1 Introduction

BV functions of a single variable were first introduced by Camille Jordan, in a paper [21] that deals with the convergence of Fourier series. Soon after Jordan’s work, many mathematicians began to study notions of bounded variation for functions of several variables. There are various approaches to the notion of variation for functions of several variables. We can mention those belonging to Vitali, Hardy, Krause, Arzela, Frechét, Tonelli, Hahn, Kronrod-Vitushkin, Minlos, and others. Functions of bounded variation in n variables (\(n>1\)) belonging to each of these classes have more or less the same properties as the functions of bounded variation of one variable. However, there are some properties in the one-dimensional case that cannot be transferred automatically to the multidimensional one (see [22]). On the other hand, functions of bounded variation in \(\mathbb {R}^n\) can be identified with n-dimensional normal currents in \(\mathbb {R}^n\). This point of view is due to Federer [18].

In the literature, the many notions of bounded variation are mainly studied for functions defined on a rectangle \(J \subset \mathbb {R}^n\). A definition of the variation in the sense of Hardy and Krause is given in [22]. Let \(f: [0,1]^n \rightarrow \mathbb {R}\). Let \(\mathbf {a}=(a_1,\ldots ,a_n)\) and \(\mathbf {b}=(b_1,\ldots ,b_n)\) be elements of \([0,1]^n\) such that \(\mathbf {a} < \mathbf {b}\) (see Sect. 2). The n-dimensional difference operator \(\Delta ^n\), which assigns to the axis-parallel rectangle \([\mathbf {a}, \mathbf {b}]\) a n-dimensional quasi-volume

$$\begin{aligned} \Delta ^{(n)} (f; [\mathbf {a}, \mathbf {b}]):= & {} \sum \limits _{j_1=0}^1 \cdots \sum \limits _{j_n=0}^1 (-1)^{j_1+ \cdots +j_n} f(b_1+j_1(a_1-b_1),\ldots , b_n \\&+ j_n(a_n-b_n)). \end{aligned}$$

For \(s= 1,\ldots , n\) let \(0 = x_0^{(s)}< x_1^{(s)} < \ldots x_{m_s}^{(s)} = 1 \) be a partition of [0, 1], and \(\mathcal {P}\) be the partition of \([0,1]^n\) which is given by

$$\begin{aligned} \mathcal {P}= \left\{ \left[ x_{l_1}^{(1)},x_{l_1+1}^{(1)} \right] \times \cdots \times \left[ x_{l_n}^{(n)},x_{l_n+1}^{(n)} \right] , l_s=0,\ldots ,m_s-1, s=1,\ldots ,n \right\} .\nonumber \\ \end{aligned}$$
(1.1)

Then the variation of f on \([0,1]^n\) in the sense of Vitali is given by

$$\begin{aligned} V^{(d)} (f; [0,1]^n) := \sup \limits _{\mathcal {P}} \sum \limits _{A \in \mathcal {P}} |\Delta ^{(n)} (f; A)|, \end{aligned}$$

where the supremum is extended over all partitions of \([0,1]^n\) into axis-parallel boxes generated by d one-dimensional partitions of [0, 1], as in (1.1).

If the same functions restricted to the various faces of \([0,1]^s\) with \(s=1,\ldots ,n\) are of bounded variation in the sense of Vitali over each of them, then f is said to be of bounded variation on \([0,1]^s\) in the sense of Hardy and Krause, that is, for \(1 \le s \le n\) and \(1 \le i_1< \cdots < i_s \le n,\) let \(V^{(s)}(f; i_1, \ldots , i_s; [0,1]^n)\) denote the s-dimensional variation in the sense of Vitali of the restriction of f to the face

$$\begin{aligned} U_d^{(i_1,\ldots ,i_s)} = \{(x_1,\ldots ,x_n) \in [0,1]^n : x_j = 1 \quad \text {for all} \quad j\ne i_1,\ldots , i_s\} \end{aligned}$$

of \([0,1]^n\). Then the variation of f on \([0, 1]^n\) in the sense of Hardy and Krause anchored at \(\mathbf {1}\), abbreviated by HK-variation, is given by

$$\begin{aligned} \mathrm{Var}_{HK} (f; [0,1]^n) = \sum \limits _{s=1}^n \sum \limits _{1 \le i_1< \cdots < i_s \le n} V^{(s)}(f; i_1,\ldots ,i_s; [0,1]^n). \end{aligned}$$
(1.2)

Note that for the definition of the HK-variation in (1.2), we add the n-dimensional variation in the sense of Vitali plus the variation in the sense of Vitali on all lower dimensional faces of \([0, 1]^n\) which are adjacent to \(\mathbf {1}\).

On the other hand, a function \(f: \mathbb {R}^n \rightarrow \mathbb {R}\) is said to be of bounded Tonelli-variation if a.e. in \((x_1, \ldots , x_{j-1}, x_{j+1},\ldots ,x_n)\) it is of bounded variation in each variable \(x_j\) for all \(1 \le j \le n\) and if these variations \(\mathrm{BV}_j(f(x)):= \mathrm{BV}_{x_j \in \mathbb {R}(f(x))}\) are Lebesgue integrable as functions of the other \(n-1\) variables \(x_1, x_2,\ldots , x_{j-1}, x_{j+1},\ldots , x_n\):

$$\begin{aligned} \mathrm{VT}(f):= \sum \limits _{j=1}^n \int _{\mathbb {R}^{n-1}} \mathrm{BV}_j(f(x)) \prod \limits _{\begin{array}{c} k=1 \\ k \ne j \end{array}}^n \mathrm{d}x_k, \end{aligned}$$

which for a smooth enough function f, it is equal to

$$\begin{aligned} \mathrm{VT}(f)= \int _{\mathbb {R}} \sum \limits _{j=1}^n \left| \dfrac{\partial f(x)}{\partial x_j} \right| \mathrm{d}x. \end{aligned}$$

Among the sources dealing with the Tonelli variation, let us mention [2, 7, 9, 16, 27].

