Proof of Mode Derivatives of Linear Combinations and of Restrictions of Twice Differentiable Functions on a Negative Sobolev Space

Each result cited is universally quantified over the data in its own statement.

Elementary arithmetic, order and absolute values of real numbers are used freely under The Real Numbers: Standing Notation and Background §background. Differentiability of a function $f:\mathbb{R}\to\mathbb{R}$ at a point $t_{0}$, and its derivative $f'(t_{0})$, are those of Single-Variable Calculus on an Interval §derivative with $I=\mathbb{R}$, as fixed in Derivatives of a Function on the Sobolev Space of Order -1 along the Fourier Modes; every point of $\mathbb{R}$ is an interior point of $\mathbb{R}$, and the condition "$t_{0}+h\in I$" in that definition is automatic. For $\varphi:H^{-1}\to\mathbb{R}$, $x\in H^{-1}$ and $k\in\mathbb{Z}^{n}$ we write $\sigma^{\varphi}_{x,k}$ for the section $t\mapsto\varphi(x+te_{k})$ of Derivatives of a Function on the Sobolev Space of Order -1 along the Fourier Modes.

Step 1 (Clause linear: combinations). Let $\varphi,\psi$, $a,b$ be as in clause 1, put $\chi=a\varphi+b\psi$, and fix $x\in H^{-1}$ and $k\in\mathbb{Z}^{n}$. Since functions are combined pointwise, $\sigma^{\chi}_{x,k}(t)=a\,\sigma^{\varphi}_{x,k}(t)+b\,\sigma^{\psi}_{x,k}(t)$ for every $t\in\mathbb{R}$, that is, $\sigma^{\chi}_{x,k}=a\sigma^{\varphi}_{x,k}+b\sigma^{\psi}_{x,k}$ as functions on $\mathbb{R}$. By Derivatives of a Function on the Sobolev Space of Order -1 along the Fourier Modes §twice, $\sigma^{\varphi}_{x,k}$ and $\sigma^{\psi}_{x,k}$ are differentiable at every $t\in\mathbb{R}$, and their derivatives are differentiable at $0$. By claim 2 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives (the sum and constant multiple rules, applied on the interval $\mathbb{R}$, first to the two multiples and then to their sum), $\sigma^{\chi}_{x,k}$ is differentiable at every $t\in\mathbb{R}$ with

$$(\sigma^{\chi}_{x,k})'(t)=a\,(\sigma^{\varphi}_{x,k})'(t)+b\,(\sigma^{\psi}_{x,k})'(t),$$

so that $(\sigma^{\chi}_{x,k})'=a(\sigma^{\varphi}_{x,k})'+b(\sigma^{\psi}_{x,k})'$ as functions on $\mathbb{R}$; and the same claim, applied at the point $0$ to these derivative functions, shows that $(\sigma^{\chi}_{x,k})'$ is differentiable at $0$ with

$$\bigl((\sigma^{\chi}_{x,k})'\bigr)'(0)=a\,\bigl((\sigma^{\varphi}_{x,k})'\bigr)'(0)+b\,\bigl((\sigma^{\psi}_{x,k})'\bigr)'(0).$$

As $x$ and $k$ were arbitrary, $\chi$ is twice differentiable along the modes (Derivatives of a Function on the Sobolev Space of Order -1 along the Fourier Modes §twice), and evaluating the first display at $t=0$ and reading both displays through Derivatives of a Function on the Sobolev Space of Order -1 along the Fourier Modes §derivatives gives the two asserted identities.

Step 2 (Clause linear: constants). Let $\varphi$ have constant value $c\in\mathbb{R}$, and fix $x$ and $k$. Then $\sigma^{\varphi}_{x,k}$ is the constant function with value $c$ on $\mathbb{R}$, which by claim 1 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives is differentiable at every $t\in\mathbb{R}$ with derivative $0$. So $(\sigma^{\varphi}_{x,k})'$ is the constant function with value $0$, which by the same claim is differentiable at $0$ with derivative $0$. Hence $\varphi$ is twice differentiable along the modes, and $\partial_{k}\varphi(x)=0$ and $\partial_{k}^{2}\varphi(x)=0$ by Derivatives of a Function on the Sobolev Space of Order -1 along the Fourier Modes §derivatives.

