A function $f \in L^1(\mathbb R^n)$ is said to be of *bounded variation* if there exists a constant $C \geq 0$ such that
$$
\int\_{\mathbb R^n} f(x) \operatorname{div} \phi(x) \; dx
\leq 
C \sup\_{ x \in \mathbb R^n } \lvert\phi(x)\rvert
$$
for all compactly supported differentiable vector fields $\phi : \mathbb R^n \rightarrow \mathbb R^n$. The smallest such constant $C$ can be called the total variation of $f$.


I am wondering about the requirement that $f \in L^1(\mathbb R^n)$. For example, we could replace the requirement “$f \in L^1(\mathbb R^n)$” by “$f \in L^\infty(\mathbb R^n)$” and still obtain a reasonable theory.


For example, compactly support smooth functions are essentially bounded and have total variation. The total variation of constant functions is zero, and constant functions are clearly in $L^\infty$ but not $L^1$.


Hence, if consider the subspace of functions in $L^p(\mathbb R^n)$ with bounded variation (defined analogously to above), then at the very least we get non-trivial vector spaces for any $1 \leq p \leq \infty$, and there is no obvious inclusion between those spaces.


**Question:** Why is $L^1$ usually required? Are there exceptions to this discussed in the literature?


I am aware of the functions of *local bounded variation*, where the function $f$ is merely expected to be Lebesgue integrable. But that big generalization already seems like a step way too far.


Among the possible reasons I can imagine are:


* Authors working on the subject are only interested in analysis over bounded domains, in which case $f \in L^1$ suffices.
* The (historical) application in elasticity actually prefers $L^1$.
* There are actual mathematical reasons to require $f \in L^1$ that are much less obvious than the simple musings above.



What happens if we consider functions of bounded variation that are not in $L^1$?
The main historical reason for which the requirement $f\in L^1$ enters in the definition of $BV$ is that functions of bounded variation (*tout court*) of several variables were introduced by Lamberto Cesari building upon previous work by Leonida Tonelli in order to solve the problem of characterization of (hyper)surfaces of finite (Lebesgue) area (see reference [1] for a complete description of their work). When measuring the surface area (i.e. the perimeter) of a bounded domain $G\in\Bbb R^n$, then $f=\chi\_G$ (where $\chi\_G$ is the characteristic function of the domain $G$), thus the requirements of simple absolute integrability of $f$ suit very naturally in that context.  

Another, more practical reason is that, by simply dropping the requirement $f\in L^1$ while keeping the other one, you get the space of functions of locally finite variation $BV\_\text{loc}$: this space is at once more general than $BV$ and $BV^p$, $p>1$ (as are called the spaces defined by requiring $f\in L^p$, $p>1$): even if it is not Banach, this space is not extremely difficult to manage, thus working in it allows more generality without the need of a really cumbersome technical machinery.


**Notes: brief (scant...) literature survey**


* The space $BV^2$ as described in [3], chapter 5, §3.1-3.7 pp. 214-227, is not defined by using functions $f\in L^2$ but functions $f\in BV$ such that $\partial\_if\in L^2$ whose (inward) trace on their domain of definition $G$ (this implies that the theory must be developed on bounded domains of finite perimeter) is square summable too. Then, from this starting point the Authors go on by defining a scalar product, proving its completeness and the compactness of the embedding operator and a few other results. However, I am not aware of other applications of this space than the ones described in this book.
* The only fields where $BV^p$ functions currently play an active role seem to be approximation theory and analysis of nonlinear integral equations (for functions of a single variable). A nice annotated bibliography, updated up to 1998 and which includes also works on functions of bounded $\phi$-variation, has been published in [2], part VI, pp. 241-272.


**References**


[1] Lamberto Cesari, *Surface area* (English), Annals of Mathematics Studies No. 35, Princeton: Princeton University Press, pp. X+595 (1956), [MR0074500](https://mathscinet.ams.org/mathscinet-getitem?mr=MR0074500), [Zbl 0073.04101](https://zbmath.org/?q=an%3A0073.04101).


[2] Richard M. Dudley, Rimas Norvaiša, *Differentiability of six operators on nonsmooth functions and $p$-variation. With the collaboration of Jinghua Qian* (English), Lecture Notes in Mathematics, 1703, Berlin: Springer, pp. viii+277 (1999), ISBN 3-540-65975-7, [MR1705318](https://mathscinet.ams.org/mathscinet-getitem?mr=MR1705318), [Zbl 0973.46033](https://zbmath.org/?q=an%3A0973.46033).


[3] Aizik Isaakovich Vol'pert and Sergei Ivanovich Hudjaev (1985), *[Analysis in classes of discontinuous functions and equations of mathematical physics](https://books.google.com/books?id=lAN0b0-1LIYC&printsec=frontcover&dq=%22Analysis+in+classes+of+discontinuous+functions%22)*, Mechanics: analysis, 8, Dordrecht–Boston–Lancaster: Martinus Nijhoff Publishers, pp. xviii+678, ISBN 90-247-3109-7, [MR 0785938](https://www.ams.org/mathscinet-getitem?mr=0785938), [Zbl 0564.46025](https://zbmath.org/?q=an%3A0564.46025).


