Chain rule for a $C^1$ function with bounded derivative

By [F24] the Axiom of Choice [F25] yields Countable Choice and Dependent Choice in ZF; only Countable Choice is used below, namely in the ACL definition [F6], the locality lemma [F8], the box-measure formula [F12], the finiteness of Lebesgue measure on bounded sets [F13], the Borel measurability of continuous maps [F18], and the countable selection of box representatives below, while the ACL characterisation [F5] is stated under the Axiom of Choice itself.

Fix a measurable representative $\widehat{u}$ of the class $u$ and, for each $i\in \{1,\dots ,n\}$, a measurable representative ${\widehat{v}}_i$ of the class $D_{i}u$; this is one representative plus finitely many others, so no infinite selection is made. By [F18] and [F19] the function $F^{\mathrm{\prime }}(\widehat{u})$ is measurable, while [F10] applied on $\mathbb{R}$ to $F$ shows $\mathrm{\mid }F(s)-F(t)\mathrm{\mid }\le L\mathrm{\mid }s-t\mathrm{\mid }$ for all real $s,t$, in particular $\mathrm{\mid }F(t)\mathrm{\mid }\le \mathrm{\mid }F(0)\mathrm{\mid }+L\mathrm{\mid }t\mathrm{\mid }$ and $\mathrm{\mid }{F}^{\mathrm{\prime }}(\widehat{u})\mathrm{\mid }\le L$ everywhere; hence [F16] gives $\mathrm{\parallel }{F}^{\mathrm{\prime }}(\widehat{u}){\mathrm{\parallel }}_{\mathrm{\infty }}\le L$, so [F2] puts $F^{\mathrm{\prime }}(\widehat{u})$ in $L^{\mathrm{\infty }}(\mathrm{\Omega })$. Define the measurable function $h_i:=F^{\mathrm{\prime }}(\widehat{u}){\widehat{v}}_i$. For $1\le p<\mathrm{\infty }$ the case $(p,q,r)=(\mathrm{\infty },p,p)$ of [F14] gives $h_i\in {\mathcal{L}}^p(\mathrm{\Omega })$ with $\mathrm{\parallel }{h}_i{\mathrm{\parallel }}_p\le L\mathrm{\parallel }{\widehat{v}}_i{\mathrm{\parallel }}_p<\mathrm{\infty }$, and for $p=\mathrm{\infty }$ its $(\mathrm{\infty },\mathrm{\infty },\mathrm{\infty })$ case gives $h_i\in L^{\mathrm{\infty }}(\mathrm{\Omega })$; in both cases [F3] and [F4] make $h_i$ an element of the quotient space $L^p(\mathrm{\Omega })$. If ${\widehat{u}}^{\mathrm{\prime }}$ and ${\widehat{v}}_i^{\mathrm{\prime }}$ are further measurable representatives of the same two classes, then ${\widehat{u}}^{\mathrm{\prime }}=\widehat{u}$ and ${\widehat{v}}_i^{\mathrm{\prime }}={\widehat{v}}_i$ almost everywhere, hence $F^{\mathrm{\prime }}({\widehat{u}}^{\mathrm{\prime }})=F^{\mathrm{\prime }}(\widehat{u})$ and $F^{\mathrm{\prime }}({\widehat{u}}^{\mathrm{\prime }}){\widehat{v}}_i^{\mathrm{\prime }}=F^{\mathrm{\prime }}(\widehat{u}){\widehat{v}}_i$ almost everywhere, and [F4] shows that the same class is obtained; we write $F^{\mathrm{\prime }}(u)D_{i}u$ for it.