Until now it seems ([20, 22]) that only the approach due to Vitali–Hardy–Krause gives a notion of variation for real-valued functions of several variables such that a complete analogue of the Helly theorem holds with respect to the pointwise convergence of extracted subsequences. However, the point of view which is nowadays accepted in the literature as most efficient generalization of the 1-dimensional theory is due to De Giorgi and Fichera (see [17, 19]).

Thus, the unvarying interest generated by the classical notion of function of bounded variation has led to some generalizations of the concept, mainly, intended to the search of bigger classes of functions whose elements have pointwise convergent Fourier series or to find applications in geometric measure theory, calculus of variations, and mathematical physics. As in the classical case, these generalizations have found many applications in the study of certain differential and integral equations (see e.g., [8]).

In this paper, we present a detailed study of the space of functions of bounded Riesz-\(\Phi \)-variation, which was introduced previously in [5, 6], for real-valued functions of several variables.

This extends the work done in [3] (resp. in [4]), in which the authors present the notion of real-valued function (resp. vector valued) of bounded Riesz-\(\Phi \)-variation, that are defined on a rectangle of \(\mathbb {R}^2\). In particular, we extend some results due to Chistyakov ([15]) and give a version of Riesz’s Lemma for the case of functions of several real variables which take values in a reflexive Banach space.

2 Notation and preliminary

To start, we give some notations and definitions that will be used throughout the rest of this paper (see [5, 12, 13]).

As usual, \(\mathbb {N}, \mathbb {N}_0\) and \(\mathbb {R}\) denotes the set of all positive integers, non-negative integers and real numbers, respectively. A typical point of \(\mathbb {X}^{n}\) (\(\mathbb {N}, \mathbb {N}_0\) or \(\mathbb {R}\)) is denoted as \( \mathbf {x} = (x_1, x_2, \ldots , x_n) := (x_i)_{i=1}^{n}\), but the canonical unit vectors of \(\mathbb {R}^{n}\) are denoted by \(\mathbf {e_j}\) (\(j = 1, 2, \dots , n\)); that is, \(\mathbf {e_j} := (e_1^{j}, e_2^{j},\ldots , e_n^{j})\) where, \( e_r^{j} := \left\{ \begin{array}{ll} 0 &{} \text { if } \, r\ne j \\ 1 &{} \text { if } \, r= j. \end{array} \right. \). The zero n-tuple \((0,0, \dots , 0)\) will be denoted by \(\mathbf {0}\), and by \(\mathbf {1}\) we will denote the n-tuple \(\mathbf {1}= (1,1, \dots , 1)\).

If \( {{\varvec{\alpha }}} = (\alpha _1, \alpha _2, \ldots , \alpha _n)\) is a n-tuple of non-negative integers then we call \( {\varvec{\alpha }}\) a multi-index ([1]).

If \( \mathbf {a}=(a_1, a_2, \dots , a_n) ,\mathbf {b}= (b_1, b_2, \dots , b_n) \in \mathbb {R}^{n}\) we use the notation \( \mathbf {a} < \mathbf {b}\) to mean that \(a_i < b_i\) for each \(i=1,\ldots , n\) and similarly are defined \( \mathbf {a} = \mathbf {b}\), \( \mathbf {a} \le \mathbf {b}\), \( \mathbf {a} \ge \mathbf {b}\) and \( \mathbf {a} > \mathbf {b}\). If \( \mathbf {a} < \mathbf {b}\), the set \(\mathbf {J}:= [ \mathbf {a}, \mathbf {b}] = \prod \nolimits _{i=1}^{n} [a_i, b_i]\) will be called a n-dimensional closed interval. The Euclidean volume of an n-dimensional closed interval will be denoted by \({{\mathrm{Vol}}}[ \mathbf {a}, \mathbf {b}] \); that is, \({{\mathrm{Vol}}}[ \mathbf {a}, \mathbf {b}] = \prod \nolimits _{i=1}^{n} (b_i - a_i). \)

In addition, for \( {\varvec{\alpha }}= (\alpha _1, \alpha _2, \ldots , \alpha _n) \in \mathbb {N}_0^{n}\) and \( \mathbf {x} = (x_1, x_2, \ldots , x_n) \in \mathbb {R}^{n}\) we will use the notations \(| {\varvec{\alpha }}| := \alpha _1 + \alpha _2 + \cdots + \alpha _n \) and \(\alpha \mathbf {x} := (\alpha _1 x_1, \alpha _2 x_2, \ldots , \alpha _n x_n).\)

In this work, we will consider functions whose domain is a n-dimensional closed interval \([ \mathbf {a}, \mathbf {b}] \) and whose range is an invariant metric semigroup \((X,d,+,\cdot )\); i.e., (X, d) is a complete metric space, d is a translation invariant metric on X, \((X,+)\) is an commutative semigroup. In particular, the triangle inequality implies that, for all \(u,v, p, q \in X\),

$$\begin{aligned}&d(u,v) \le d(p,q) + d(u + p, v + q) \quad \text {and} \nonumber \\&\quad (u + p, v + q) \le d(u,v) + d(p,q). \end{aligned}$$
(2.1)

Clearly any normed space is a metric semigroup.