Step 3 (Clause sobolev: setup). Let $m$ and $\Phi$ be as in clause 2, and write $\langle\cdot,\cdot\rangle$ and $|\cdot|$ for the inner product and norm of $H^{-m}=H^{-m}(\mathbb{T}^{n})$. Both $H^{-1}$ and $H^{-m}$ are linear subspaces of $\mathrm{Map}(\mathbb{Z}^{n},\mathbb{R})$ under the pointwise operations (The Negative-Order Sobolev Spaces of the Torus §space), and $H^{-1}\subseteq H^{-m}$ as recorded in the statement; so for $x\in H^{-1}$ and $t\in\mathbb{R}$ the vector $x+te_{k}$ is the same family whether formed in $H^{-1}$ or in $H^{-m}$, and the section of $\Phi|_{H^{-1}}$ at $x$ along $k$ is $\sigma(t)=\Phi(x+te_{k})$. The set $H^{-m}$ is open in itself (Real Hilbert Spaces: Standing Notation and Background §topology). Since $\Phi$ is of class $C^{2}$ on $H^{-m}$, it is of class $C^{1}$ there, hence differentiable at every point of $H^{-m}$, and has a second derivative at every point of $H^{-m}$ (The Classes $C^1$ and $C^2$ on an Open Subset of a Real Inner Product Space §c2, The Classes $C^1$ and $C^2$ on an Open Subset of a Real Inner Product Space §c1). Fix $x\in H^{-1}$ and $k\in\mathbb{Z}^{n}$, and put $c=|e_{k}|$, a nonnegative real number; then $0<c+1$ and $0<c^{2}+1$.

Step 4 (Clause sobolev: the first derivative). Let $t_{0}\in\mathbb{R}$ and $y=x+t_{0}e_{k}\in H^{-m}$, and let $p=D\Phi(y)$, which by Fréchet Differentiability and the Gradient on an Open Subset of a Real Inner Product Space §gradient has the property of Fréchet Differentiability and the Gradient on an Open Subset of a Real Inner Product Space §differentiable at $y$. We show that $\sigma$ is differentiable at $t_{0}$ with derivative $\langle p,e_{k}\rangle$. Let $\varepsilon>0$ be given; the choices are made in the order $\varepsilon'$, then $\delta$, then $\delta'$. Put $\varepsilon'=\varepsilon/(c+1)$, which is positive, and let $\delta>0$ be the radius that Fréchet Differentiability and the Gradient on an Open Subset of a Real Inner Product Space §differentiable provides for $\Phi$ at $y$ with gradient $p$ and $\varepsilon'$ in place of $\varepsilon$; put $\delta'=\delta/(c+1)>0$. Let $h\in\mathbb{R}$ with $0<|h|<\delta'$, and $z=he_{k}$. By Elementary Identities in a Real Inner Product Space §homogeneity, $|z|=|h|\,c\le|h|(c+1)<\delta$, so

$$\bigl|\Phi(y+z)-\Phi(y)-\langle p,z\rangle\bigr|\le\varepsilon'\,|h|\,c .$$

Here $y+z=x+(t_{0}+h)e_{k}$ by the vector space axioms, so $\Phi(y+z)=\sigma(t_{0}+h)$ and $\Phi(y)=\sigma(t_{0})$, and $\langle p,z\rangle=h\langle p,e_{k}\rangle$ by Elementary Identities in a Real Inner Product Space §bilinear. Dividing by $|h|>0$,

$$\Bigl|\frac{\sigma(t_{0}+h)-\sigma(t_{0})}{h}-\langle p,e_{k}\rangle\Bigr|\le\varepsilon'c=\varepsilon\,\frac{c}{c+1}<\varepsilon .$$