Fix one coordinate direction and a rational box $Q={\prod }_{j=1}^n(a_j,b_j)$ with $\bar{Q}\subseteq \mathrm{\Omega }$, as in [F6], and write $Q=Q_{\widehat{i}}\times I_{i,Q}$ for the splitting along the chosen direction. By [F12] $Q$ has finite measure ${\lambda }_n(Q)={\prod }_j(b_j-a_j)$, and the restriction $u{\mathrm{\mid }}_Q$ of the class $u$ lies in $L^p(Q)$ (in $L^{\mathrm{\infty }}(Q)$ when $p=\mathrm{\infty }$). Put $q:=p$ when $p<\mathrm{\infty }$ and $q:=1$ when $p=\mathrm{\infty }$. Then $u{\mathrm{\mid }}_Q$ lies in $L^q(Q)$: for finite $p$ this is $q=p$, and for $p=\mathrm{\infty }$ the class $u{\mathrm{\mid }}_Q\in L^{\mathrm{\infty }}(Q)$ is converted into ${\mathcal{L}}^1(Q)$ by the second clause of [F15]; and by [F8] each weak derivative restricts, $D_j(u{\mathrm{\mid }}_Q)=(D_{j}u){\mathrm{\mid }}_Q$ on $Q$, with $(D_{j}u){\mathrm{\mid }}_Q\in L^q(Q)$ by the same two clauses. By [F1] this says $u{\mathrm{\mid }}_Q\in W^{1,q}(Q;\mathbb{R})$, and since $1\le q<\mathrm{\infty }$ the ACL characterisation [F5] applies on the open box $Q$: it provides one measurable representative $u_Q^{\ast }$ of the class $u{\mathrm{\mid }}_Q$, ACL in every direction, whose classical coordinate derivatives ${\mathrm{\partial }}_j{u}_Q^{\ast }$ exist almost everywhere on $Q$, are measurable, lie in ${\mathcal{L}}^q(Q)$, and represent $D_j(u{\mathrm{\mid }}_Q)$ almost everywhere for every $j$. Since $u_Q^{\ast }$ and $\widehat{u}$ both represent $u{\mathrm{\mid }}_Q$, and ${\mathrm{\partial }}_j{u}_Q^{\ast }$ and ${\widehat{v}}_j$ both represent $(D_{j}u){\mathrm{\mid }}_Q$, we have $u_Q^{\ast }=\widehat{u}$ and ${\mathrm{\partial }}_j{u}_Q^{\ast }={\widehat{v}}_j$ almost everywhere on $Q$ for every $j$, so in particular $F^{\mathrm{\prime }}(u_Q^{\ast }){\mathrm{\partial }}_i{u}_Q^{\ast }=h_i$ almost everywhere on $Q$. The assignment of one representative $u_Q^{\ast }$ to each rational box $Q$ is a selection from countably many nonempty sets, licensed by the Countable Choice of step 1.1.

We verify the membership and representative clauses for the class $F\circ u{\mathrm{\mid }}_Q$ on $Q$, whose exponent $q$ satisfies $1\le q<\mathrm{\infty }$. First $F\circ u{\mathrm{\mid }}_Q\in L^q(Q)$: by [F10] $\mathrm{\mid }F(s)\mathrm{\mid }\le \mathrm{\mid }F(0)\mathrm{\mid }+L\mathrm{\mid }s\mathrm{\mid }$ for every real $s$, so $\mathrm{\mid }F(u)\mathrm{\mid }\le \mathrm{\mid }F(0)\mathrm{\mid }+L\mathrm{\mid }u\mathrm{\mid }$ pointwise; the bound function lies in ${\mathcal{L}}^q(Q)$ because the constant belongs to ${\mathcal{L}}^q(Q)$ by the second clause of [F15] applied to the finite-measure box of [F12], the multiple $L\mathrm{\mid }u\mathrm{\mid }$ belongs to ${\mathcal{L}}^q(Q)$ by [F17], and sums of ${\mathcal{L}}^q$ functions belong to ${\mathcal{L}}^q(Q)$ by [F17]; monotonicity [F22] of the nonnegative integral then gives ${\int }_Q\mathrm{\mid }F(u){\mathrm{\mid }}^q<\mathrm{\infty }$, so [F3] and [F4] make $F\circ u{\mathrm{\mid }}_Q$ an element of $L^q(Q)$. Second, $\mathrm{\Phi }:=F\circ u_Q^{\ast }$ is a measurable representative of that class: it is measurable by [F19] applied to the Borel function $F$ and the measurable $u_Q^{\ast }$ of step 3.1, and it agrees with $F\circ u$ almost everywhere on $Q$ because $u_Q^{\ast }=u$ almost everywhere there. Third, $\mathrm{\Phi }$ is ACL: by [F6] the sections $g_y$ of $u_Q^{\ast }$ in each coordinate direction are absolutely continuous in the sense of [F7] on every compact subinterval of the corresponding side interval, for every transverse parameter outside a null set depending on the direction, and [F10] makes $F$ Lipschitz on the whole real line with constant $L$, so [F9] makes each composite section $F\circ g_y$ absolutely continuous on every compact subinterval; the same null exceptional sets therefore serve for $\mathrm{\Phi }$.