The following standard notation (see [14]) will be used: \(\mathcal {N}\) will denote the set of all continuous convex functions \(\Phi :[0,+\infty ) \rightarrow [0,+\infty )\) such that \(\Phi (t)= 0\) if and only if \(t=0\), and \(\mathcal {N}_{\infty }\) the set of all functions \(\Phi \in \mathcal {N}\), for which the Orlicz condition (also called \(\infty _1 \) condition) holds: \( \lim \nolimits _{t \rightarrow \infty } \dfrac{\Phi (t)}{t} = + \infty \). Following [23], functions from \(\mathcal {N}\) are called \(\varphi \)-functions. Any function \(\Phi \in \mathcal {N}\) is strictly increasing, and so, its inverse \(\Phi ^{-1}\) is continuous and concave; besides, the functions   \(t \longmapsto \dfrac{\Phi (t)}{t}\)   and   \(t \longmapsto t \Phi ^{-1}\left( \dfrac{1}{t} \right) \)   are nondecreasing for \(t > 0\).

One says that a function \(\Phi \in \mathcal {N}\) satisfies a \(\Delta _2\) condition, and writes \(\Phi \in \Delta _2\), if there are constants \(\, K > 2\, \) and \(\, t_0 >0 \in \mathbb {R} \, \) such that

$$\begin{aligned} \Phi (2 t) \le K \Phi (t) \, \, \text { for all } \, \, t \ge t_0. \, \end{aligned}$$
(2.2)

  For instance, if \( \Phi (x) := t^p, \, p > 1, \) one may choose the optimal constant \(K= 2^p.\)

Now we define two sets that will play an important role in this work:

$$\begin{aligned} \mathcal {E}(n):= & {} \{ \theta \in \mathbb {N}_0^{n} : \mathbf {\theta } \le \mathbf {1} \, \text { and } \, |\mathbf {\theta }| \, \text { is even}\}, \\ \mathcal {O}(n):= & {} \{ \theta \in \mathbb {N}_0^{n} : \mathbf {\theta }\le \mathbf {1 } \, \text { and } \, |\mathbf {\theta }| \, \text { is odd}\}. \end{aligned}$$

Notice that these sets are related in one-to-one correspondence; indeed, if \(\theta = (\theta _1, \ldots , \theta _n) \in \mathcal {E}(n)\) then we can define \(\widetilde{\mathbf {\theta }} := (\theta _1, \ldots ,\theta _{i-1}, 1 - \theta _i,\theta _{i+1},\ldots \theta _n) \in \mathcal {O}(n)\), and this operation is clearly invertible.

Definition 2.1

[10, 11, 26] Given \(f : [ \mathbf {a}, \mathbf {b}] \rightarrow X\), we define the n-dimensional Vitali difference of f over an n-dimensional interval \([ \mathbf {x}, \mathbf {y}] \subseteq [ \mathbf {a}, \mathbf {b}] \), by

$$\begin{aligned} \Delta _n(f,[ \mathbf {x}, \mathbf {y}] ) := d\left( \underset{\theta \in \mathcal {E}(n)}{\sum } f( \mathbf {\theta } \, \mathbf {x} + (\mathbf {1} - \mathbf {\theta }) \mathbf {y}), \underset{ \mathbf {\theta } \in \mathcal {O}(n)}{\sum } f( \mathbf {\theta } \, \mathbf {x} + (\mathbf {1} - \mathbf {\theta }) \mathbf {y}) \right) .\nonumber \\ \end{aligned}$$
(2.3)

Note that in the case \(n=2\), we have \(\mathcal {E}(2):= \{(0,0),(1,1)\}\) and \(\mathcal {O}(2)=\{(1,0),(0,1)\},\) thus \( \Delta _2(f,[ \mathbf {x}, \mathbf {y }] ) = d(f(x_1,x_2) + f(y_1,y_2), f(y_1, x_2) + f(x_1,y_2)).\)

Even when \(\Delta _n (f, [ \mathbf {x}, \mathbf {y}] ),\) in (2.3), is defined for \( \mathbf {x} < \mathbf {y}\), note that if \(x_i = y_i\) for some i, then the right-hand side of (2.3) is equal to zero for all maps \(f : [ \mathbf {a}, \mathbf {b}] \rightarrow X\). This difference is also called mixed difference and it is usually associated to the names of Vitali, Lebesgue, Hardy, Krause, Fréchet and De la Vallée Poussin ([10, 11, 20]).

Now, we are going to define the \(\Phi \)-variation of a function \(f:[ \mathbf {a}, \mathbf {b}] \rightarrow X\) (see [5, 6]). To that end, we consider net partitions of \([ \mathbf {a}, \mathbf {b}] = \prod \nolimits _{i=1}^{n} [a_i, b_i]\); that is, partitions of the kind

$$\begin{aligned} \mathbf {\xi } = \mathbf {\xi } _1 \times \mathbf {\xi } _2 \times \cdots \times \mathbf {\xi } _n&\text {with}&\mathbf {\xi } _i:= \left\{ t_j^{(i)}\right\} _{j=0}^{k_i}, \, \, \, i=1, \ldots , n, \end{aligned}$$
(2.4)

where \(\{k_i\}_{i=1}^n \subset \mathbb {N}\) and for each i,   \(\mathbf {\xi } _i \) is a partition of \([a_i, b_i]\). The set of all net partitions of an interval \([ \mathbf {a}, \mathbf {b}]\) will be denoted by \( \Lambda ([ \mathbf {a}, \mathbf {b}])\).

A point in a net partition \(\mathbf {\xi } \) is called a node ([25]) and it is of the form

$$\begin{aligned} \mathbf {t}_{{ {\varvec{\alpha }}} } :=(t_{\alpha _1}^{(1)}, \, t_{\alpha _2}^{(2)}, \, t_{\alpha _3}^{(3)}, \ldots , \, t_{\alpha _n}^{(n)}), \end{aligned}$$

where \(\mathbf 0 \le {\varvec{\alpha }} \, = \, (\, \alpha _i \,)_{i=1}^{n} \, \le \, \mathbf {\kappa } \), with \( \mathbf {\kappa } := (k_i)_{i=1}^{n},\) as a result, \(t_{\alpha _i}^{(j)} \in [a_j, b_j]\).

