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2020 Mathematics Subject Classification: Primary: 26A45 [MSN][ZBL] (Functions of one variable)

2020 Mathematics Subject Classification: Primary: 26B30 Secondary: 28A1526B1549Q15 [MSN][ZBL] (Functions of severable variables)

Functions of one variable

Classical definition

Let $I\subset \mathbb R$ be an interval. A function $f: I\to \mathbb R$ is said to have bounded variation if its total variation is bounded. The total variation is defined in the following way.

Definition 1 Let $I\subset \mathbb R$ be an interval and consider the collection $\Pi$ of ordered $2N$-ples points $a_1<b_1<a_2< b_2 < \ldots < a_N<b_N\in I$, where $N$ is an arbitrary natural number. The total variation of a function $f: I\to \mathbb R$ is given by \begin{equation}\label{e:TV} TV\, (f) := \sup \left\{ \sum_{i=1}^N |f(b_i)-f(a_i)| : (a_1, \ldots, b_N)\in\Pi\right\}\, \end{equation} (cp. with Section 4.4 of [Co] or Section 10.2 of [Ro]).

Generalizations

The definition of total variation of a function of one real variable can be easily generalized when the target is a metric space $(X,d)$: it suffices to substitute $|f(b_i)-f(a_i)|$ with $d (f(a_i), f(b_i))$ in \ref{e:TV}. Consequently, one defines functions of bounded variation taking values in an arbitrary metric space. Observe that, if $f:I\to X$ is a function of bounded variation and $\varphi:X\to Y$ a Lipschitz map, then $\varphi\circ f$ is also a function of bounded variation and \[ TV\, (\varphi\circ f) \leq {\rm Lip (\varphi)}\, TV\, (f)\, , \] where ${\rm Lip}\, (\varphi)$ denotes the Lipschitz constant of $\varphi$.

As a corollary we derive

Proposition 2 A function $(f^1, \ldots, f^k) = f: I\to \mathbb R^k$ is of bounded variation if and only if each coordinate function $f^j$ is of bounded variation.

General properties

Jordan decomposition

A fundamental characterization of functions of bounded variation of one variable is due to Jordan.

Theorem 3 Let $I\subset \mathbb R$ be an interval. A function $f: I\to\mathbb R$ has bounded variation if and only if it can be written as the difference of two bounded nondecreasing functions.

(Cp. with Theorem 4 of Section 5.2 in [Ro]). Indeed it is possible to find a canonical representation of any function of bounded variation as difference of nondecreasing functions.

Theorem 4 If $f:[a,b] \to\mathbb R$ is a function of bounded variation then there is a pair of nondecreasing functions $f^+$ and $f^-$ such that $f= f^+- f^-$ and $TV (f) = f^+ (b)-f^+ (a) + f^- (b)- f^- (a)$. The pair is unique up to addition of a constant, i.e. if $g^+$ and $g^-$ is a second pair with the same property, then $g^+-g^-=f^+-f^-\equiv {\rm const}$.

(Cp. with Theorem 3 of Section 5.2 in [Ro]). The latter representation of a function of bounded variation is also called Jordan decomposition.

Continuity

It follows immediately from Theorem 3 that

Proposition 5 If $f:I\to [a,b]$ is a function of bounded variation, then

• The right and left limits

\[ f (x^+) :=\lim_{y\downarrow x} f (y) \qquad f (x^-):= \lim_{y\uparrow x} f(y) \] exist at every point $x\in I$;

• The set of points of discontinuity of $f$ is at most countable.

Warning 6 However, according to the definitions given above, it may happen that at a goven point right and left limits coincide, but nonetheless the function $f$ is discontinuous. For instance the function $f:\mathbb R\to\mathbb R$ given by \[ f (x) =\left\{\begin{array}{ll} 1 \qquad &\mbox{if '"`UNIQ-MathJax34-QINU`"'}\\ 0 \qquad &\mbox{otherwise} \end{array}\right. \] is a function of bounded variation

Precise representative

In order to avoid patologies as in Warning 6 it is customary to postulate some additional assumptions for functions of bounded variations. Two popular choices are

• the imposition of right (resp. left) continuity, i.e. at any point $x$ we impose $f(x)=f (x^+)$ (resp. $f(x)=f(x^-$), cp. with Section 4.4 of [Co];
• at any point $x$ we impose $f(x) =\frac{1}{2} (f(x^+) + f(x^-))$.