Fourth, the classical coordinate derivatives of $\mathrm{\Phi }$ exist almost everywhere and agree almost everywhere with the measurable function $G:=F^{\mathrm{\prime }}(u_Q^{\ast }){\mathrm{\partial }}_i{u}_Q^{\ast }$. Indeed, at a point $(y,t)\in Q$ where ${\mathrm{\partial }}_i{u}_Q^{\ast }(y,t)$ exists, the section $g_y$ is differentiable at $t$ with $g_y^{\mathrm{\prime }}(t)={\mathrm{\partial }}_i{u}_Q^{\ast }(y,t)$ — the partial derivative of a function at a point is by definition the derivative of its coordinate section there — and $F$ is differentiable at $g_y(t)$, so [F11] gives $(F\circ g_y{)}^{\mathrm{\prime }}(t)=F^{\mathrm{\prime }}(g_y(t))g_y^{\mathrm{\prime }}(t)=G(y,t)$; since $[F5]$ gives that ${\mathrm{\partial }}_i{u}_Q^{\ast }$ exists almost everywhere on $Q$, the derivative ${\mathrm{\partial }}_i\mathrm{\Phi }$ exists almost everywhere on the box and equals $G$ there. The function $G$ is measurable and lies in ${\mathcal{L}}^q(Q)$: $u_Q^{\ast }$ is measurable with $\mathrm{\mid }{F}^{\mathrm{\prime }}(u_Q^{\ast })\mathrm{\mid }\le L$ everywhere, so [F18] and [F19] make $F^{\mathrm{\prime }}(u_Q^{\ast })$ measurable and [F16] gives $\mathrm{\parallel }{F}^{\mathrm{\prime }}(u_Q^{\ast }){\mathrm{\parallel }}_{\mathrm{\infty }}\le L$, whence [F2] puts $F^{\mathrm{\prime }}(u_Q^{\ast })$ in $L^{\mathrm{\infty }}(Q)$; with ${\mathrm{\partial }}_i{u}_Q^{\ast }\in {\mathcal{L}}^q(Q)$ from step 3.1, the case $(p,q,r)=(\mathrm{\infty },q,q)$ of [F14] gives $G\in {\mathcal{L}}^q(Q)$. All clauses of the characterisation [F5] hold for the class $F\circ u{\mathrm{\mid }}_Q$ with representative $\mathrm{\Phi }$, so $F\circ u{\mathrm{\mid }}_Q\in W^{1,q}(Q;\mathbb{R})$ and the weak derivative $D_i(F\circ u{\mathrm{\mid }}_Q)$ is represented almost everywhere by $G$. By the comparison of step 3.1, $G=h_i$ almost everywhere on $Q$, so $D_i(F\circ u{\mathrm{\mid }}_Q)$ is represented almost everywhere by $h_i$. The same argument applies to every coordinate direction simultaneously, because the single representative $u_Q^{\ast }$ of step 3.1 is ACL in every direction and its classical coordinate derivatives represent all $D_j(u{\mathrm{\mid }}_Q)$.

When $p=\mathrm{\infty }$ the exponent used so far is $q=1$, and we upgrade the conclusion of step 5.1 to $W^{1,\mathrm{\infty }}(Q)$ for this box. The class $F\circ u{\mathrm{\mid }}_Q$ lies in $L^{\mathrm{\infty }}(Q)$: $\mathrm{\mid }F(u)\mathrm{\mid }\le \mathrm{\mid }F(0)\mathrm{\mid }+L\mathrm{\mid }u\mathrm{\mid }$ pointwise and $\mathrm{\mid }u\mathrm{\mid }\le \mathrm{\parallel }u{\mathrm{\mid }}_Q{\mathrm{\parallel }}_{\mathrm{\infty }}$ almost everywhere on $Q$ by [F16], so $F(u)$ is bounded almost everywhere by $\mathrm{\mid }F(0)\mathrm{\mid }+L\mathrm{\parallel }u{\mathrm{\mid }}_Q{\mathrm{\parallel }}_{\mathrm{\infty }}$ and [F16] with [F2] gives $F\circ u{\mathrm{\mid }}_Q\in L^{\mathrm{\infty }}(Q)$. Each weak derivative class $D_j(F\circ u{\mathrm{\mid }}_Q)$ lies in $L^{\mathrm{\infty }}(Q)$: by step 5.1 it is represented by $h_j$, and $\mathrm{\mid }{h}_j\mathrm{\mid }=\mathrm{\mid }{F}^{\mathrm{\prime }}(\widehat{u})\mathrm{\mid }\mathrm{\mid }{\widehat{v}}_j\mathrm{\mid }\le L\mathrm{\parallel }{\widehat{v}}_j{\mathrm{\parallel }}_{\mathrm{\infty }}$ almost everywhere on $Q$ by [F16], so $h_j{\mathrm{\mid }}_Q\in L^{\mathrm{\infty }}(Q)$ by [F2]. Thus the class lies in $L^{\mathrm{\infty }}(Q)$ and every first weak derivative class has an $L^{\mathrm{\infty }}$ representative, which is membership in $W^{1,\mathrm{\infty }}(Q;\mathbb{R})$ by the definition [F1].

