Proof of The Chain Rule for Weak Derivatives on the Torus

Each result cited below is universally quantified over the data appearing in its own statement, and is applied here to the data named in the step in question. For a real number $y$ we write $y^{2}$ for $y\,y$.

Preliminary remarks.

(R1) For $y\in\mathbb{R}$ with $0\le y$ one has $|y|=y$: by claim 1 of Properties of the Absolute Value in an Ordered Field either $|y|=y$ or $|y|=-y$, and in the second case $0\le|y|=-y$ gives $y\le0$ by claim 4 of Elementary Order Arithmetic in an Ordered Field, whence $y=0$ by antisymmetry of the order and $|y|=-0=0=y$ by claim 4 of Additive Cancellation and Elementary Additive Identities in a Field.

(R2) For $t,\delta\in\mathbb{R}$ with $0\le t$ and $0<\delta$ one has $t^{2}<\delta^{2}$ if and only if $t<\delta$. Suppose $t<\delta$: claim 5 of Elementary Arithmetic in an Ordered Field, with the nonnegative multiplier $t$, gives $t\,t\le\delta\,t$, and claim 10 of Elementary Order Arithmetic in an Ordered Field, with the positive multiplier $\delta$, gives $\delta\,t<\delta\,\delta$; claim 2 of the latter lemma combines these into $t^{2}<\delta^{2}$. If instead $t<\delta$ fails then $\delta\le t$, the order of $\mathbb{R}$ being a total order by Ordered Field, and the same two steps with the roles of $t$ and $\delta$ exchanged give $\delta^{2}\le t^{2}$, so that $t^{2}<\delta^{2}$ fails.

(R3) If $(a_{m})_{m\in\mathbb{N}}$ is a sequence of nonnegative real numbers such that $(a_{m}^{2})_{m\in\mathbb{N}}$ converges to $0$, then $(a_{m})_{m\in\mathbb{N}}$ converges to $0$. Indeed, let $\eta\in\mathbb{R}$ be positive; then $\eta^{2}$ is positive by claim 5 of Elementary Order Arithmetic in an Ordered Field, so there is a natural number $K$ with $|a_{m}^{2}-0|<\eta^{2}$ for every $m$ with $K\le m$; since $a_{m}^{2}$ is nonnegative, (R1) turns this into $a_{m}^{2}<\eta^{2}$, and (R2) into $a_{m}<\eta$, that is $|a_{m}-0|<\eta$.

Claim 1. Let $s,t\in\mathbb{R}$. Both sides of the asserted inequality are unchanged when $s$ and $t$ are interchanged, because $|y|=|-y|$ for every real $y$ by claim 2 of Properties of the Absolute Value in an Ordered Field; and the order of $\mathbb{R}$ is total, as recalled in (R2). We may therefore assume $s\le t$. If $s=t$ both sides are $0$ and the inequality holds. So assume $s<t$, and let $p=s-1$ and $q=t+1$. Since $0<1$ by claim 6 of Elementary Order Arithmetic in an Ordered Field, claim 1 of that lemma gives $p<s$ and $t<q$; hence $s$ and $t$ belong to the open interval $(p,q)$. By An Open Interval is an Interval All of Whose Points Are Interior that open interval is an interval all of whose points are interior points of it, so by claim 2 of Restriction Stability of Continuity and of the Derivative, applied with $I=\mathbb{R}$ and $J=(p,q)$, the restriction $\phi|_{(p,q)}$ is differentiable at every point $x_{0}$ of $(p,q)$ with derivative $\phi'(x_{0})$. Hence Mean Value Theorem on an Open Interval applies to $\phi|_{(p,q)}$ with $a=s$ and $b=t$ and yields a $c\in(s,t)$ with

$$\phi(t)-\phi(s)=\phi'(c)\,(t-s).$$

Taking absolute values, claim 4 of Properties of the Absolute Value in an Ordered Field gives $|\phi(t)-\phi(s)|=|\phi'(c)|\,|t-s|$, and since $|\phi'(c)|\le M$, claim 5 of Elementary Arithmetic in an Ordered Field, applied with the nonnegative multiplier $|t-s|$, gives $|\phi(t)-\phi(s)|\le M\,|t-s|$, which is the assertion.