For the sake of simplicity in notation, we will simply write \(\mathbf {\xi } =\{\mathbf {t}_{{ {\varvec{\alpha }}} }\}, \) to refer to all nodes determined by a given partition \(\mathbf {\xi } .\)

A cell of an n-dimensional interval \([ \mathbf {a}, \mathbf {b}]\) is an n-dimensional subinterval of the form \([ \mathbf {t}_{ {\varvec{\alpha }}- \mathbf {1} }, \mathbf {t}_{ {\varvec{\alpha }} }], \) for \(\mathbf 0 < {\varvec{\alpha }}\le \mathbf {\kappa }\).

Note that

$$\begin{aligned} \mathbf {t}_{ \mathbf {0}}= & {} (t^{(1)}_{0}, t^{(2)}_{0},\ldots , t^{(n)}_{0}) = (a_1, a_2, \ldots , a_n) \text { and } \mathbf {t}_{ \mathbf {\kappa }} = (t^{(1)}_{k_1}, t^{(2)}_{k_2},\ldots , t^{(n)}_{k_n})\\= & {} (b_1, b_2, \ldots , b_n). \end{aligned}$$

3 \( \mathrm{RV}^{n}_{\Phi } ([ \mathbf {a}, \mathbf {b}] ; X)\)

Now we introduce the Riesz-\(\Phi \)-variation of a function f.

Definition 3.1

Let \(f : [ \mathbf {a}, \mathbf {b}] \rightarrow X\) and \(\Phi \in \mathcal {N} \). The \(\Phi \)-variation, in the sense of Vitali–Riesz of f is defined as

$$\begin{aligned} \mathrm{RV}_{\Phi }^{n}(f,[ \mathbf {a}, \mathbf {b}] ):= & {} \sup _{ \mathbf {\xi } \in \Lambda ([ \mathbf {a}, \mathbf {b}]) } \, \mathrm{RV}_{\Phi }^{n}(f,[ \mathbf {a}, \mathbf {b}],\mathbf {\xi } ), \end{aligned}$$
(3.1)

where \(\mathbf {\xi } =\{\mathbf {t}_{{ {\varvec{\alpha }}} }\}, \) and

$$\begin{aligned} \mathrm{RV}_{\Phi }^{n}(f,[ \mathbf {a}, \mathbf {b}],\mathbf {\xi } ):= & {} \, \sum \limits _{\mathbf {1 } \le {\varvec{\alpha }}\le \mathbf {\kappa } } \, \Phi \left( \dfrac{\Delta _n \left( f, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}] \right) }{\mathrm{Vol} \, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}]} \right) \mathrm{Vol} \, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}]. \end{aligned}$$

The main objective of this section is to define the Riesz-\(\Phi \)-variation of a function f. To do that it will be necessary to define the variation of a function when we consider that certain variables are fixed, thus as it was done in [11], we now define the truncation of a point, of an interval and of a function, by a given multi-index \(\mathbf {0}\le \eta \le \mathbf {1}\), with \(\mathbf {0} \ne \eta \). Notice that in this case, the entries of a such \(\eta \) are either 0 or 1.

  • The truncation of a point \( \mathbf {x} \in \mathbb {R}^{n}\) by a multi-index \(\mathbf {0}\le \eta \le \mathbf {1}\) with \(\mathbf {0} \ne \eta \), which is denoted by \( \mathbf {x} \lfloor \mathbf {\eta }\), is defined as the \(| \mathbf {\eta }|\)-tuple that is obtained if we suppress from \(\mathbf {x}\) the entries for which the corresponding entries of \(\eta \) are equal to 0. That is, \( \mathbf {x}\lfloor \mathbf {\eta } = (x_i : i \in \{1,2,\ldots ,n\}, \eta _i = 1) \). For instance, if \( \mathbf {x} = (x_1, x_2, x_3, x_4, x_5)\) and \( \mathbf {\eta } = ( 0, 1, 1, 0, 1)\) then \( \mathbf {x}\lfloor \mathbf {\eta } = ( x_2, x_3, x_5).\)

  • The truncation of an n-dimensional interval \([ \mathbf {a}, \mathbf {b}]\) by a multi-index \(\mathbf {0}\le \eta \le \mathbf {1}\) with \(\mathbf {0} \ne \eta \), is defined as \([ \mathbf {a}, \mathbf {b}] \lfloor \mathbf {\eta } := [{ \mathbf {a}\lfloor \mathbf {\eta }},{{ \mathbf {b}\lfloor \mathbf {\eta } }}] \).

  • Given a function \(f: [ \mathbf {a}, \mathbf {b}] \rightarrow X\), a multi-index \(\mathbf {0}\le \eta \le \mathbf {1}\) with \(\mathbf {0} \ne \eta \) and a point \( \mathbf {z} \in [ \mathbf {a}, \mathbf {b}] \), we define \(f_{ \mathbf {\eta }}^{ \mathbf {z}} : [ \mathbf {a}, \mathbf {b}] \lfloor \mathbf {\eta } \rightarrow X \), the truncation of f by \(\eta \), by the formula

    $$\begin{aligned} f_{ \mathbf {\eta }}^{ \mathbf {z}}( \mathbf {x}\lfloor \mathbf {\eta }) : = f( \mathbf {\eta } \mathbf {x} + (\mathbf 1 - \mathbf {\eta } ) \mathbf {z}), \, \, \, x \in [ \mathbf {a}, \mathbf {b}]. \end{aligned}$$

Note that the function \(f_{ \mathbf {\eta }}^{ \mathbf {z}}\) depends only on the \(| \mathbf {\eta }|\) variables \(x_i\) for which \(\eta _i = 1\).