We patch the box conclusions into a global weak derivative identity. For the fixed direction $i$ and any test function $\phi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$, cover $\mathrm{\Omega }$ by the rational boxes of [F6] and use [F20] to choose a locally finite smooth partition of unity $({\chi }_k)$ subordinate to that cover, with compact supports. Only finitely many ${\chi }_k$ meet the compact support of $\phi$, and $\phi ={\sum }_k{\chi }_k\phi$; each summand ${\chi }_k\phi$ has compact support contained in some box $Q_k$ with $\overline{Q_k}\subseteq \mathrm{\Omega }$, so it is a test function on $Q_k$. step 5.1 and step 6.1 give, for every $\psi \in C_c^{\mathrm{\infty }}(Q_k)$, ${\int }_{Q_k}F(u){\mathrm{\partial }}_i\psi =-{\int }_{Q_k}{h}_i\psi$; applying this to the finitely many summands and summing with the linearity of the integral [F21] gives ${\int }_{\mathrm{\Omega }}F(u){\mathrm{\partial }}_i\phi =-{\int }_{\mathrm{\Omega }}{h}_i\phi$. Both sides are integrable: $u\in L^p(\mathrm{\Omega })$ is locally integrable, $F\circ u\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1(\mathrm{\Omega })$ because $\mathrm{\mid }F(u)\mathrm{\mid }\le \mathrm{\mid }F(0)\mathrm{\mid }+L\mathrm{\mid }u\mathrm{\mid }$ with a constant function on finite-measure pieces ([F12] and [F17]), and $h_i\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^p(\mathrm{\Omega })$ — for finite $p$ because $[h_i]\in L^p(\mathrm{\Omega })$ by step 2.1 and for $p=\mathrm{\infty }$ because $\mathrm{\mid }{h}_i\mathrm{\mid }\le L\mathrm{\parallel }{\widehat{v}}_i{\mathrm{\parallel }}_{\mathrm{\infty }}$ almost everywhere by [F16]. Since $\phi$ was arbitrary and the argument applies to every direction, the definition [F1] gives $D_i(F\circ u)=[h_i]=F^{\mathrm{\prime }}(u)D_{i}u$ weakly on $\mathrm{\Omega }$ for every $i$, hence almost everywhere on $\mathrm{\Omega }$.

We record the local membership. Let $U\subseteq \mathrm{\Omega }$ be open with $\bar{U}$ compact in $\mathrm{\Omega }$; then $U$ is bounded, so ${\lambda }_n(U)<\mathrm{\infty }$ by [F13]. The class $F\circ u{\mathrm{\mid }}_U$ lies in $L^p(U)$: for $p<\mathrm{\infty }$ the pointwise bound $\mathrm{\mid }F(u)\mathrm{\mid }\le \mathrm{\mid }F(0)\mathrm{\mid }+L\mathrm{\mid }u\mathrm{\mid }$, the second clause of [F15] applied to the constant and to the $L^{\mathrm{\infty }}$ case, and the vector-space clauses of [F17] show first that the bound function lies in ${\mathcal{L}}^p(U)$, and then monotonicity [F22] gives ${\int }_U\mathrm{\mid }F(u){\mathrm{\mid }}^p<\mathrm{\infty }$; for $p=\mathrm{\infty }$ the bound $\mathrm{\mid }F(u)\mathrm{\mid }\le \mathrm{\mid }F(0)\mathrm{\mid }+L\mathrm{\parallel }u{\mathrm{\mid }}_U{\mathrm{\parallel }}_{\mathrm{\infty }}$ holds almost everywhere on $U$ by [F16], so $F\circ u{\mathrm{\mid }}_U\in L^{\mathrm{\infty }}(U)$ by [F2]. Each weak derivative $D_i(F\circ u{\mathrm{\mid }}_U)$ is the restriction $(D_i(F\circ u)){\mathrm{\mid }}_U=[h_i{\mathrm{\mid }}_U]$ by the locality lemma [F8], and $h_i{\mathrm{\mid }}_U\in L^p(U)$: for finite $p$ this is the restriction of the class $[h_i]\in L^p(\mathrm{\Omega })$ from step 2.1, and for $p=\mathrm{\infty }$ we have $\mathrm{\mid }{h}_i\mathrm{\mid }\le L\mathrm{\parallel }{\widehat{v}}_i{\mathrm{\parallel }}_{\mathrm{\infty }}$ almost everywhere by [F16]. By the definition [F1], $F\circ u{\mathrm{\mid }}_U\in W^{1,p}(U;\mathbb{R})$ for every such $U$, that is, $F\circ u\in W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,p}(\mathrm{\Omega })$, and the identity of step 7.1, $D_i(F\circ u)=F^{\mathrm{\prime }}(u)D_{i}u$ almost everywhere on $\mathrm{\Omega }$, holds for every $i$.