Step 2: $\phi$ and $\phi'$ are sequentially continuous on $\mathbb{R}$. Since $\phi$ is of class $C^{1}$ on $\mathbb{R}^{1}$, claim 3 of Euclidean Space Is Open in Itself shows that its single coordinate function, namely $\phi$ itself, is continuous in the Euclidean sense at every point of $\mathbb{R}^{1}$; and clause 1 of Functions of Class Cᵏ on an Open Subset of ℝⁿ includes that the function $\partial_{1}\phi$, which is $\phi'$, is continuous in that same sense at every point of $\mathbb{R}^{1}$. By Euclidean, Metric and Sequential Continuity of a Real Function of a Real Variable (iv), both $\phi$ and $\phi'$ are therefore sequentially continuous on $\mathbb{R}$ in the sense required by Sequentially Continuous Functions of Measurable Euclidean Maps are Measurable.

Claim 2. Let $U\in L^{2}(\mathbb{T}^{n})$ and let $u$ be a representative of $U$. By Power-Integrable Functions and the p-Seminorm (ii) the function $u:Q\to\mathbb{R}$ is measurable with respect to $\mathcal{B}_{Q}$ and the Borel $\sigma$-algebra of the real line. Applying Sequentially Continuous Functions of Measurable Euclidean Maps are Measurable with the measurable space $(Q,\mathcal{B}_{Q})$, with $d=1$ and $E=\mathbb{R}$, with $f=u$ and with the sequentially continuous function $\phi$ of step 2, we conclude that $\phi\circ u$ is measurable.

By claim 1, $|\phi(u(x))-\phi(0)|\le M\,|u(x)|$ for every $x\in Q$, so the triangle inequality, claim 5 of Properties of the Absolute Value in an Ordered Field, gives

$$|\phi(u(x))|\le M\,|u(x)|+|\phi(0)|\qquad\text{for every }x\in Q .$$

The function $|u|$ is measurable by claim 4 of Arithmetic of Measurable Real- and [0,∞]-Valued Maps and belongs to $\mathcal{L}^{2}(\mathbb{T}^{n})$ with $\lVert\,|u|\,\rVert_{2}=\lVert u\rVert_{2}$, by two applications of Elementary Properties of the p-Seminorm (iii); hence $M\,|u|\in\mathcal{L}^{2}(\mathbb{T}^{n})$ by Elementary Properties of the p-Seminorm (ii). The function on $\mathbb{R}^{n}$ with constant value $|\phi(0)|$ is smooth by claim 2 of Constants, Affine Functions, Sums, Products and Quotients of Cᵏ Functions, hence continuous at every point by claim 3 of Euclidean Space Is Open in Itself, and it is periodic because its values at any two points coincide; so it lies in $C_{\mathrm{per}}$ by Lattice-Periodic Functions and the Periodic Function Classes (iii) and its restriction to $Q$ lies in $\mathcal{L}^{2}(\mathbb{T}^{n})$ by Continuous Periodic Functions are Power-Integrable and Dense on the Torus (i). By Minkowski's Inequality the pointwise sum $M\,|u|+|\phi(0)|$ therefore lies in $\mathcal{L}^{2}(\mathbb{T}^{n})$; it is nonnegative, so by (R1) it equals its own absolute value, and Elementary Properties of the p-Seminorm (iii) gives $\phi\circ u\in\mathcal{L}^{2}(\mathbb{T}^{n})$.

Finally, if $\tilde{u}$ is a further representative of $U$ then $u$ and $\tilde{u}$ agree almost everywhere by The Lebesgue Space Lᵖ(μ) (i); at every point where they agree so do $\phi\circ u$ and $\phi\circ\tilde{u}$, so these two functions agree almost everywhere, and The Lebesgue Space Lᵖ(μ) (i) gives $[\phi\circ u]=[\phi\circ\tilde{u}]$.

