I'm currently studying the function $\log|x|$, which is locally integrable and defined on $\mathbb R \setminus \{0\}$. As a consequence, it can be regarded as a distribution on the space $C_c^\infty(\mathbb R\setminus\{0\}$). However, I'm interested in examining log|x| as a distribution on the space $C_c^\infty (\mathbb R)$.
Could anyone provide insights or methodology on how to successfully extend $\log|x|$ to a distribution on $C_c^\infty (\mathbb R)$?
Any references to relevant literature or theory would also be greatly appreciated. Thank you!
The function $\log|x|$ is locally integrable over all of $\mathbb{R}$, and as such it defines a distribution. Indeed, the only possible issue is at $x=0$, and we have $$\int_{-1}^{1}\big|\log|x|\big|dx=2.$$ This is enough information to show that integrating $\big|\log|\cdot|\big|$ returns a finite value over any compact subset of $\mathbb{R}$. (The fact that $\log$ is undefined/unspecified at zero is not important as $\{0\}$ is a null set, and so the value at zero does not change $\log|\cdot|$ as an element of $L^1_{\mathrm{loc}}$.)
Edit: Sorry for not including this in the original answer but I feel like there's a bit more to be said here. Specifically that what I've described above is not the only way that you can extend $\log|x|$ to the whole of $C_c^\infty(\mathbb{R})$. In order to keep track of things let me write $$T:C_c^\infty(\mathbb{R}\setminus\{0\})\to\mathbb{R}$$ for the original functional that integrates functions from $C_c^\infty(\mathbb{R}\setminus\{0\})$ against $\log|x|$. An extension $$S:C_c^\infty(\mathbb{R})\to\mathbb{R}$$ of $T$ to $C_c^\infty(\mathbb{R})$ should have the property that $$S(\phi)=T(\phi|_{\mathbb{R}\setminus\{0\}})$$ whenever $\phi\in C_c^\infty(\mathbb{R}\setminus\{0\})\subsetneqq C_c^\infty(\mathbb{R})$.
Recall that the dirac delta $\delta_0$ is defined as $$\delta_0:C_c^\infty(\mathbb{R})\to\mathbb{R},\phi\mapsto\phi(0)$$ and that the distributional derivatives $\delta^{(j)}_0$ of $\delta_0$ are defined by $\delta^{(j)}_0(\phi)=(-1)^{j}\phi^{(j)}(0)$ for $j\in\mathbb{N}$. As $\delta_0$ and its derivatives return $0$ for all $\phi\in C_c^\infty(\mathbb{R}\setminus\{0\})\to\mathbb{R}$ we get that $$S+\sum_{j=0}^nc_j\delta^{(j)}_0,c_j\in\mathbb{R}$$ is an extension of $T$ whenever $S$ is an extension of $T$. This means that $S$ is not unique at all.
Indeed, we can show that this exhausts all possibilities for $S$: if $S_0$ is an extension of $T$, like the one given in my original answer, then every other exetension takes the form $$S_0+\sum_{j=0}^nc_j\delta^{(j)}_0,c_j\in\mathbb{R}.$$ This is because if $S$ is another extension of $T$, then for any $\psi\in C_c^\infty(\mathbb{R}\setminus\{0\})$ we have that $$S(\psi)-S_0(\psi)=T(\psi)-T(\psi)=0.$$ To use the language of distribution theory, this means that $S-S_0$ is supported in $\{0\}$, or simply $\mathrm{supp}(S-S_0) \subseteq\{0\}$. There is a plausible, but somewhat technical theorem about "distributions with point support" that says that such distributions are within the $\mathbb{R}$-linear span of $\{\delta_0^{(j)}:j\in\mathbb{N}\}$ and so the claim follows, and we've shown that $$\log|\cdot|+\sum_{j=0}^nc_j\delta^{(j)}_0,$$ where $c_j\in\mathbb{R}$ are the only possible distributional extensions of $\log|\cdot|$ from $C_c^\infty(\mathbb{R}\setminus\{0\})$ to $C_c^\infty(\mathbb{R})$.
Hopefully there were enough keywords in this explanation to allow you to do some further reading. But two good references for this topic are "Distribution Theory and Fourier Transforms" by Strichartz, and "Distributions: Theory and Applications" by Duistermaat and Kolk. The first book provides a somewhat informal overview of the subject, and the second goes into more detail with many worked examples. Both of them are worth looking into.