The global membership criterion. If $F\circ u\in W^{1,p}(\mathrm{\Omega })$, then $F\circ u\in L^p(\mathrm{\Omega })$ by the definition [F1], and $F\circ u$ is the class of $F(u)$. Conversely, if $F(u)\in L^p(\mathrm{\Omega })$, then the class $F\circ u\in L^p(\mathrm{\Omega })$, and each derivative class $D_i(F\circ u)=[h_i]$ lies in $L^p(\mathrm{\Omega })$ by step 2.1; the definition [F1] then gives $F\circ u\in W^{1,p}(\mathrm{\Omega })$. This proves the equivalence of clause 2 of the Statement.

The automatic cases and the closing discussion. If $F(0)=0$, then $\mathrm{\mid }F(u)\mathrm{\mid }\le L\mathrm{\mid }u\mathrm{\mid }$ pointwise by [F10], so ${\int }_{\mathrm{\Omega }}\mathrm{\mid }F(u){\mathrm{\mid }}^p\le L^p{\int }_{\mathrm{\Omega }}\mathrm{\mid }u{\mathrm{\mid }}^p<\mathrm{\infty }$ by monotonicity [F22] when $p<\mathrm{\infty }$ and $\mathrm{\parallel }F(u){\mathrm{\parallel }}_{\mathrm{\infty }}\le L\mathrm{\parallel }u{\mathrm{\parallel }}_{\mathrm{\infty }}$ by [F16] when $p=\mathrm{\infty }$; in both cases $F(u)\in L^p(\mathrm{\Omega })$ by [F2], [F3] and [F4], and clause 2 of the Statement follows from step 8.2. If $p<\mathrm{\infty }$ and ${\lambda }_n(\mathrm{\Omega })<\mathrm{\infty }$, so that the restricted Lebesgue measure is finite in the sense of [F23], the constant $\mathrm{\mid }F(0)\mathrm{\mid }$ lies in $L^{\mathrm{\infty }}(\mathrm{\Omega })$ and hence in ${\mathcal{L}}^p(\mathrm{\Omega })$ by the second clause of [F15], while $L\mathrm{\mid }u\mathrm{\mid }\in {\mathcal{L}}^p(\mathrm{\Omega })$ by [F17]; the sum $\mathrm{\mid }F(0)\mathrm{\mid }+L\mathrm{\mid }u\mathrm{\mid }\in {\mathcal{L}}^p(\mathrm{\Omega })$ by [F17] dominates $\mathrm{\mid }F(u)\mathrm{\mid }$ pointwise, so ${\int }_{\mathrm{\Omega }}\mathrm{\mid }F(u){\mathrm{\mid }}^p<\mathrm{\infty }$ by [F22] and $F(u)\in L^p(\mathrm{\Omega })$. If $p=\mathrm{\infty }$, then $\mathrm{\mid }F(u)\mathrm{\mid }\le \mathrm{\mid }F(0)\mathrm{\mid }+L\mathrm{\parallel }u{\mathrm{\parallel }}_{\mathrm{\infty }}$ almost everywhere on $\mathrm{\Omega }$ by [F16], so $F(u)\in L^{\mathrm{\infty }}(\mathrm{\Omega })$ by [F2] and again clause 2 holds by step 8.2. Together with step 8.1 this proves clause 1, and step 8.2 and the present step prove clause 2. The case $n=1$ is included: the ACL definition [F6] then reads that one representative is absolutely continuous on every compact subinterval, and no transverse parameter occurs. If $\mathrm{\Omega }=\mathrm{\varnothing }$, then the rational-box family of [F6] is empty, the only class is zero, $F\circ u$ is the zero class, and all displayed assertions hold vacuously. The finite-many representative selections of step 2.1 need no choice principle; the only countable selection is that of the box representatives in step 3.1, licensed by the Countable Choice obtained in step 1.1 from the Axiom of Choice. $\mathrm{\square }$