Claim 3, the product bound. Let $U\in H^{1}(\mathbb{T}^{n})$ with representative $u$, let $j\in[n]$ and let $g_{j}$ be a representative of $\partial_{j}U$. By step 2 and Sequentially Continuous Functions of Measurable Euclidean Maps are Measurable, $\phi'\circ u$ is measurable, and so is the pointwise product $(\phi'\circ u)\,g_{j}$ by claim 3 of Arithmetic of Measurable Real- and [0,∞]-Valued Maps. For every $x\in Q$, claim 4 of Properties of the Absolute Value in an Ordered Field and claim 5 of Elementary Arithmetic in an Ordered Field, applied with $|\phi'(u(x))|\le M$ and the nonnegative multiplier $|g_{j}(x)|$, give

$$\bigl|(\phi'\circ u)(x)\,g_{j}(x)\bigr|=|\phi'(u(x))|\,|g_{j}(x)|\le M\,|g_{j}(x)| .$$

As above $M\,|g_{j}|\in\mathcal{L}^{2}(\mathbb{T}^{n})$ with $\lVert M\,|g_{j}|\,\rVert_{2}=M\,\lVert g_{j}\rVert_{2}$, by Elementary Properties of the p-Seminorm (iii) and Elementary Properties of the p-Seminorm (ii) together with $|M|=M$ from (R1). Since $M\,|g_{j}|$ is nonnegative it equals its absolute value by (R1), so Elementary Properties of the p-Seminorm (iii) gives $(\phi'\circ u)\,g_{j}\in\mathcal{L}^{2}(\mathbb{T}^{n})$ with $\lVert(\phi'\circ u)\,g_{j}\rVert_{2}\le M\,\lVert g_{j}\rVert_{2}$.

Step 5: the mollified approximations. Keep $U$, $u$ and the $g_{j}$ of claim 3. Let $\rho$ be a mollifier kernel of radius $1$ on $\mathbb{R}^{n}$, which exists by Existence of Mollifier Kernels of Every Radius, and for a natural number $k$ let $u_{k}=\rho_{1/k}\star u$ and $U_{k}=[u_{k}|_{Q}]$ be as fixed in The Sobolev Space on the Torus is a Separable Hilbert Space Densely Embedded in the Square-Integrable Space (iii). By The Sobolev Space on the Torus is a Separable Hilbert Space Densely Embedded in the Square-Integrable Space (iv) we have $u_{k}\in C^{\infty}_{\mathrm{per}}$ and $U_{k}\in H^{1}(\mathbb{T}^{n})$. By Elementary Properties of the Weak Partial Derivative on the Torus (i), applied to $u_{k}$, which lies in $C^{1}_{\mathrm{per}}$ because a smooth map is of class $C^{1}$ by Smooth Functions on an Open Subset of ℝⁿ and periodicity is the same condition for the two classes by Lattice-Periodic Functions and the Periodic Function Classes (iii), the class $g_{k,j}=[(\partial_{j}u_{k})|_{Q}]$ is the $j$-th weak partial derivative of $U_{k}$ in $L^{2}(\mathbb{T}^{n})$; by the uniqueness recorded in The Weak Partial Derivative on the Torus (ii) it is $\partial_{j}U_{k}$, and $(\partial_{j}u_{k})|_{Q}$ is a representative of it. By The Sobolev Space on the Torus is a Separable Hilbert Space Densely Embedded in the Square-Integrable Space (vi) the sequence $(U_{k})_{k\in\mathbb{N}}$ converges to $U$ in $H^{1}(\mathbb{T}^{n})$, so by the distance inequalities of The Sobolev Space on the Torus is a Separable Hilbert Space Densely Embedded in the Square-Integrable Space (i) together with claim 3 of Order Properties of Limits of Real Sequences the sequences $(U_{k})_{k\in\mathbb{N}}$ and $(\partial_{j}U_{k})_{k\in\mathbb{N}}$ converge to $U$ and to $\partial_{j}U$ in $L^{2}(\mathbb{T}^{n})$, for every $j\in[n]$.