The latter is perhaps more popular because of the Jordan criterion (see Theorem 11 below) and it is often called precise representative.

Differentiability

Functions of bounded variation of one variable are classically differentiable at a.e. point of their domain of definition, cp. with Corollary 5 of Section 5.2 in [Ro]. It turns out that such derivative is always a summable function (see below in the section Structure theorem). However, the fundamental theorem of calculus does not apply in this case, i.e. there are functions $f:[a,b]\to\mathbb R$ of bounded variation such that the identity \[ f(b') - f(a') =\int_{a'}^{b'} f' (t)\, dt \] fails for a set of pairs $(b', a')\in I\times I$ of positive measure (see below in the section Examples).

Measure theoretic characterization

Classically right-continuous functions of bounded variations can be mapped one-to-one to signed measures. More precisely, consider a signed measure $\mu$ on (the Borel subsets of) $\mathbb R$ with finite total variation (see Signed measure for the definition). We then define the function \begin{equation}\label{e:F_mu} F_\mu (x) := \mu (]-\infty, x])\, . \end{equation}

Theorem 7

• For every signed measure $\mu$ with finite total variation, $F_\mu$ is a right-continuous function of bounded variation such that $\lim_{x\to -\infty} F_mu (x) = 0$ and $TV (f)$ equals the total variation of $|\mu|$.
• For every right-continuous function $f:\mathbb R\to \mathbb R$ of bounded variations such that $\lim_{x\to-\infty} f (x) = 0$ there is a unique signed measure $\mu$ such that $f=F_\mu$

For a proof see Section 4 of Chapter 4 in [Co]. Obvious generalizations hold in the case of different domains of definition.

Distributional derivatives: modern definition

The measure $\mu$ is indeed the generalized derivatie of the function $f=F_\mu$ in the sense of distributions. More precisely \begin{equation}\label{e:distrib} \int f(t)\varphi' (t)\, dt = -\int \varphi (t)\, d\mu (t) \qquad \forall \varphi\in C^\infty_c (\mathbb R)\, . \end{equation} This identty is the starting point for the modern definition of functions of bounded variation, cp. with {{Cite|AFP} or Chapter 5 of [EG].

Definition 8 Let $I\subset\mathbb R$ be a bounded open interval. A function $f\in L^1 (E)$ is said to be of bounded variation if \begin{equation}\label{e:variation_modern} \sup \left\{ \int \varphi' (t) f(t)\, dt \;:\; \varphi\in C^\infty_c (I), \|\varphi\|_{C^0} \leq 1\right\} <\infty\, . \end{equation}

The following theorem links the classical and the modern definitions. See section 3.2 of [AFP] for a proof.

Theorem 9 Let $f$ and $I$ be as in Definition 8. Then there is a function $\tilde{f}:I\to\mathbb R$ and a signed measure $\mu$ on $I$ such that

• $\mu$ is the derivative, in the sense of distributions, of $f$, i.e. \eqref{e:distrib} holds
• $F_\mu = \tilde{f} = f$ almost everywhere
• $\tilde{f}$ is a function of bounded variation in the sense of Definition 1
• $TV (\tilde{f})$ equals the total variation of the measure $\mu$ which in turn is equal to the supremum in \eqref{e:variation_modern}.

Similar definitions and properties can be given for more general domains. However some caution is needed for unbounded domains since then functions of bounded variation are, in general, only locally summable.

