Has any $L^1(\mathbb{R})$ function a distributional derivative?

Question

If I remember correctly, $L^1(\mathbb{R})$ functions can be identified with distributions via $$L^1(\mathbb{R}) \hookrightarrow D'(\mathbb{R})$$ defined as $f\mapsto T_f\in D'(\mathbb{R})$ and then $f=T_f$ as abuse of notation. It acts as follows: $$\langle T_f,\varphi\rangle= \int_{\mathbb{R}} f(x)\varphi(x)dx, \quad \varphi\in D(\mathbb{R}).$$

My question is, any function $f\in L^1(\mathbb{R})$ seen as a distribution, does it have a distributional derivative? If so, how does it act or look like?

Thanks!

Answer 1

Every distribution has a derivative, so every function in $L^1$ has a derivative in the sens of distributions. It is generally not a function, but a distribution, and it is defined as $$\langle T'_f, \phi \rangle = - \langle T_f, \phi' \rangle $$ Here's an example : take $f = \mathbf{1}_{[0,1]}$. Then, straightforward computation gives $\langle T'_f , \phi \rangle = -(\phi(1) - \phi(0) )$, so $$T'_f = \delta_0 - \delta_1 $$ where $\delta_a$ is the dirac distribution at the point $a$. This example is instructive, because here the derivative of the function $f$ (in the sense of distributions) IS NOT a function : you can't find a function $g \in L^1$ (or in any other space included in $L^1_{loc}$) such that $T_g = T'_f$.

Answer 2

Yes, in fact every function in $L^1_{loc}(\mathbb{R})$ (which is a larger space) has a distributional derivative. In general, the derivative will be simply a linear functional which acts on test functions and will not have a real-valued function as a representative.

The example of Herbert Quain does not have a real-valued function as a representative, but it is represented by a measure (Dirac measure). Often, it is not even the case that you can represent the derivative by a measure (try taking the weak derivative of a sample path of a Brownian motion for example).

Do you know anything about Sobolev spaces? For example the Sobolev space $W^{k,1}(\mathbb{R})$ can be viewed as a subspace of $L^1(\mathbb{R})$ where each function has $k$ weak derivatives each of which lie in $L^1(\mathbb{R})$.