> [!definition] > > Let $U \subset \real^d$ be open, $V$ be a vector space, and $f: U \to V$ be a [[Function|function]]. For any $r > 0$, $1 \le j \le d$ and $h \in (-r, r)$, define > $ > D^h_jf(x) = \frac{f(x + he_j) - f(x)}{h} > $ > on $V = \bracs{x \in U: x + \eps e_j \in U \forall \eps \in (-r, r)}$ as the $j$-th **difference quotient** of size $h$. Define > $ > D^h f = (D^h_1f, \cdots, D^h_df) > $ > on $V = \bracs{x \in U: x + \eps e_j \in U \forall \eps \in (-r, r), 1 \le j \le d}$ as the **difference quotient** of $f$. > [!theorem] > > Let $f \in C^1(U)$, then $D_j^hf \to D_j f$ [[Uniform Convergence on Compact Sets|uniformly on compact sets]]. > [!theorem] > > Let $f \in \cd(U)$ be a [[Space of Test Functions|test function]], then $D^h_j f \to D_jf$ in $\cd(U)$ as $h \to 0$. > > *Proof.* For sufficiently small $h$, all the difference quotients are compactly supported in the same set. The difference quotient of $C^1$ functions converges [[Uniform Convergence on Compact Sets|uniformly on compact sets]], so the above convergence is uniform. Moreover, for any [[Multi-Index|multi-index]] $\alpha$, > $ > \partial^{\alpha + e_j}\phi(x) = \limv{h}\frac{\partial^\alpha \phi(x + he_j) - \partial^\alpha \phi(x)}{h} > $ > with the same uniform convergence. > [!theorem] > > Let $f \in \loci(U)$ and $\phi \in \cd(U)$ be a test function, then for all $h \in \real$ such that $\supp{\phi} + he_j \subset U$, > $ > \angles{D^h_jf, \phi} = -\anglesn{f, D^{-h}_j\phi} > $ > *Proof*. > $ > \begin{align*} > \int_U \phi(x) \frac{f(x + he_j) - f(x)}{h}dx &= \frac{1}{h}\int_U \phi(x)[f(x + he_j) - f(x)]dx \\ > &= \frac{1}{h}\braks{\int_U \phi(x) f(x + he_j)dx - \int_U \phi(x)f(x)dx} \\ > &= \frac{1}{h}\braks{\int_U \phi(x - he_j)f(x)dx - \int_U \phi(x)f(x)dx} \\ > &= -\frac{1}{-h}\braks{\int_U \phi(x - he_j)f(x)dx - \int_U \phi(x)f(x)dx} \\ > &= -\anglesn{f, D_j^{-h}\phi} > \end{align*} > $ > [!theorem] > > Let $f, g: U \to V$ be functions, then > $ > D_j^h(fg) = (\tau_{-he_j}f)D_j^hg + gD_j^hf > $ > *Proof*. > $ > \begin{align*} > \frac{f(x + he_j)g(x + he_j) - f(x)g(x)}{h} &= \frac{f(x + he_j)[g(x + he_j) - g(x)]}{h} \\ > &+ \frac{g(x)[f(x + he_j) - f(x)]}{h} \\ > &= (\tau_{-he_j}f)D_j^hg + gD_j^hf > \end{align*} > $ > [!theorem] > > Let $f \in L^p_{\text{loc}}(U)$, $1 \le p < \infty$, and $1 \le j \le d$, then the following are equivalent: > 1. $D_j f$ exists and is in $L^p_{\text{loc}}(U)$. > 2. $D^h_jf$ converges in $L^p_{\text{loc}}(U)$ as $h \to 0$. > 3. For each $V \subset \subset U$, $D_j^h$ converges [[Weak Topology|weakly]] in $L^p(V)$ as $h \to 0$. > > where $D^h_j f \to D_jf$ in $(2)$ and $(3)$. If $p \ne 1$, then the following is equivalent to the above: > > 4. For each $V \subset \subset U$, there exists $r > 0$ such that $\sup_{h \in (-r, r)}\normn{D_j^hf}_{L^p(V)} < \infty$. > > > *Proof*. Suppose that $(1)$ holds. Let $V \subset \subset U$, then there exists $r > 0$ such that $V' = \bigcup_{h \in (-r, r)}(V + he_j) \subset \subset U$. Suppose that $f \in C^1(V')$, then by the [[Mean Value Theorem]] and [[Jensen's Inequality]], > $ > \begin{align*} > {f(x + he_j) - f(x)} &= h\int_0^1{D_jf(x + the_j)}dt \\ > D^h_jf(x) &= \int_0^1 D_jf(x + the_j)dt \\ > \abs{D_j^hf(x) - D_jf(x)} &\le \int_0^1 \abs{D_jf(x + the_j) - D_jf(x)}dt \\ > \norm{D_j^hf - D_jf}_{L^p(V)}^p &\le \int_0^1 \norm{\tau_{-the_j}D_jf - D_jf}_{L^p(V)}^pdt \\ > \norm{D_j^hf - D_jf}_{L^p(V)} &\le \sup_{t \in [0, 1]}\norm{\tau_{-the_j}D_jf - D_jf}_{L^p(V)} \le 2\norm{D_jf}_{L^p(U)} > \end{align*} > $ > By continuity of translation in $L^p$, $D_j^hf \to D_jf$ in $L^p$ as $h \to 0$. Suppose that $f \in W^{1, p}(U)$ is arbitrary, then there exists $\seq{f_n} \subset C^1(V')$ such that $f_n \to f$ in $W^{1, p}(V')$, in which case, > $ > \begin{align*} > \norm{D_j^hf - D_jf}_{L^p(V)} &\le \norm{D^h_jf_n - D^h_jf}_{L^p(V)} + \norm{D_jf_n - D_jf}_{L^p(V)} \\ > &+ \norm{D^h_jf_n - D_jf_n}_{L^p(V)} \\ > &\le 2\norm{D_jf}_{L^p(U)} + \norm{D^h_jf_n - D^h_jf}_{L^p(V)} \\ > &+ \norm{D_jf_n - D_jf}_{L^p(V)} > \end{align*} > $ > Sending $n \to \infty$ yields the same bound. > > $(2) \Rightarrow (3) \Rightarrow (4)$ directly. > > Suppose that $(3)$ holds. Let $V \subset \subset U$ and $g$ be the weak limit of $D^h_jf$ in $L^p(V)$, then using integration by parts, for any $\phi \in \cd(V)$, > $ > \begin{align*} > \angles{f, D_j \phi} &= \lim_{h \to 0}\angles{f, D_j^h \phi} = -\lim_{h \to 0}\anglesn{D^{-h}_jf, \phi} \\ > &= -\lim_{h \to 0}\anglesn{D^{h}_jf, \phi} = -\angles{g, \phi} > \end{align*} > $ > so $g = D_jf$. Since limits are unique a.e. in the above topologies, convergence in $(2)$, $(3)$, and $(4)$ are all to $D_jf$. > > Suppose that $(4)$ holds, then since $L^p$ is reflexive for $1 < p < \infty$, by [[Tychonoff's Theorem]], there exists a convergent subsequence, converging to $D_jf$.