# Is there any nontrivial characterization of weakly differentiable functions?

When $$f\in L_\text{loc}^1$$, it's distributional derivative can be defined as $$D_{f'}\in\mathfrak{D}'$$, such that $$D_{f'}(\varphi)=-\int f\varphi'$$ for all $$\varphi\in\mathfrak{D}$$, where $$\mathfrak{D}$$ is the space of test functions. Then from what I understood, $$f$$ is said weakly differentiable, if there exists $$f'\in L_\text{loc}^1$$ such that $$D_{f'}(\varphi)=\int f'\varphi$$ for all $$\varphi\in\mathfrak{D}$$. It is also well-known fact that not all locally integrable functions are weakly differentiable, because $$L_\text{loc}^1$$ is proper subset (in sense of isomorphism) of $$\mathfrak{D}'$$.

I tried to picture these definitions into my head in visual ways (which is not necessary), but having hard time to fully characterize the definition of weak differentiability. What I can say is $$f$$ must not make any jumps. I also tried to understand the space on the space of distributions : if we let $$D(L_\text{loc}^1) \subset \mathfrak{D}'$$ as the set of all distributions such that there exists it's representation $$f\in L_\text{loc}^1$$, and $$d(L_\text{loc}^1)\subset\mathfrak{D}'$$ a set of all distributional derivatives, then the space of weakly differentiable distributions (which means it's representation in $$L_\text{loc}^1$$ is weakly differentiable) will be $$W(L_\text{loc}^1):=D(L_\text{loc}^1)\cap d(L_\text{loc}^1)$$. But I don't have any idea to characterize this space in other ways.

Here are the questions : For any given $$f\in L_\text{loc}^1$$,

1. Is there any sufficient condition (which is not too trivial, for example, $$f$$ is differentiable in classical sense) for $$f$$ to be weakly differentiable?
2. Is there any necessary condition (again, not trivial one) for $$f$$ to be weakly differentiable?
3. In other words, is there any nontrivial characterization of $$W(L_\text{loc}^1)$$?

• In one dimension, $f'$ (distributional derivative) being locally integrable is equivalent to $f$ being absolutely continuous. The result is discussed in my distribution lecture notes here: math.ou.edu/~cremling/teaching/ln.html – Christian Remling May 20 at 17:40
• The Sobolev space $W^{1,1}$ of $f\in L^1$ such that the distribution derivative $f'$ belongs also to $L^1$ is a nice space (by the way included in $L^{\frac{d}{d-1}}$). The space $BV$ of $f\in L^1$ such that the distribution derivative $f'$ is a Radon measure is larger (and nice as well). – Bazin May 20 at 19:48

Definition. If $$U\subset\mathbb{R}$$ is open, we say that $$u\in {AC}(U)$$ if $$u$$ is absolutely continuous on every compact interval in $$U$$. Let $$\Omega\subset\mathbb{R}^n$$. We say that $$u$$ is absolutely continuous on lines, $$u\in {ACL}(\Omega)$$, if the function $$u$$ is Borel measurable and for almost every line $$\ell$$ parallel to one of the coordinate axes, $$u|_\ell\in AC(\Omega\cap\ell)$$. Since absolutely continuous functions in dimension one are differentiable a.e., $$u\in {ACL}(\Omega)$$ has partial derivatives a.e.

Theorem. $$f\in L^1_{\rm loc}(\Omega)$$ has weak derivative $$\nabla f\in L^1_{\rm loc}(\Omega)$$ if and only if $$f\in ACL(\Omega)$$. Moreover the pointwise derivative of $$f$$ which exists a.e., equals to the weak derivative.

Remark. Functions that are equal a.e. are identified so by writing $$f\in ACL(\Omega)$$ we mean that $$f$$ equals a.e. to a function that belongs to $$ACL(\Omega)$$.

The above result is Theorem 4.21 in [EG] or Theorem 2.23 in [H] or Theorem 1 p. 4 and Theorem 2 p. 6 in [M].

[EG] L. C. Evans, R. F. Gariepy, Measure theory and fine properties of functions. Revised edition. Textbooks in Mathematics. CRC Press, Boca Raton, FL, 2015.

[M] V. Maz'ya, Sobolev spaces with applications to elliptic partial differential equations. Second, revised and augmented edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 342. Springer, Heidelberg, 2011.

• This is amazing! This is exactly what I was looking for. Can I ask for the reference of the theorem? – Jingeon An May 21 at 15:01
• @JingeonAn I added references. – Piotr Hajlasz May 21 at 16:58
• Thank you so much. – Jingeon An May 21 at 16:58
• @JingeonAn I have just changes the link. The previous one was not correct. – Piotr Hajlasz May 21 at 16:59