Remark 3.2

Given a function \(f:[\mathbf {a},\mathbf {b}] \rightarrow X \) and a multi-index \(\eta \ne \mathbf {0}\), then the \(|\eta |\)-dimensional Vitali difference for \(f_{\eta }^{\mathbf {a}}\) (cf. (2.3)), is given by

$$\begin{aligned}&\Delta _{|\eta |}(f_{\eta }^{\mathbf {a}},[ \mathbf {x}, \mathbf {y}] )\\&\quad := d\left( \underset{{\begin{array}{c} \theta \in \mathcal {E}(n) \\ \theta \le \eta \end{array}}}{\sum } f\left( \eta ( \theta \mathbf {x} + (\mathbf {1} - \theta ) \mathbf {y} ) + (\mathbf {1}-\eta ) \mathbf {a}\right) , \underset{{\begin{array}{c} \mathbf {\theta } \in \mathcal {O}(n) \\ \theta \le \eta \end{array}}}{\sum } f ( \eta ( \theta \mathbf {x} + (\mathbf {1} - \theta ) \mathbf {y} ) + (\mathbf {1}-\eta ) \mathbf {a})\right) . \end{aligned}$$

The Vitali-type nth variation of \(f: [\mathbf {a},\mathbf {b} ] \rightarrow X\) is defined by

$$\begin{aligned} V_n(f,[\mathbf {a},\mathbf {b} ]) := \sup \sum \limits _{\mathbf {1} \le \eta \le \kappa } \Delta _{|\eta |}(f_{\eta }^{\mathbf {a}},[ \mathbf {x}, \mathbf {y}]), \end{aligned}$$

the supremum being taken over all multiindices \(\kappa \) and all net partitions of \([\mathbf {a},\mathbf {b} ]\).

The total variation of \(f: [\mathbf {a}, \mathbf {b}]\rightarrow X\) in the sense of Hildebrandt and Leonov (see [20, 22]) is defined by

$$\begin{aligned} \mathrm{TV}(f,[\mathbf {a},\mathbf {b} ] ) := \sum \limits _{\mathbf {0} \ne \alpha \le \mathbf {1}} V_{|\alpha |}(f_{\alpha }^{\mathbf {a}},[\mathbf {a},\mathbf {b} ]\lfloor \alpha ), \end{aligned}$$

the summations here and throughout the paper being taken over n-dimensional multiindices in the ranges specified under the summation sign.

Definition 3.3

Let \(\Phi \in \mathcal {N}\) and let \(f: [ \mathbf {a}, \mathbf {b}] \rightarrow X\) be a function. We call

$$\begin{aligned} \mathrm{TRV}_{\Phi } (f,[ \mathbf {a}, \mathbf {b}] ) := \sum \limits _{\mathbf {0} \ne \mathbf {\eta } \le \mathbf {1} } \, \mathrm{RV}_{\Phi }^{| \mathbf {\eta }|} (f_{ \mathbf {\eta }}^{ \mathbf {a}}, [ \mathbf {a}, \mathbf {b}] \lfloor \mathbf {\eta }) \end{aligned}$$

the \(\Phi \)-variation of f in the sense of Vitali–Hardy–Riesz, briefly: Riesz-\(\Phi \)-variation of f, in \([ \mathbf {a}, \mathbf {b}]\). The set of all functions f satisfying the condition \( \mathrm{TRV}_{\Phi } (f,[ \mathbf {a}, \mathbf {b}] ) < \infty \) will be denoted by \( \mathrm{RV}^{n}_{\Phi } ([ \mathbf {a}, \mathbf {b}] ; X)\).

It is easy to check that if f is a constant function then \( \Delta _n(f,[ \mathbf {x}, \mathbf {y}] )= 0\) and consequently \(\mathrm{TRV}_{\Phi } (f,[ \mathbf {a}, \mathbf {b}] )=0\). In fact, \(\mathrm{TRV}_{\Phi } (f,[ \mathbf {a}, \mathbf {b}] )=0\) if and only if f is constant, as we show next.

Theorem 3.4

\(\mathrm{TRV}_{\Phi } (f,[ \mathbf {a}, \mathbf {b}] )=0\) if and only if f is a constant function.

Proof

We just prove the necessity of the condition. Suppose \(\mathrm{TRV}_{\Phi } (f,[ \mathbf {a}, \mathbf {b}] )=0\) and let \( \mathbf {x}= (x_1,\ldots , x_n)\) be a point in \([ \mathbf {a}, \mathbf {b}]\). Then, \(\mathbf {x}\) determines, for each \(1 \le i \le n\), the partitions \(\mathbf {\xi } _i := \{a_i, x_i, b_i\}:= \{t_0^{(i)},t_1^{(i)}, t_2^{(i)}\} \).

Since \(\mathrm{TRV}_{\Phi } (f,[ \mathbf {a}, \mathbf {b}] ) =0\), we must have   \( \Delta _{|\eta |}(f_{\eta }^{\mathbf {a}},[ \mathbf {t}_{\alpha -\mathbf {1}}, \mathbf {t}_{\alpha }] ) = 0 \)  for every \(\mathbf {1} \le {\varvec{\alpha }}\le \mathbf {2} \) and every \(\mathbf {0} \ne \eta \le \mathbf {1} \). Consequently, if \(\eta = \mathbf {e}_i\) and \(\alpha = \mathbf {1}\) we obtain

$$\begin{aligned} 0 = d\left( f( \eta \mathbf {t}_{ \mathbf {1}} + (\mathbf {1}-\eta ) \mathbf {a}), f(\mathbf {a}) \right) . \end{aligned}$$

Hence,

$$\begin{aligned} f(a_1, \ldots , x_i, a_{i+1}, \ldots , a_n)= f( \eta \mathbf {t}_{ \mathbf {1}} + (\mathbf {1}-\eta ) \mathbf {a}) = f(\mathbf {a}),&\text {for all}&1 \le i \le n.\nonumber \\ \end{aligned}$$
(3.2)