Structure theorem

It is possible to relate the pointwise properties of a function $f: I\to \mathbb R$ of bounded variation with the properties of its generalized derivative $\mu$. More pecisely, using the Radon-Nikodym theorem we write $\mu = g \lambda + \mu_s$, where $\mu_s$ is a singular measure with respect to the Lebesgue measure $\mu$. We further follow the discussion of Section 3.2 of [AFP] and decompose $\mu_s = \mu_c +\mu_j$, where $\mu_c$ is the non-atomic part of the measure $\mu_s$, i.e. \[ \mu_c (\{x\}) = 0\qquad \mbox{for every '"`UNIQ-MathJax76-QINU`"'}\, \] and $\mu_j$ is the purely atomic part of $\mu_s$, that is, there is a set $J$ at most countable and weights $c_x\in \mathbb R, x\in J$ such that \[ \mu_j (E) = \sum_{x\in J\cap E} c_x\, . \] If we denote by $\delta_x$ the Dirac mass at the point $x$, then $\mu_j = \sum_{x\in J} c_x \delta_x$. We then have the following theorem (cp. with Section 3.2 of [AFP]), which is often referred to as BV structure theorem fur functions of one variable.

Theorem 10 Let $I = ]a,b[$, $f:I\to \mathbb R$ a right-continuous function of bounded variation and $\mu = g\lambda + \mu_c + \mu_j$ its generalized derivative.

• If $J$ denotes the set of points of discontinuity of $f$, then

\[ \mu_j = \sum_{x\in J} (f(x^+) - f(x^-)) \delta_x\, . \]

• At $\lambda$-a.e. $x$ the function $f$ is differentiable and $f(x) = g(x)$.

Lebesgue decomposition

Observe also that, if we define the functions

• $f_a (x) := f(a)+ \int_a^x g(t)\, dt$,
• $f_j (x) := \mu_j (]a, x])$,
• $f_c (x) := \mu_c (]a, x])$,

then

• $f_a$ is an absolutely continuous function
• $f_c$ is a singular function
• $f_j$ is a jump function.

Then $f=f_a+f_c+f_j$ is called the Lebesgue decomposition of the function $f$ and it is unique up to constants. For such funct

Examples

Smooth functions

If $f: I\to\mathbb R$ is smooth, then we have the identity \begin{equation}\label{e:smooth_var} TV (f) = \int_I |f'(t)|\, dt\, . \end{equation}

Absolutely continuous functions

Absolutely continuous functions are functions of bounded variation and indeed they are the largest class of functions of bounded variation for which \eqref{e:smooth_var} hold. Indeed absolutely continuous functions can be characterized as those functions of bounded variation such that their generalized derivative is an absolutely continuous measure.

Jump functions

The indicator function of the half line, also called Heaviside function \[ {\bf 1}_{[a, \infty[} (x) := \left\{\begin{array}{ll} 0 \qquad &\mbox{if '"`UNIQ-MathJax102-QINU`"'}\\ 1 \qquad &\mbox{if '"`UNIQ-MathJax103-QINU`"'} \end{array}\right. \] is a function of bounded variation (on $\mathbb R$) with total variation equal to $1$. Its generalized derivative is the [[Delta-function|Dirac mass] $\delta_a$. Obviously the Heaviside function is differentiable a.e. with derivative $0$ but its total variationis $1$, thereby showing that \eqref{e:smooth_var} fails for general functions of bounded variation.

The Heaviside function is a prototype of jump function in the sense of the Lebesgue decomposition. If $f$ is a jump function on $\mathbb R$ with $\lim_{x\to\infty} f(x) = 0$, then there are two (at most) countable collections $\{c_i\}, \{a_i\}\subset\mathbb R$ such that \[ f = \sum_i c_i {\bf 1}_{[a_i, \infty[}\, . \]

Cantor ternary function

The Cantor ternary function, also called Devil's staircase (and Cantor-Vitali functions, by some Italian authors) is the most famous example of a continuous function of bounded variation for which \eqref{e:smooth_var} fails (which was first pointed out by Vitali in [Vi]). In fact it is a nondecreasing function such that its derivative vanishes almost everywhere. Its generalized derivative $\mu$ vanishes on the complement of the Cantor set and the function is the prototype of singular function in the Lebesgue decomposition.

Historical remark

Functions of bounded variation were introduced for the first time by C. Jordan in [Jo] to study the pointwise convergence of Fourier series. In particular Jordan proved the following generalization of Dirichlet theorem on the convergence of Fourier series, called Jordan criterion

Theorem 11 Let $f: \mathbb R\to\mathbb R$ be a $2\pi$ periodic summable function.

