Weak first derivatives and Riemannian chart norm comparison

Suppose first that $n\ge 1$. We establish the coordinate transformation of weak derivatives locally. Let $T:A\to B$ be a smooth coordinate transition, write $z$ for the coordinates on $A$, put $J=\mathrm{\mid }\det DT\mathrm{\mid }$, and for a fixed $j$ set $b_i=J(D{T}^{-1}{)}_{ij}$, where $D{T}^{-1}$ denotes the inverse matrix at $z$. Then ${\sum }_i{\mathrm{\partial }}_i{b}_i=0$: on each connected component the sign of $\det DT$ is constant, and $b_i$ is that sign times the $(j,i)$ cofactor of $DT$. In its determinant expansion, differentiating a factor ${\mathrm{\partial }}_{\mathrm{\ell }}{T}_a$ in the cofactor for index $i$ pairs with the term differentiating ${\mathrm{\partial }}_i{T}_a$ in the cofactor for index $\mathrm{\ell }$; the signs are opposite and mixed second partials are equal. All terms cancel (for $n=1$ the cofactor is the constant one). For a coordinate function $u\in W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,1}(A)$, set $v=u\circ T^{-1}$. Change of variables on compact subsets makes $v$ and the proposed derivatives locally integrable. For $\phi \in C_c^{\mathrm{\infty }}(B)$, the smooth functions $b_i(\phi \circ T)$ have compact support in $A$. The ordinary chain rule gives ${\sum }_i{b}_i{\mathrm{\partial }}_i(\phi \circ T)=J({\mathrm{\partial }}_j\phi )\circ T$. Applying the weak identity to these tests, summing, and using the zero divergence just proved gives ${\int }_{B}v{\mathrm{\partial }}_j\phi dy=-{\int }_B({\sum }_i{u}_i(T^{-1}(y)){\mathrm{\partial }}_j(T^{-1}{)}_i(y))\phi (y)dy\mathrm{.}$ Thus $v_j={\sum }_i(u_i\circ T^{-1}){\mathrm{\partial }}_j(T^{-1}{)}_i$ weakly. Uniqueness identifies this with any other weak derivative, so the covector expressions agree on overlaps. Applying change of variables to null-set indicators, with $J>0$, and then to the inverse transition shows that both directions preserve null sets; hence these statements are well defined for almost-everywhere classes.

For the compactly supported $u$ of the Statement, its restriction to $\mathrm{\Omega }\setminus x(K)$ is zero almost everywhere. Its weak derivatives there are zero by the test identity and uniqueness. Therefore $u_i=0$ almost everywhere off $x(K)$. Step 1.1 patches the local differential within $U$; each point outside $U$ has a neighborhood disjoint from the closed set $K$, on which the zero extension is zero. Hence the extension has the asserted global weak differential and no boundary contribution. If $K$ is nonempty, compactness gives numbers $0<r\le s$ and $0<a\le b$ such that on $x(K)$, $r\le \rho \le s$ and $a\mathrm{\mid }q{\mathrm{\mid }}_2\le \sqrt{q^T{G}^{-1}q}\le b\mathrm{\mid }q{\mathrm{\mid }}_2$. Indeed the positive continuous density and the positive continuous function $\sqrt{q^T{G}^{-1}q}$ on the compact set $x(K)\times \{q:\mathrm{\mid }q{\mathrm{\mid }}_2=1\}$ have positive lower and finite upper bounds, by taking finite subcovers of neighborhoods where each has such bounds. No bound on the closure of the whole chart is used. Since $\mathrm{\mid }q{\mathrm{\mid }}_2\le {\sum }_i\mathrm{\mid }{q}_i\mathrm{\mid }\le \sqrt{n}\mathrm{\mid }q{\mathrm{\mid }}_2$, substituting [F1] and [F2] into the integral proves the inequalities with $c=r\min (1,a\mathrm{/}\sqrt{n})$ and $C=s\max (1,b)$. If $K$ is empty, $u$ and its derivatives are zero almost everywhere and $c=C=1$ suffices.

Apply [F4] locally to the coordinate expressions of any locally Lipschitz scalar function. Locally Lipschitz here means locally Lipschitz in smooth coordinates; this property is invariant because smooth transitions have bounded derivative on sufficiently small compact coordinate boxes. On each such box the function and its weak derivatives are integrable, so it is $W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,1}$. At points of classical differentiability the ordinary chain rule gives exactly the transformation in step 1.1. The null-set invariance proved there makes the resulting differential independent of chart as an almost-everywhere field. Second countability and [A1] give a countable subatlas, so these local null exceptions can be combined into one null set. For an arbitrary $W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,1}$ coordinate function and smooth $\chi$, use $\chi \phi$ as the test in [F3]. Expanding ${\mathrm{\partial }}_i(\chi \phi )$ gives $\int \chi u{\mathrm{\partial }}_i\phi dx=-\int (\chi u_i+u{\mathrm{\partial }}_i\chi )\phi dx\mathrm{.}$ The right-hand coefficient is locally integrable because the smooth factors are bounded on the support of each test. Thus $D_i(\chi u)=\chi u_i+u{\mathrm{\partial }}_i\chi$, and step 1.1 makes this the stated global weak product rule.

The positive density $\rho$ has exactly the same null sets as coordinate Lebesgue measure: on each compact coordinate subset it is bounded above and below, and countably many relatively compact coordinate boxes cover $\mathrm{\Omega }$. Therefore essential supremum norms agree. The equality of pointwise suprema for a specified representative follows directly from bijectivity of $x$. If that representative is continuous, any strict lower bound on its absolute value attained at a point holds on a neighborhood of that point, which has positive volume; hence its pointwise supremum is also its essential supremum, including the possible infinite value.

For $n=0$, a nonempty coordinate chart is a single point, its tangent and cotangent spaces are zero, and [F2] assigns that point volume one. Under the stated zero-dimensional convention both norms are $\mathrm{\mid }u\mathrm{\mid }$, all differential terms vanish, and the product rule is $0=0$. Null sets for this measure contain no chart point, so essential and pointwise suprema agree. The empty chart gives only zero norms. This proves all remaining cases.