On the other hand, if \(\eta = \mathbf {e}_i + \mathbf {e}_j\) with \(i < j\) then

$$\begin{aligned} d(f( \eta \mathbf {t}_{ \mathbf {1}} + (\mathbf {1}-\eta ) \mathbf {a} )+ f(\mathbf {a}), f(\mathbf {e}_j \mathbf {t}_1 + (\mathbf {1}-\mathbf {e}_j)\mathbf {a}) + f(\mathbf {e}_i \mathbf {t}_1 + (\mathbf {1}-\mathbf {e}_i)\mathbf {a}))=0.\nonumber \\ \end{aligned}$$
(3.3)

Thus, using (3.3) and (3.2) we have

$$\begin{aligned} f( \eta \mathbf {t}_{ \mathbf {1}} + (\mathbf {1}-\eta ) \mathbf {a} )+ f(\mathbf {a})= & {} f(\mathbf {e}_j \mathbf {t}_1 + (\mathbf {1}-\mathbf {e}_j)a) + f(\mathbf {e}_i \mathbf {t}_1 + (\mathbf {1}-\mathbf {e}_i)a) \nonumber \\ f( \eta \mathbf {t}_{ \mathbf {1}} + (\mathbf {1}-\eta ) \mathbf {a} )+ f(\mathbf {a})= & {} f(\mathbf {a}) + f(\mathbf {a}). \end{aligned}$$

Equivalently

$$\begin{aligned} f(\eta \mathbf {x} + (\mathbf {1} - \eta )\mathbf {a}) = f(\mathbf {a}). \end{aligned}$$
(3.4)

Now, suppose that (3.4) holds for any multi-index \(\eta \), \(\mathbf {0} \le \eta \le \mathbf {1}\), \(\mathbf {0} \ne \eta \), with k non-zero entries.

Then, if \( \lambda \) is a multi-index such that \(\mathbf {0} \le \lambda \le \mathbf {1}\), \(\mathbf {0} \ne \lambda \), with \(k+1\) non-zero entries and \(\Delta _{|\lambda |}(f_{\lambda }^{\mathbf {a}},[ \mathbf {t}_{\alpha -\mathbf {1}}, \mathbf {t}_{\alpha }] ) = 0 \) for all \(\mathbf {1} \le {\varvec{\alpha }}\le \mathbf {2} \), then

$$\begin{aligned} \underset{\begin{array}{c} \theta \in \mathcal {E}(n) \\ \theta \le \lambda \end{array}}{\sum } f\left( \lambda ( \theta \mathbf {t}_\mathbf {0} + (\mathbf {1} - \theta ) \mathbf {t}_{\mathbf {1}} ) + (\mathbf {1}-\lambda ) \mathbf {a}\right) =\underset{\begin{array}{c} \mathbf { \theta } \in \mathcal {O}(n) \\ \theta \le \lambda \end{array}}{\sum } f \left( \lambda ( \theta \mathbf {t}_\mathbf {0} + (\mathbf {1} - \theta ) \mathbf {t}_{\mathbf {1}} ) + (\mathbf {1}-\lambda ){\mathbf {a}}\right) .\nonumber \\ \end{aligned}$$
(3.5)

Notice that if   \(\theta \ne \mathbf {0}\)   then   \( \lambda ( \theta \mathbf {t}_\mathbf {0} + (\mathbf {1} - \theta ) \mathbf {t}_{\mathbf {1}}) + (\mathbf {1}-\lambda ) \mathbf {a} \)  has at most k entries equal to the corresponding entries of \( \mathbf {x}\) and the remaining entries equal to the corresponding entries of \(\mathbf {a}.\) In this case, (3.4) implies that \(f( \lambda ( \theta \mathbf {t}_\mathbf {0} + (\mathbf {1} - \theta ) \mathbf {t}_{\mathbf {1}}) + (\mathbf {1}-\lambda ) \mathbf {a}) = f(\mathbf {a})\). Hence, since \( \mathcal {E}(n) \) has the same number of elements as \(\mathcal {O}(n)\), it follows from identity (3.4) that \(f\left( \lambda \mathbf {t}_{\mathbf {1}} + (\mathbf {1}-\lambda ) \mathbf {a}\right) = f(\mathbf {a})\). We conclude that f is a constant function. \(\square \)

Remark 3.5

Note that if X is a normed space, then (2.3) can be replaced by

$$\begin{aligned} \Delta _n ( f,[ \mathbf {x}, \mathbf {y}] ) := \left\| \underset{\theta \le \mathbf {1}}{\sum } (-1)^{|\theta |} \, f( \theta \, \mathbf {x} + (\mathbf {1} - \theta ) \mathbf {y}) \right\| . \end{aligned}$$

Example 3.6

Let \(c_0\) be the space of all null sequences with the \({\infty }\)-norm, and let \(\, f: [0, 1] \times [0, 1] \times [0, 1] \rightarrow c_0\)   be defined by

$$\begin{aligned} f( \mathbf {x}):= \left\{ {\sum \limits _{i=1}^{3}(-1)^{i+1} \frac{x_i}{n}}\right\} _{n \ge 1}. \end{aligned}$$