Step 6: the chain rule for the approximations. Fix a natural number $k$. The set $\mathbb{R}^{n}$ is open in itself, and so is $\mathbb{R}^{1}$, by claim 1 of Euclidean Space Is Open in Itself; the map $u_{k}$ is smooth on $\mathbb{R}^{n}$, hence of class $C^{1}$ there by Smooth Functions on an Open Subset of ℝⁿ, and takes its values in $\mathbb{R}^{1}$. Applying Composites of Cᵏ Functions Are of Class Cᵏ with $F=u_{k}$, with $G=\phi$, and with $m=p=1$: claim 2 of that theorem gives that $\phi\circ u_{k}$ is of class $C^{1}$ on $\mathbb{R}^{n}$, and claim 1, the sum over the single index being its one term by claim 1 of Peeling, Splitting, and Interchange for Sums over a Finite Index Set, gives

$$\partial_{j}(\phi\circ u_{k})(x)=\partial_{1}\phi\bigl(u_{k}(x)\bigr)\,\partial_{j}u_{k}(x)=\phi'\bigl(u_{k}(x)\bigr)\,\partial_{j}u_{k}(x)$$

for every $j\in[n]$ and every $x\in\mathbb{R}^{n}$. Moreover $\phi\circ u_{k}$ is periodic, since $(\phi\circ u_{k})(x+z)=\phi(u_{k}(x+z))=\phi(u_{k}(x))$ for every $x\in\mathbb{R}^{n}$ and every $z$ in the lattice, by Lattice-Periodic Functions and the Periodic Function Classes (ii) applied to $u_{k}$. Hence $\phi\circ u_{k}\in C^{1}_{\mathrm{per}}$ by Lattice-Periodic Functions and the Periodic Function Classes (iii), and Elementary Properties of the Weak Partial Derivative on the Torus (i) shows that, for every $j\in[n]$, the class

$$H_{k,j}=\bigl[\bigl((\phi'\circ u_{k})\,\partial_{j}u_{k}\bigr)|_{Q}\bigr]$$

is the $j$-th weak partial derivative of $[(\phi\circ u_{k})|_{Q}]$ in $L^{2}(\mathbb{T}^{n})$. Since $u_{k}|_{Q}$ is a representative of $U_{k}$, the class $[(\phi\circ u_{k})|_{Q}]$ is the class $\phi(U_{k})$ of claim 2.

Step 7: passage to the limit. Write $h_{j}=(\phi'\circ u)\,g_{j}$, which lies in $\mathcal{L}^{2}(\mathbb{T}^{n})$ by claim 3, and $h_{k,j}=\bigl((\phi'\circ u_{k})\,\partial_{j}u_{k}\bigr)|_{Q}$, so that $H_{k,j}=[h_{k,j}]$.

(i) $\phi(U_{k})\to\phi(U)$ in $L^{2}(\mathbb{T}^{n})$. By claim 1, $|\phi(u_{k}(x))-\phi(u(x))|\le M\,|u_{k}(x)-u(x)|$ for every $x\in Q$; the function $\phi\circ u_{k}|_{Q}-\phi\circ u$ is measurable, because claim 2, applied to $U_{k}$ with representative $u_{k}|_{Q}$ and to $U$ with representative $u$, places $\phi\circ u_{k}|_{Q}$ and $\phi\circ u$ in $\mathcal{L}^{2}(\mathbb{T}^{n})$, hence in the measurable functions by Power-Integrable Functions and the p-Seminorm (ii), and differences of measurable functions are measurable by claim 2 of Arithmetic of Measurable Real- and [0,∞]-Valued Maps; the function $u_{k}|_{Q}-u$ lies in $\mathcal{L}^{2}(\mathbb{T}^{n})$ by Minkowski's Inequality (ii), and hence so does $M\,(u_{k}|_{Q}-u)$, so Elementary Properties of the p-Seminorm (iii), Elementary Properties of the p-Seminorm (ii) and (R1) give

$$\bigl\lVert\phi\circ u_{k}|_{Q}-\phi\circ u\bigr\rVert_{2}\le M\,\bigl\lVert u_{k}|_{Q}-u\bigr\rVert_{2}.$$

