Positive, negative, and truncated Sobolev functions

By [F15] and [F16] the Axiom of Choice yields Countable Choice and Dependent Choice in ZF, so the choice hypotheses of [F3], [F5], [F7], [F9] and [F23] are in force, while [F4] is stated under the Axiom of Choice itself. Representatives of $u$ and of $D_{i}u$ are measurable and finite almost everywhere by [F17]–[F19]; the functions $u^{+}$, $u^{-}$, $\mathrm{\mid }u\mathrm{\mid }$, $\mathrm{sgn}(u)$, $1_{\{u>0\}}$, $1_{\{u<0\}}$, $1_{\{\mathrm{\mid }u\mathrm{\mid }<M\}}$ and $T_{M}u$ are measurable by [F2], [F21] and [F22], because $\mathrm{sgn}$, the indicators and $T_M$ are Borel functions of a measurable function; and by [F18] and [F23] every class formed below is independent of the chosen representatives.

The corner family. For $\epsilon >0$ define $P_{\epsilon }:\mathbb{R}\to \mathbb{R}$ by $P_{\epsilon }(t):=0$ for $t\le 0$, $P_{\epsilon }(t):=t^2\mathrm{/}(2\epsilon )$ for $0\le t\le \epsilon$, and $P_{\epsilon }(t):=t-\epsilon \mathrm{/}2$ for $t\ge \epsilon$. On each of the three closed pieces $P_{\epsilon }$ is a polynomial function, so [F12] and [F13] make it differentiable at every interior point with $P_{\epsilon }^{\mathrm{\prime }}(t)=0$ for $t<0$, $P_{\epsilon }^{\mathrm{\prime }}(t)=t\mathrm{/}\epsilon$ for $0<t<\epsilon$ and $P_{\epsilon }^{\mathrm{\prime }}(t)=1$ for $t>\epsilon$; at the junction $t=0$ the two one-sided difference quotients are identically $0$ on the left and $h\mathrm{/}(2\epsilon )$ on the right, with limits $0$ and $0$, and at $t=\epsilon$ they are $1+h\mathrm{/}(2\epsilon )$ on the left and $1$ on the right, with limits $1$ and $1$, so [F14] gives differentiability at both junctions with the same formula; the derivative is continuous there since $t\mathrm{/}\epsilon \to 0$ and $t\mathrm{/}\epsilon \to 1$. Hence $P_{\epsilon }\in C^1(\mathbb{R})$ with $0\le P_{\epsilon }^{\mathrm{\prime }}\le 1$, the pointwise bounds $0\le P_{\epsilon }(t)\le t^{+}\le \mathrm{\mid }t\mathrm{\mid }$ and $\mathrm{\mid }{P}_{\epsilon }(t)-t^{+}\mathrm{\mid }\le \epsilon \mathrm{/}2$ hold for every real $t$, and $P_{\epsilon }^{\mathrm{\prime }}(t)\to 1_{\{t>0\}}$ as $\epsilon \downarrow 0$ for every real $t$, the value at $t=0$ being $0$ on both sides.

The chain rule on a relatively compact piece. Let $U\subseteq \mathrm{\Omega }$ be open with $\bar{U}$ compact and $\bar{U}\subseteq \mathrm{\Omega }$, and let $w\in W^{1,p}(U;\mathbb{R})$. Fix $k\ge 1$ and put $\epsilon :=1\mathrm{/}k$, $u_k:=P_{\epsilon }\circ w$ and $v_k:=P_{\epsilon }^{\mathrm{\prime }}(w)D_{i}w$. By step 1.2 the pointwise bounds $\mathrm{\mid }{P}_{\epsilon }(w)\mathrm{\mid }\le \mathrm{\mid }w\mathrm{\mid }$ and $\mathrm{\mid }{P}_{\epsilon }^{\mathrm{\prime }}(w)\mathrm{\mid }\le 1$ hold, so $u_k\in L^p(U)$ and $v_k\in L^p(U)$ by [F10], [F11] and [F17] for finite $p$ and by [F19] and [F20] for $p=\mathrm{\infty }$. Since $P_{\epsilon }\in C^1(\mathbb{R})$ has bounded derivative, clause 2 of [F4] applied on $U$ to the class $w$ gives $u_k\in W^{1,p}(U)$, and clause 1 gives $D_i{u}_k=v_k$ almost everywhere on $U$ for every $i$.