If   \(\mathbf {\xi } = \mathbf {\xi } _1 \times \mathbf {\xi } _2 \times \mathbf {\xi } _3 \), where \(\mathbf {\xi } _i := \{t_1^{(i)},t_2^{(i)},\ldots , t_{k_i}^{(i)} \}\), \(i=1,2,3\). Then,

$$\begin{aligned} \left\| \underset{\theta \le \mathbf {e}_i}{\sum } (-1)^{|\theta |} \, f( \mathbf {e}_i ( \theta \mathbf {t}_{\alpha - \mathbf {1}} + (\mathbf {1} - \theta ) \mathbf {t}_{\alpha } ) + (\mathbf {1}- \mathbf {e}_i) \mathbf {0})\right\| _{\infty }= & {} \left\| \, f(\mathbf {e}_i \mathbf {t}_{\alpha }) - f(\mathbf {e}_i \, \mathbf {t}_{\alpha - \mathbf {1}} ) \right\| _{\infty }\\= & {} \left\| \left( \dfrac{ t_{\alpha _i}^{(i)} - t_{\alpha _{i-1}}^{(i)} }{n}\right) _{n \in \mathbb {N}}\right\| _{\infty }. \end{aligned}$$

Hence,

$$\begin{aligned}&\sum \limits _{\mathbf {1 } \le {\varvec{\alpha }}\le \mathbf {\kappa } } \,\Phi \left( \dfrac{\Delta _1 \left( f_{\mathbf {e}_i}^{0}, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}]\lfloor \mathbf {e}_i \right) }{\mathrm{Vol} \, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}]\lfloor \mathbf {e}_i} \right) \mathrm{Vol} \, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}]\lfloor \mathbf {e}_i\nonumber \\&\quad = \sum \limits _{\mathbf {1} \le {\varvec{\alpha }}\le \mathbf {\kappa } } \, \Phi \left( \dfrac{ \left\| \left( \dfrac{ t_{\alpha _i}^{(i)} - t_{\alpha _{i-1}}^{(i)} }{n}\right) _{n \in \mathbb {N}}\right\| _{\infty }}{t_{\alpha _i}^{(i)} - t_{\alpha _{i-1}}^{(i)} } \right) (t_{\alpha _i}^{(i)} - t_{\alpha _{i-1}}^{(i)} )\nonumber \\&\quad = \sum \limits _{\mathbf {1} \le {\varvec{\alpha }}\le \mathbf {\kappa } } \, \Phi \left( \dfrac{ ( t_{\alpha _i}^{(i)} - t_{\alpha _{i-1}}^{(i)})\left\| \left( \dfrac{1 }{n}\right) _{n \in \mathbb {N}}\right\| _{\infty }}{t_{\alpha _i}^{(i)} - t_{\alpha _{i-1}}^{(i)} } \right) (t_{\alpha _i}^{(i)} - t_{\alpha _{i-1}}^{(i)} )\nonumber \\&\quad = \sum \limits _{\mathbf {1} \le {\varvec{\alpha }}\le \mathbf {\kappa } } \Phi (1) \, (t_{\alpha _i}^{(i)} - t_{\alpha _{i-1}}^{(i)} ) = \Phi (1). \end{aligned}$$
(3.6)

Consequently, for each \( i=1,2,3 \) we must have \( \mathrm{RV}_{\Phi }^{1}(f_{\mathbf {e}_i}^{0},[ \mathbf {a}, \mathbf {b}] ) := \Phi (1).\) In addition,

$$\begin{aligned}&\left\| \underset{\theta \le \mathbf {e}_1 + \mathbf {e}_2}{\sum } (-1)^{|\theta |} \, f( (\mathbf {e}_1 + \mathbf {e}_2) ( \theta \mathbf {t}_{\alpha - \mathbf {1}} + (\mathbf {1} - \theta ) \mathbf {t}_{\alpha } ) + (\mathbf {1}- (\mathbf {e}_1 + \mathbf {e}_2)) \mathbf {0})\right\| _{\infty }\\&\quad = \left\| \left( \dfrac{t_{\alpha _1}^{(1)} - t_{\alpha _2}^{(2)}}{n}\right) - \left( \dfrac{t_{\alpha _1-1}^{(1)} -t_{\alpha _2}^{(2)} }{n}\right) -\left( \dfrac{t_{\alpha _1}^{(1)} -t_{\alpha _2-1}^{(2)} }{n}\right) + \left( \dfrac{t_{\alpha _1-1}^{(1)} - t_{\alpha _2-1}^{(2)} }{n}\right) \right\| =0, \end{aligned}$$

and therefore, \( \mathrm{RV}_{\Phi }^{2}(f_{(1,1,0)}^{0},[ \mathbf {a}, \mathbf {b}] )= 0.\)

It can be verified similarly that \(\mathrm{RV}_{\Phi }^{2}(f_{\mathbf {e}_1+ \mathbf {e}_2+\mathbf {e}_3}^{\mathbf {0}},[ \mathbf {a}, \mathbf {b}] )=\mathrm{RV}_{\Phi }^{2}(f_{\mathbf {e}_i+ \mathbf {e}_j}^{\mathbf {0}},[ \mathbf {a}, \mathbf {b}] )= 0\), where \(i, j =1,2,3\) with \(i\ne j\).

From (3.2) we conclude that \(\mathrm{TRV}_{\Phi } (f,[ \mathbf {a}, \mathbf {b}] ) = 3 \Phi (1).\)

A proof of the following lemma can be found in [5] or in [6].

Lemma 3.7

[5, 6] If X is a normed space, then the functional \( \mathrm{TRV}_{\Phi } (\cdot ,[ \mathbf {a}, \mathbf {b}] )\) is convex.

4 Further properties

Theorem 4.1

Let X be a normed space. If \(f \in \mathrm{RV}_{\Phi } ([ \mathbf {a}, \mathbf {b}] ; X)\), then f is bounded.

