What happens if we consider functions of bounded variation that are not in $L^1$?

Question

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.

Answer 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, Zbl 0073.04101.

[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, Zbl 0973.46033.

[3] Aizik Isaakovich Vol'pert and Sergei Ivanovich Hudjaev (1985), Analysis in classes of discontinuous functions and equations of mathematical physics, Mechanics: analysis, 8, Dordrecht–Boston–Lancaster: Martinus Nijhoff Publishers, pp. xviii+678, ISBN 90-247-3109-7, MR 0785938, Zbl 0564.46025.

Answer 2

Although I agree with the above answer, I would like to point out the important scaling properties linked to $BV$ functions. Let us consider a function $u$ in $L^1_{\text{loc}}(\mathbb R^n)$ such that its distribution gradient is a Radon measure with finite total mass. It is then possible to prove that $$ u\in L^{\frac{n}{n-1}}(\mathbb R^n), $$ and the isoperimetric inequality is even giving that $$ \Vert u\Vert_{L^{\frac{n}{n-1}}(\mathbb R^n)}\le C_n\times\text{total mass}(\nabla u), \quad C_n=(\vert \mathbb B^n\vert_n)^{\frac{n-1}{n}}/\vert \mathbb S^{n-1}\vert_{n-1}.\label{1}\tag{1} $$ As a consequence, the initial hypothesis on $u$ is here to make sure that the distribution derivative of $u$ makes sense and I guess that the above properties are fulfilled assuming that $u$ is a distribution on $\mathbb R^n$ whose distribution gradient is a Radon measure with finite total mass. It is then natural to consider in fact as $BV$ functions the functions in $L^{\frac{n}{n-1}}(\mathbb R^n)$ whose distribution gradient is a Radon measure with finite total mass. The inequality \eqref{1} is important and its scaling is not negotiable: the total mass of the gradient is controlling the $L^{\frac{n}{n-1}}$ norm and no other $L^p$ norm, at least if you do not want that the constant depends on the size of the support of $u$.