• If $f$ has bounded variation in an open interval $I$ then its Fourier series converges to $\frac{1}{2} (f (x^+) + f(x^-))$ at every $x\in I$.
• If in addition $f$ is continuous in $I$ then its Fourier series converges uniformly to $f$ on every closed interval $J\subset I$.

For a proof see Section 10.1 and Exercises 10.13 and 10.14 of [Ed]. The criterion is also called Jordan-Dirichlet test, see [Zy]

Functions of several variables

Historical remarks

After the introduction by Jordan of functions of bounded variations of one real variable, several authors attempted to generalize the concept to functions of more than one variable. The first attempt was made by Arzelà and Hardy in 1905, see Arzelà variation and Hardy variation, followed by Vitali, Fréchet, Tonelli and Pierpont, cp. with Vitali variation, Fréchet variation, Tonelli plane variation and Pierpont variation (moreover, the definition of Vitali variation was also considered independently by Lebesgue and De la Vallée-Poussin). However, the point of view which became popular and it is nowadays accepted in the literature as most efficient generalization of the $1$-dimensional theory is due to De Giorgi and Fichera (see [DG] and [Fi]). Though with different definitions, the functions of bounded variation defined by De Giorgi and Fichera are equivalent (and very close in spirit) to the distributional theory described below. A promiment role in the further developing of the theory was also played by Fleming and Federer. Moreover, Krickeberg and Fleming showed, independently, that the current definition of functions of bounded variation is indeed equivalent to a slight modification of Tonelli's one [To], proposed by Cesari [Ce], cp. with the section Tonelli-Cesari variation below. We refer to Section 3.12 of [AFP] for a thorough discussion of the topic.

Definition

Following Section 3.1 of [AFP],

Definition 12 Let $\Omega\subset \mathbb R^n$ be open. $u\in L^1 (\Omega)$ is a function of bounded variation if the generalized partial derivatives of $u$ in the sense of distributions are signed measures, i.e. if for every $i\in \{1, \ldots, n\}$ there is a signed measure $\mu_i$ (with finite total variation) on the $\sigma$-algebra of Borel sets of $\Omega$ such that \begin{equation}\label{e:distrib2} \int_\Omega u \frac{\partial \varphi}{\partial x_i}\, d\lambda = - \int_\Omega \varphi\, d\mu_i \qquad \forall \varphi\in C^\infty_c (\Omega)\, . \end{equation} The vector measure $\mu := (\mu_1, \ldots, \mu_n)$ will be denoted by $Du$ and its variation measure (see Signed measure for the definition) will be denoted by $|Du|$. The vector space of all functions of bounded variations on $\Omega$ is denoted by $BV (\Omega)$.

We assume $u\in L^1 (\Omega)$ to keep the technicalities at a minimum. However, it is possible to relax this assumption, as it is possible to define the spacel $BV_{loc}$ of functons of bounded local variation (see [AFP]).

Total variation

Some authors use instead the following alternative road (cp. with Section 5.1 of [EG]).

Definition 13 Let $\Omega\subset \mathbb R^n$ be open. The total variation of $u\in L^1 (\Omega)$ is given by \begin{equation}\label{e:diverg} V (u, \Omega) := \sup \left\{ \int_\Omega u\, {\rm div}\, \psi : \psi\in C^\infty_c (\Omega, \mathbb R^n), \, \|\psi\|_{C^0}\leq 1\right\}\, . \end{equation}

As a consequence of the Radon-Nikodym theorem we then have

Prposition 14 A function $u\in L^1 (\Omega)$ is a function of bounded variation if and only if $V(u, \Omega)<\infty$ and moreover $V (u,\Omega) = |Du| (\Omega)$.