Hence $\sigma$ is differentiable at every $t_{0}\in\mathbb{R}$, and

$$\sigma'(t_{0})=\langle D\Phi(x+t_{0}e_{k}),e_{k}\rangle\qquad(t_{0}\in\mathbb{R});$$

in particular $\partial_{k}(\Phi|_{H^{-1}})(x)=\sigma'(0)=\langle D\Phi(x),e_{k}\rangle$ by Derivatives of a Function on the Sobolev Space of Order -1 along the Fourier Modes §derivatives.

Step 5 (Clause sobolev: the second derivative). Since $\Phi$ has a second derivative at $x$, The Second Derivative and the Hessian on an Open Subset of a Real Inner Product Space §expansion provides a form $b\in\mathrm{Sym}(H^{-m})$ and a positive $\rho$ such that $b$ is a second derivative of $\Phi$ at $x$ (with $B_{d}(x,\rho)\subseteq H^{-m}$ and $\Phi$ differentiable on that ball); by The Second Derivative and the Hessian on an Open Subset of a Real Inner Product Space §hessian, $b=D^{2}\Phi(x)$. We show that $\sigma'$ is differentiable at $0$ with derivative $b(e_{k},e_{k})$. Let $\varepsilon>0$ be given; the choices are made in the order $\varepsilon'$, then $\delta$, then $\delta'$. Put $\varepsilon'=\varepsilon/(c^{2}+1)>0$, let $\delta$ (positive, with $\delta\le\rho$) be the radius that The Second Derivative and the Hessian on an Open Subset of a Real Inner Product Space §expansion provides for $b$ with $\varepsilon'$ in place of $\varepsilon$, and put $\delta'=\delta/(c+1)>0$. Let $h\in\mathbb{R}$ with $0<|h|<\delta'$, and $w=he_{k}$; as in Step 4, $|w|=|h|c<\delta$. Applying the expansion inequality with this $w$ and with $e_{k}$ in the second slot,

$$\bigl|\langle D\Phi(x+w)-D\Phi(x),e_{k}\rangle-b(he_{k},e_{k})\bigr|\le\varepsilon'\,|h|\,c\cdot c .$$

By Step 4 and Elementary Identities in a Real Inner Product Space §bilinear, $\langle D\Phi(x+w)-D\Phi(x),e_{k}\rangle=\sigma'(h)-\sigma'(0)$; and $b(he_{k},e_{k})=h\,b(e_{k},e_{k})$ by the homogeneity condition of Bounded Symmetric Bilinear Forms on a Real Inner Product Space: Norm, Order, Identity Form and Restriction §form. Dividing by $|h|>0$,

$$\Bigl|\frac{\sigma'(h)-\sigma'(0)}{h}-b(e_{k},e_{k})\Bigr|\le\varepsilon'c^{2}=\varepsilon\,\frac{c^{2}}{c^{2}+1}<\varepsilon .$$

So $\sigma'$ is differentiable at $0$ with $(\sigma')'(0)=b(e_{k},e_{k})=D^{2}\Phi(x)(e_{k},e_{k})$.

Step 6 (Conclusion of clause sobolev). Steps 4 and 5 hold for every $x\in H^{-1}$ and $k\in\mathbb{Z}^{n}$, so $\Phi|_{H^{-1}}$ is twice differentiable along the modes (Derivatives of a Function on the Sobolev Space of Order -1 along the Fourier Modes §twice), and by Derivatives of a Function on the Sobolev Space of Order -1 along the Fourier Modes §derivatives

$$\partial_{k}(\Phi|_{H^{-1}})(x)=\langle D\Phi(x),e_{k}\rangle_{H^{-m}},\qquad\partial_{k}^{2}(\Phi|_{H^{-1}})(x)=D^{2}\Phi(x)(e_{k},e_{k}).$$