Limits along ${\epsilon }_k=1\mathrm{/}k$. On the same $U$, step 1.2 gives $\mathrm{\mid }{u}_k-w^{+}\mathrm{\mid }\le {\epsilon }_k\mathrm{/}2$ pointwise, so $u_k\to w^{+}$ in $L^p(U)$: for $p<\mathrm{\infty }$, [F10] and [F11] give ${\int }_U\mathrm{\mid }{u}_k-w^{+}{\mathrm{\mid }}^p\le ({\epsilon }_k\mathrm{/}2{)}^p{\int }_{U}1$, and ${\int }_{U}1<\mathrm{\infty }$ because $U$ is bounded, hence of finite measure, by [F9] with the constant function converted by [F8]; for $p=\mathrm{\infty }$, [F20] gives $\mathrm{\parallel }{u}_k-w^{+}{\mathrm{\parallel }}_{\mathrm{\infty }}\le {\epsilon }_k\mathrm{/}2$. Also $v_k\to v:=1_{\{w>0\}}{D}_{i}w$ in $L^q(U)$, where $q:=p$ for $p<\mathrm{\infty }$ and $q:=1$ for $p=\mathrm{\infty }$: by step 1.2, $P_{{\epsilon }_k}^{\mathrm{\prime }}(w)\to 1_{\{w>0\}}$ pointwise, so $v_k\to v$ pointwise and $\mathrm{\mid }{v}_k-v\mathrm{\mid }\le 2\mathrm{\mid }{D}_{i}w\mathrm{\mid }$; and $(2\mathrm{\mid }{D}_{i}w\mathrm{\mid }{)}^q$ is integrable on $U$, by [F10] and [F11] from ${\int }_U\mathrm{\mid }{D}_{i}w{\mathrm{\mid }}^p<\mathrm{\infty }$ when $p<\mathrm{\infty }$, and by [F20] together with $\lambda (U)<\mathrm{\infty }$ when $p=\mathrm{\infty }$ (in which case $D_{i}w\in L^1(U)$ by [F8]). Hence [F6] applied to the functions $\mathrm{\mid }{v}_k-v{\mathrm{\mid }}^q$ gives ${\int }_U\mathrm{\mid }{v}_k-v{\mathrm{\mid }}^q\to 0$. The class $v$ is well defined by [F18] and [F23].

Weak stability and membership. The hypotheses of [F5] hold on $U$ with exponents $p$ and $q$, sequences $(u_k)$ and $(v_k)$, limits $w^{+}$ and $v$, and $\alpha =e_i$: all four classes lie in the required local $L^p$ and $L^q$ spaces with $\mathrm{\mid }{v}_k\mathrm{\mid },\mathrm{\mid }v\mathrm{\mid }\le \mathrm{\mid }{D}_{i}w\mathrm{\mid }$, the two convergences are those of step 2.2, and each $v_k$ is a weak derivative of $u_k$ by step 2.1. Hence $D_i{w}^{+}=v$ weakly on $U$. Moreover $w^{+}\in L^p(U)$ because $\mathrm{\mid }{w}^{+}\mathrm{\mid }\le \mathrm{\mid }w\mathrm{\mid }$, and $v\in L^p(U)$ because $\mathrm{\mid }v\mathrm{\mid }\le \mathrm{\mid }{D}_{i}w\mathrm{\mid }$, both by [F10], [F11] and [F17] for finite $p$ and by [F19], [F20] for $p=\mathrm{\infty }$; so [F1] gives $w^{+}\in W^{1,p}(U)$ with $D_i{w}^{+}=v$ almost everywhere on $U$.