Consistency with the one variable theory

By Theorem 9, Definition 13 is consistent, in the case $n=1$, with Definition 1. More precisely, if $I\subset \mathbb R$ is a bounded open interval and $f:I\to \mathbb R$ a right-continuous $L^1$ function, then $V(f, I) = TV (f)$ (in particular, if $TV (f)<\infty$, then necessarily $f\in L^1 (I)$ and $V (f, I)<\infty$). Viceversa, if $f\in L^1 (I)$ and $V(f, I)$, then there is a right-continuous function $\tilde{f}$ which coincides $\lambda$-a.e. with $f$ and such that $TV (\tilde{f}) = V (f, I)$. Similar assertions can be proved for more general intervals. However some technical adjustments are needed if the domain is unbounded because a function of bounded variation in the sense of Definition 1 is not necessarily summable.

Generalizations

Let $\Omega\subset \mathbb R^n$ be an open set. $f:\Omega\to\R^m$ belongs to the space $BV (\Omega, \mathbb R^m)$ if each component function is an element in $BV (\Omega)$. A far-reaching generalization for general metric targets has been introduced by Ambrosio in [Am]:

Definition 14 Let $\Omega\subset \mathbb R^n$ be a bounded set and $(X,d)$ a metric space. A Lebesgue measurable map $f:\Omega \to X$ is a generalized function of bounded variation if

• $\varphi\circ f\in BV (\Omega)$ for every Lipschitz function $\varphi:X\to\mathbb R$.
• There is a measure $\mu$ such that $|D (\varphi\circ f)|\leq {\rm Lip}\, (\varphi) \mu$ for every Lipschitz function $\varphi:X\to\mathbb R$.

This definition, which found recently quite important applications, is consistent with the one-dimensional theory and with the case $X=\mathbb R^m$ given above (for the latter see the section Volpert chain rule).

Functional properties

The space $BV (\Omega)$ enjoys several properties that are typical of the Sobolev spaces $W^{1,p} (\Omega)$.

Banach space structure

The norm $\|u\|_{BV} := \|u\|_{L^1} + V (u, \Omega)$ endows $BV (\Omega)$ with a Banach space structure. $BV (\Omega)$ is not reflexive but it is the dual of a separable space (see Remark 3.12 of Section 3.1 in [AFP]). $BV (\Omega)$ contains $W^{1,1} (\Omega)$ and the norm $\|\cdot\|_{BV}$ restricted to $W^{1,1}$ coincides with the $\|\cdot\|_{W^{1,1}}$ norm. In fact $W^{1,1} (\Omega)$ is a closed subspace of $BV (\Omega)$ (see Example 1 of Section 5.1 in [EG]).

Semicontinuity of the variation

If a sequence of functions $\{u_n\}\in L^1 (\Omega)$ converges strongly to $L^1 (\Omega)$, then \[ \liminf_{n\to\infty} V (u_n, \Gamma)\geq V (u, \Gamma) \] for every open set $\Gamma\subset\Omega$ (cp. with Remark 3.5 of [AFP]). In particular, if $\liminf V (u_n,\Omega)<\infty$, then $u\in BV (\Omega)$.

Approximation with smooth functions

Theorem 15 A function $u$ belongs to $BV (\Omega)$ if and only if there exists a sequence of smooth functions $\{u_n\}$ such that

• $\|u_n-u\|_{L^1 (\Omega)} \to 0$
• $\liminf_n V (u_n, \Omega) < \infty$.

Moreover, for every $u\in BV (\Omega)$ there is an approximating sequence $\{u_n\}\in C^\infty\cap BV (\Omega)$ converging strongly to $u$ and such that $V (u_n, \Omega)\to V (u, \Omega)$ (therefore $\|u_n\|_{BV}\to \|u\|_{BV}$.

Cp. with Theorem 3.9 of Section 5.1 in [AFP]. However, differently from the usual Sobolev spaces, the space $C^\infty (\Omega)$ is not dense in the strong topology: its closure is instead $W^{1,1} (\Omega)$.

Weak$^\star$ convergence

A sequence $\{u_n\}$ converges weakly$^\star$ in $BV (\Omega)$ to $u$ if $u_n\to u$ strongly in $L^1 (\Omega)$ and $Du_h$ converges weakly$^\star$ in the sense of measures to $Du$ (cp. with Convergence of measures). In fact a sequence converges weakly$^\star$ if and only if it converges in $L^1$ and it is bounded in the $BV$ norm (cp. with Proposition 3.13 of Section 3.1 in [AFP]

In fact, closed and bounded convex subsets of $BV (\Omega)$ are weakly$^\star$ compact (cp. with Theorem 3.23 in Section 3.1 of [AFP]).

