triangle inequality gives $\mathrm{\parallel }f{\mathrm{\parallel }}_{\tilde{\mathcal{A}},r}\le C\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathcal{A},r}$, and reversing the atlases gives the other inequality. If $f_j$ is Cauchy in the atlas norm, the order-zero comparison in step 1.1 makes it Cauchy in $L^2(S;\mathbb{C})$, and [F6] supplies a limit $f$ there; each localized representative converges also in complete $H^r$, whose continuous injection into $L^2$ identifies its limit with the localized representative of $f$. Hence $f$ belongs to the space and the atlas norm convergence holds, proving completeness.

For a whole-space $g\in H^{1\mathrm{/}2}({\mathbb{R}}^{n-1})$, the Fourier derivative identity shows that $g$ and all ${\mathrm{\partial }}_{a}g$ are in $H^{1\mathrm{/}2}$ exactly when $g\in H^{3\mathrm{/}2}$. Indeed the sum of their squared norms is $\int ⟨\xi ⟩(1+4{\pi }^2\mathrm{\mid }\xi {\mathrm{\mid }}^2)\mathrm{\mid }\widehat{g}{\mathrm{\mid }}^2$, whose weight is bounded above and below by positive constants times $⟨\xi {⟩}^3$. Apply this to every $f_i$ and sum, obtaining the displayed characterization in both directions, including distributional derivative membership rather than presuming it.

In any compatible chart the coordinate vector fields are tangent to $S$, so these first derivatives are local tangential weak derivatives. On an overlap the chain rule gives a smooth transition Jacobian multiplying the coordinate derivatives; partition Leibniz terms add smooth coefficients multiplying the undifferentiated representative. To justify this identity for the present functions, approximate each localized representative by Schwartz functions in $H^{3\mathrm{/}2}$, localize with a fixed larger chart cutoff, and pass to the limit using bounded overlap pullbacks on $H^{3\mathrm{/}2}$ and bounded coefficient multipliers on $H^{1\mathrm{/}2}$. Differentiation $H^{3\mathrm{/}2}\to H^{1\mathrm{/}2}$ is continuous by the preceding Fourier estimate. The limiting distributional identity follows, so the local derivative description is intrinsic without a global frame.