By The Riesz–Fischer Theorem (i) the two sides are the $L^{2}$ distances from $\phi(U_{k})$ to $\phi(U)$ and from $U_{k}$ to $U$, multiplied by $M$ in the second case. The latter converges to $0$ by step 5, hence so does $M$ times it by Arithmetic of Limits of Real Sequences, and therefore the former converges to $0$ by claim 3 of Order Properties of Limits of Real Sequences. Thus $(\phi(U_{k}))_{k\in\mathbb{N}}$ converges to $\phi(U)$ in $L^{2}(\mathbb{T}^{n})$.

(ii) A subsequence along which the derivative terms converge. By The Riesz–Fischer Theorem (iii), applied to the sequence $(U_{k})_{k\in\mathbb{N}}$ converging to $U$ in $L^{2}(\mathbb{T}^{n})$ with the representatives $u_{k}|_{Q}$ and $u$, there are natural numbers $k_{1}<k_{2}<\dots$ and a null set $N$ such that for every $x\in Q\setminus N$ the sequence $(u_{k_{m}}(x))_{m\in\mathbb{N}}$ converges to $u(x)$. Fix $j\in[n]$ and let $x\in Q\setminus N$. By Euclidean, Metric and Sequential Continuity of a Real Function of a Real Variable (i) the Euclidean distance from $u_{k_{m}}(x)$ to $u(x)$ is $|u_{k_{m}}(x)-u(x)|$, so the hypothesis of Euclidean, Metric and Sequential Continuity of a Real Function of a Real Variable (iv) is met and step 2 gives that $(\phi'(u_{k_{m}}(x)))_{m\in\mathbb{N}}$ converges to $\phi'(u(x))$. By Arithmetic of Limits of Real Sequences and claim 4 of Order Properties of Limits of Real Sequences, the sequence

$$\Bigl(\bigl|\bigl(\phi'(u_{k_{m}}(x))-\phi'(u(x))\bigr)\,g_{j}(x)\bigr|^{2}\Bigr)_{m\in\mathbb{N}}$$

then converges to $0$. This holds for almost every $x\in Q$. Moreover, for every $x\in Q$ and every $m$, claim 4 of Properties of the Absolute Value in an Ordered Field, the triangle inequality and $|\phi'|\le M$ give $\bigl|\bigl(\phi'(u_{k_{m}}(x))-\phi'(u(x))\bigr)g_{j}(x)\bigr|\le 2M\,|g_{j}(x)|$, whence, by two applications of claim 5 of Elementary Arithmetic in an Ordered Field to nonnegative numbers, the $m$-th term above is at most $(2M)^{2}\,|g_{j}(x)|^{2}$. The function $|g_{j}|^{2}$ is measurable by claims 3 and 4 of Arithmetic of Measurable Real- and [0,∞]-Valued Maps and integrable, because $\int_{\mathbb{T}^{n}}|g_{j}|^{2}\,dx=(\lVert g_{j}\rVert_{2})^{2}$ is a real number by Elementary Properties of the p-Seminorm (i); hence so is $(2M)^{2}|g_{j}|^{2}$, by claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion. Each term of the sequence is itself measurable: the representative $u_{k_{m}}|_{Q}$ of $U_{k_{m}}$ is measurable by Power-Integrable Functions and the p-Seminorm (ii), so $\phi'\circ u_{k_{m}}|_{Q}$ is measurable by step 2 and Sequentially Continuous Functions of Measurable Euclidean Maps are Measurable, and claims 2, 3 and 4 of Arithmetic of Measurable Real- and [0,∞]-Valued Maps then give measurability of the difference with $\phi'\circ u$, of its product with $g_{j}$, of the absolute value and of the square. By The Lebesgue Integral and Null Sets (vii), applied with these data and with the limit function $0$,

$$\int_{\mathbb{T}^{n}}\bigl|\bigl(\phi'\circ u_{k_{m}}|_{Q}-\phi'\circ u\bigr)\,g_{j}\bigr|^{2}\,dx\longrightarrow0 .$$

