I'm studying weak derivatives, with the following definition:
$f \in L_{loc}^1(\mathbb{R}^n)$ has $j$-th weak derivative if there is a function $g$ such that:
$$
\int g(x) \phi(x)\mathrm{d}x = – \int f(x) \partial_j\phi(x)\mathrm{d}x \quad\forall \phi \in \mathscr{D}(\mathbb{R}^n)
$$
where $\partial_j\triangleq\frac{\partial}{\partial x_j}$ is the usual partial derivative respect to the $x_j$ variable.
And I've seen examples when $n=1$ that there are functions in $L_{loc}^1(\mathbb{R})$ that don't have a weak derivative.
I was wondering if there are such explicit examples in $L^p(\mathbb{R}^n)$.
If there are I would appreciate to see them or a proof that they always have weak derivative if that's the case.
Best Answer
A simple example can be constructed as follows: consider the positive orthant $$ \Bbb R_+^n=\times_1^n[0,\infty[ $$ and the associated characteristic function, i.e. $$ \chi_{\Bbb R_+^n} (x)= \begin{cases} 1 & x\in \Bbb R_+^n\\ 0 & x\notin \Bbb R_+^n \end{cases}. $$ Then the function $$ g(x)= \chi_{\Bbb R_+^n} (x) e^{-\| x\|^2} $$ belongs to $L^p(\Bbb R^n)$ for all $1\le p\le\infty$ and also it is has no weak derivative, since for all $j=1,\ldots,n$, we have $$ \begin{split} \int\limits_{\Bbb R^n} g(x)\partial_j\phi(x)\mathrm{d}x &= \int\limits_{\Bbb R^n_+} e^{-\| x\|^2} \partial_j\phi(x)\mathrm{d}x \\ &= \int\limits_{\Bbb R^{n-1}_+} \mathrm{d}x_{\mathbf{n-j}} \int\limits_{0}^{+\infty} e^{-\| x\|^2} \partial_j\phi(x)\mathrm{d}x\\ &= \int\limits_{\Bbb R^{n-1}_+} \mathrm{d}x_{\mathbf{n-j}}\left[\phi\big(x_1,\ldots,\underset{j}{0},\ldots,x_n\big)e^{-\bigg(\sum\limits_{\overset{i=1}{ i\neq j}}^nx_i^2\bigg)}-\int\limits_{0}^{+\infty} \phi(x) \partial_j e^{-\| x\|^2} \mathrm{d}x \right]\\ &=\int\limits_{\Bbb R^{n-1}_+} \mathrm{d}x_{\mathbf{n-j}}\left[\delta_{x_j}\!\big(\phi(x)e^{-\| x\|^2}\big)-\int\limits_{0}^{+\infty} \phi(x) \partial_j e^{-\| x\|^2} \mathrm{d}x \right]\\ &\neq \int\limits_{\Bbb R^{n}} f(x)\phi(x)\mathrm{d}x \qquad \forall f \in L^p(\Bbb R^n),\; \end{split} $$ where