Truncation inputs on a relatively compact piece. Fix $U$ as in step 2.1. By [F9] $\lambda (U)<\mathrm{\infty }$, so the constant function $M$ lies in $L^p(U)$ by [F8] (trivially for $p=\mathrm{\infty }$), and its classical derivative $0$ is its weak derivative on $U$ by [F7]. By [F1] and [F3], $u{\mathrm{\mid }}_U\in W^{1,p}(U)$ with $D_i(u{\mathrm{\mid }}_U)=(D_{i}u){\mathrm{\mid }}_U$; hence the classes $u-M$ and $-M-u$ lie in $W^{1,p}(U)$ and satisfy $D_i(u-M)=(D_{i}u){\mathrm{\mid }}_U$ and $D_i(-M-u)=-(D_{i}u){\mathrm{\mid }}_U$ weakly on $U$ by [F3]; since $U$ was arbitrary, $u-M$ and $-M-u$ lie in $W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,p}(\mathrm{\Omega })$ with $D_i(u-M)=D_{i}u$ and $D_i(-M-u)=-D_{i}u$ weakly on $\mathrm{\Omega }$.

Local form. Every class $w\in W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,p}(\mathrm{\Omega };\mathbb{R})$ has $w{\mathrm{\mid }}_U\in W^{1,p}(U)$ for every $U$ as in step 2.1 by [F1], so step 2.1, step 2.2 and step 3.1 apply to $w{\mathrm{\mid }}_U$ and, $U$ being arbitrary, yield $w^{+}\in W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,p}(\mathrm{\Omega })$ with $D_i{w}^{+}=1_{\{w>0\}}{D}_{i}w$ almost everywhere on $\mathrm{\Omega }$. If in addition $w\in W^{1,p}(\mathrm{\Omega })$, then $\mathrm{\mid }{w}^{+}\mathrm{\mid }\le \mathrm{\mid }w\mathrm{\mid }\in L^p(\mathrm{\Omega })$ and $\mathrm{\mid }{1}_{\{w>0\}}{D}_{i}w\mathrm{\mid }\le \mathrm{\mid }{D}_{i}w\mathrm{\mid }\in L^p(\mathrm{\Omega })$, so [F1] upgrades the membership to $w^{+}\in W^{1,p}(\mathrm{\Omega })$ with the same derivative class.

Negative part, absolute value and the level set. The class $-u$ lies in $W^{1,p}(\mathrm{\Omega })$ with $D_i(-u)=-D_{i}u$ by [F3]. Applying step 4.1 to $w:=u$ and to $w:=-u$ gives $u^{+},u^{-}\in W^{1,p}(\mathrm{\Omega })$ with $D_i{u}^{+}=1_{\{u>0\}}{D}_{i}u$ and $D_i{u}^{-}=1_{\{-u>0\}}{D}_i(-u)=-1_{\{u<0\}}{D}_{i}u$ almost everywhere on $\mathrm{\Omega }$. By [F2], $\mathrm{\mid }u\mathrm{\mid }=u^{+}+u^{-}$ and $u=u^{+}-u^{-}$ pointwise, and both identities hold as classes; $\mathrm{\mid }u\mathrm{\mid }\in L^p(\mathrm{\Omega })$ because $\mathrm{\mid }\mathrm{\mid }u\mathrm{\mid }\mathrm{\mid }=\mathrm{\mid }u\mathrm{\mid }$, so [F1] and [F3] give $\mathrm{\mid }u\mathrm{\mid }\in W^{1,p}(\mathrm{\Omega })$ with $D_i\mathrm{\mid }u\mathrm{\mid }=D_i{u}^{+}+D_i{u}^{-}=(1_{\{u>0\}}-1_{\{u<0\}})D_{i}u=\mathrm{sgn}(u)D_{i}u$ almost everywhere, and $D_{i}u=D_i{u}^{+}-D_i{u}^{-}=(1_{\{u>0\}}+1_{\{u<0\}})D_{i}u=(1-1_{\{u=0\}})D_{i}u$ almost everywhere, that is, $1_{\{u=0\}}{D}_{i}u=0$ almost everywhere on $\mathrm{\Omega }$; the same argument applies verbatim to every class of $W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,p}(\mathrm{\Omega })$, with all identities holding almost everywhere on $\mathrm{\Omega }$ and membership in $W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,p}(\mathrm{\Omega })$.