By Elementary Properties of the p-Seminorm (i) the left-hand side is the square of $\bigl\lVert(\phi'\circ u_{k_{m}}|_{Q}-\phi'\circ u)\,g_{j}\bigr\rVert_{2}$, so by (R3) that seminorm converges to $0$.

(iii) Conclusion. Fix $j\in[n]$. For every $m$,

$$h_{k_{m},j}-h_{j}=\bigl(\phi'\circ u_{k_{m}}|_{Q}\bigr)\bigl((\partial_{j}u_{k_{m}})|_{Q}-g_{j}\bigr)+\bigl(\phi'\circ u_{k_{m}}|_{Q}-\phi'\circ u\bigr)\,g_{j}.$$

Write $v_{m}=(\partial_{j}u_{k_{m}})|_{Q}-g_{j}$, which lies in $\mathcal{L}^{2}(\mathbb{T}^{n})$ by Minkowski's Inequality (ii). The first summand is $(\phi'\circ u_{k_{m}}|_{Q})\,v_{m}$; it is measurable, by the chain of measurability recorded in (ii), and at every $x\in Q$ claim 4 of Properties of the Absolute Value in an Ordered Field and claim 5 of Elementary Arithmetic in an Ordered Field give $\bigl|(\phi'\circ u_{k_{m}}|_{Q})(x)\,v_{m}(x)\bigr|\le M\,|v_{m}(x)|$, so Elementary Properties of the p-Seminorm (iii) and Elementary Properties of the p-Seminorm (ii), with $|M|=M$ from (R1), bound its $\mathcal{L}^{2}$ seminorm by $M\,\lVert v_{m}\rVert_{2}$; that quantity is $M$ times the $L^{2}$ distance from $\partial_{j}U_{k_{m}}$ to $\partial_{j}U$ and converges to $0$, because $(\partial_{j}U_{k})_{k}$ converges to $\partial_{j}U$ by step 5 and a subsequence of a convergent sequence has the same limit by A Subsequence of a Convergent Sequence Has the Same Limit. The second summand has seminorm converging to $0$ by (ii). By Minkowski's Inequality and claim 3 of Order Properties of Limits of Real Sequences, $\lVert h_{k_{m},j}-h_{j}\rVert_{2}$ converges to $0$, that is, $(H_{k_{m},j})_{m\in\mathbb{N}}$ converges to $[h_{j}]$ in $L^{2}(\mathbb{T}^{n})$.

By step 6, $H_{k_{m},j}$ is the $j$-th weak partial derivative of $\phi(U_{k_{m}})$ for every $m$; by (i) and A Subsequence of a Convergent Sequence Has the Same Limit the sequence $(\phi(U_{k_{m}}))_{m}$ converges to $\phi(U)$ in $L^{2}(\mathbb{T}^{n})$. Hence Elementary Properties of the Weak Partial Derivative on the Torus (v) applies and shows that $[h_{j}]$ is the $j$-th weak partial derivative of $\phi(U)$ in $L^{2}(\mathbb{T}^{n})$. As $j\in[n]$ was arbitrary, $\phi(U)\in H^{1}(\mathbb{T}^{n})$ by The Sobolev Space of Once Weakly Differentiable Square-Integrable Classes on the Torus (i), and $\partial_{j}\phi(U)=[h_{j}]=[(\phi'\circ u)\,g_{j}]$ for every $j\in[n]$.