Extension theorems

If $\Omega$ is an open set with compact Lipschitz boundary, then any function $u\in BV (\Omega)$ can be extended to a function $u\in BV (\mathbb R^n)$ (cp with Theorem 3.21 of Section 3.1 in [AFP]). Not all bounded open subsets possess this extension property: however the class of extension domains is larger than the class of open sets with compact Lipschitz boundary.

Sobolev inequality

The usual Sobolev inequality which holds for $W^{1,1}$ functions extends to $BV$ functions as well. Namely, there are constants $C(n)$ depending only on $n\in\mathbb N\setminus \{0\}$ such that:

• $\|f\|_{L^\infty}\leq C(1) TV (f)$ for any $f\in BV (\mathbb R)$;
• $\|f\|_{L^{n/(n-1)}}\leq V (u,\mathbb R^n)$ for any $f\in BV (\mathbb R^n)$ for any $n\geq 2$.

In the case $n=1$ the optimal constant is indeed $C(1)=1$ and the inequality follows easily from the considerations in the section Measure theoretic characterization. For the case $n\geq 2$ we refer to Theorem 1 of Section 5.6 in [EG] or Theorem 3.47 of Section 3.4 of [AFP]). The Sobolev inequality combined with the extension theorems give the embeddings $BV (\Omega)\subset L^p (\Omega)$ for any extension domain $\Omega$ and every $p\in [1, \frac{n}{n-1}]$. Such embedding is compact if $\Omega$ is bounded and $p<\frac{n}{n-1}$ (cp. with Corollary 3.49 of {{Cite.

Poincaré inequality

The usual Poincaré inequality for $W^{1,1}$ extends as well to $BV$ functions., Namely, there is a constant $C(n)$ such that, for $n\geq 2$, \[ \left(\int_{B_r (x)} |u (y)-\bar{u}|^{\frac{n-1}{n}}\right)^{\frac{n-1}{n}}\, \;\leq\; C (n) \, V (u, B_r (x)) \qquad \mbox{for every '"`UNIQ-MathJax239-QINU`"'} \] where $\bar{u}$ denotes the average of $u$ on $B_r (x)$ (and $B_r (x)\subset \mathbb R^n$ is the open ball with radius $r$ and center $x$). See Theorem 1 of Section 5.6 in [EG] or Remark 3.50 of Section 3.4 on [AFP]. In fact such inequalities hold also on more general domains $\Omega$, with constants depending on the specific geometry of $\Omega$.

Trace

For functions of bounded variations a suitable extension of the classical theory of traces of Sobolev spaces holds as well. In what follows we denote by $\mathcal{H}^{n-1}$ the Hausdorff $n-1$-dimensional measure.

Theorem 16 Assume $\Omega$ is open and bounded, with $\partial \Omega$ of class $C^1$. Then there exists a bounded linear mapping \[ T:BV (\Omega)\to L^1 (\partial \Omega, \mathcal{H}^{n-1}) \] such that the following identity holds for any test field $\varphi\in C^\infty (\mathbb R^n,\mathbb R^n)$: \[ \int_\Omega f (x)\, {\rm div} \, \varphi (x)\, dx = -\int_\Omega \varphi (x)\cdot d\mu (x) + \int_{\partial \Omega} (\varphi (x)\cdot \nu (x))\, Tf (x)\, d\mathcal{H}^{n-1} (x) \] (where $\nu$ denotes the exterior unit normal to $\partial \Omega$). In particular, if $f\in C^1 (\overline{\Omega})$, then $Tf$ is simply the restriction of $f$ to $\partial \Omega$.

The theorem holds also for Lipschitz domains (cp. with Theorem 1 of Section 5.3 in [EG]). By a Theorem of Gagliardo, see [Ga], the trace operator is in fact onto, even when restricted to $W^{1,1} (\Omega)$.

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