Truncation. By step 3.2 the classes $w_1:=u-M$ and $w_2:=-M-u$ lie in $W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,p}(\mathrm{\Omega })$ with $D_i{w}_1=D_{i}u$ and $D_i{w}_2=-D_{i}u$ weakly on $\mathrm{\Omega }$, so step 4.1 applied to them gives $w_1^{+}=(u-M{)}^{+},w_2^{+}=(-M-u{)}^{+}\in W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,p}(\mathrm{\Omega })$ with $D_i(u-M{)}^{+}=1_{\{u>M\}}{D}_{i}u$ and $D_i(-M-u{)}^{+}=-1_{\{u<-M\}}{D}_{i}u$ almost everywhere on $\mathrm{\Omega }$; the level-set argument of step 5.1 applied to $w_1$ and $w_2$ gives $1_{\{u=M\}}{D}_{i}u=0$ and $1_{\{u=-M\}}{D}_{i}u=0$ almost everywhere on $\mathrm{\Omega }$. By [F2], $T_{M}u=u-(u-M{)}^{+}+(-M-u{)}^{+}$ pointwise, so by [F3] $T_{M}u\in W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,p}(\mathrm{\Omega })$ with $D_i{T}_{M}u=(D_{i}u)-1_{\{u>M\}}{D}_{i}u-1_{\{u<-M\}}{D}_{i}u=1_{\{-M\le u\le M\}}{D}_{i}u$ almost everywhere on $\mathrm{\Omega }$; since $1_{\{-M\le u\le M\}}-1_{\{\mathrm{\mid }u\mathrm{\mid }<M\}}=1_{\{\mathrm{\mid }u\mathrm{\mid }=M\}}$ and $\{\mathrm{\mid }u\mathrm{\mid }=M\}=\{u=M\}\cup \{u=-M\}$, the two level-set identities turn this into $D_i{T}_{M}u=1_{\{\mathrm{\mid }u\mathrm{\mid }<M\}}{D}_{i}u$ almost everywhere on $\mathrm{\Omega }$.

Global membership, cases and accounting. The pointwise bound $\mathrm{\mid }{T}_{M}u\mathrm{\mid }\le \mathrm{\mid }u\mathrm{\mid }$ holds by the case check $u\le -M$, $-M\le u\le M$, $u\ge M$, and $\mathrm{\mid }{1}_{\{\mathrm{\mid }u\mathrm{\mid }<M\}}{D}_{i}u\mathrm{\mid }\le \mathrm{\mid }{D}_{i}u\mathrm{\mid }$; hence $T_{M}u\in L^p(\mathrm{\Omega })$ and $1_{\{\mathrm{\mid }u\mathrm{\mid }<M\}}{D}_{i}u\in L^p(\mathrm{\Omega })$ by [F10] and [F11] for finite $p$ and by [F19], [F20] for $p=\mathrm{\infty }$, and [F1] with the identity of step 6.1 gives $T_{M}u\in W^{1,p}(\mathrm{\Omega })$ with $D_i{T}_{M}u=1_{\{\mathrm{\mid }u\mathrm{\mid }<M\}}{D}_{i}u$ almost everywhere on $\mathrm{\Omega }$. Together with step 5.1 this proves every membership and derivative assertion. For $M=0$, [F2] gives $T_{0}u=u-u^{+}+(-u{)}^{+}=u-u^{+}+u^{-}=0$, and the derivative formula reads $0=0$. The exponents $p=1$ and $p=\mathrm{\infty }$ were included, with $q=p$ for finite $p$ and $q=1$ for $p=\mathrm{\infty }$, and the case $n=1$ needs no change. If $\mathrm{\Omega }=\mathrm{\varnothing }$, the only class is the zero class and all assertions hold vacuously by [F1]. The only selection made is that of finitely many representatives, and the approximation uses the single sequence ${\epsilon }_k=1\mathrm{/}k$, so the Axiom of Choice is used exactly through [F15] and the stated hypotheses of [F3], [F4], [F5], [F7], [F9] and [F23]. $\mathrm{\square }$