Proof

It is well known that if \(f: [a,b] \rightarrow X, \, [a,b] \subset \mathbb {R}, \) is a function of bounded Riesz-\(\Phi \)-variation, then f is bounded (see [15]). Hence, if \(f: [ \mathbf {a}, \mathbf {b}] \rightarrow X\) is a function of bounded \(\Phi \)-variation on \([ \mathbf {a}, \mathbf {b}]:= [a_1, b_1] \times [a_2,b_2] \) then there are constants \(C_1\) and \(C_2\) such that

$$\begin{aligned} \Vert f_{\mathbf {e}_i}^{ \mathbf {a}}( \mathbf {x} \lfloor \mathbf {e}_i)\Vert _X \le C_i,&(i=1,2)&\text {for all} \quad \mathbf {x} \in [ \mathbf {a}, \mathbf {b}] . \end{aligned}$$

Suppose that f is not bounded. Then, for each \(m \in \mathbb {N}\) that satisfies \(m > C_1 + C_2\), there exists \( \mathbf {x}_m = (x_1^{m}, x_2^m) \in [ \mathbf {a}, \mathbf {b}]\) such that \(\Vert f( \mathbf {x}_m) + f( \mathbf {a})\Vert \ge m\). Hence, for all \(m > C_1 + C_2\),

$$\begin{aligned}&\Vert f( \mathbf {x}) + f( \mathbf {a}) - f(x_1^{m}, a_2) - f(a_1, x_2^{m})\Vert \nonumber \\&\quad \ge \Vert f( \mathbf {x}) + f( \mathbf {a})\Vert - \Vert f(x_1^{m}, a_2)\Vert - \Vert f(a_1, x_2^{m})\Vert \nonumber \\&\quad \ge \Vert f( \mathbf {x}) + f( \mathbf {a})\Vert - C_1 - C_2 \nonumber \\&\quad \ge m - C_1 - C_2. \end{aligned}$$
(4.1)

If \(\mathrm{Vol} \, [ \mathbf {a}, \mathbf {x}] \le 1\), then (4.1) implies

$$\begin{aligned} \Phi (m - C_1 - C_2)\le & {} \Phi \left( \mathrm{Vol} \, [ \mathbf {a}, \mathbf {x}] \dfrac{ \Vert f( \mathbf {x}) + f( \mathbf {a}) - f(x_1^{m}, a_2) - f(a_1, x_2^{m})\Vert }{\mathrm{Vol} \, [ \mathbf {a}, \mathbf {x}] } \right) \\\le & {} \Phi \left( \dfrac{ \Vert f( \mathbf {x}) + f( \mathbf {a}) - f(x_1^{m}, a_2) - f(a_1, x_2^{m})\Vert }{\mathrm{Vol} \, [ \mathbf {a}, \mathbf {x}] } \right) \mathrm{Vol} \, [ \mathbf {a}, \mathbf {x}] \\\le & {} \mathrm{RV}_{\Phi }^{2}(f,[ \mathbf {a}, \mathbf {b}] ), \end{aligned}$$

a contradiction (since \( \lim \nolimits _{m\rightarrow \infty }\Phi (m - C_1 - C_2) =\infty \)).

On the other hand, if \(\mathrm{Vol} \, [ \mathbf {a}, \mathbf {x}]> 1\) then from (4.1) we obtain

$$\begin{aligned} \Phi \left( \dfrac{m - C_1 - C_2}{\mathrm{Vol} \, [ \mathbf {a}, \mathbf {x}] }\right)\le & {} \Phi \left( \dfrac{ \Vert f( \mathbf {x}) + f( \mathbf {a}) - f(x_1^{m}, a_2) - f(a_1, x_2^{m})\Vert }{\mathrm{Vol} \, [ \mathbf {a}, \mathbf {x}] } \right) \\\le & {} \Phi \left( \dfrac{ \Vert f( \mathbf {x}) + f( \mathbf {a}) - f(x_1^{m}, a_2) - f(a_1, x_2^{m})\Vert }{\mathrm{Vol} \, [ \mathbf {a}, \mathbf {x}] } \right) \mathrm{Vol} \, [ \mathbf {a}, \mathbf {x}] \\\le & {} \mathrm{RV}_{\Phi }^{2}(f,[ \mathbf {a}, \mathbf {b}] ), \end{aligned}$$

which again leads to a contradiction; thus, f is a bounded function.

Suppose that the result holds for all the cases in which \([ \mathbf {a}, \mathbf {b}]\) is a k-dimensional interval with \(k < n\).

Consider now the case in which \(f: [ \mathbf {a}, \mathbf {b}] \rightarrow X\) where \([ \mathbf {a}, \mathbf {b}]\) is an n-dimensional interval. Then, for all multi-index \(\mathbf {0}< \mathbf {\eta }\le \mathbf {1} (\mathbf {0}< \mathbf {\eta })\) that satisfies \(| \mathbf {\eta }| < n\) there exists a constant \(M_{\eta }\) such that

$$\begin{aligned} \Vert f_{\eta }^{ \mathbf {a}}( \mathbf {x} \lfloor \eta ) \Vert \le M_{\eta }. \end{aligned}$$

In this case, if we suppose that f is not bounded, then for all \(m > \sum \nolimits _{\mathbf {0}< \mathbf {\eta }< \mathbf {1}} M_{\eta }\) there is a point \( \mathbf {x}_m = (x_1^{m}, x_2^{m}, \ldots , x_n^{m})\) such that

$$\begin{aligned} \Vert f( \mathbf {x}_m) + (-1)^{n}f( \mathbf {a})\Vert > m, \end{aligned}$$

and hence

$$\begin{aligned} \Delta _n(f,[ \mathbf {a}, \mathbf {x}_m] )= & {} \left\| \sum \limits _{\theta \le \mathbf {1}} (-1)^{|\theta |} f( \theta \, \mathbf {a} + (\mathbf {1} - \theta ) \mathbf {x}_m)\right\| \\\ge & {} \left\| f( \mathbf {x}_m) + (-1)^{n} f( \mathbf {a})\right\| - \sum \limits _{\begin{array}{c} \mathbf {0} \ne \theta \le \mathbf {1} \\ \theta \ne \mathbf {1} \end{array}} \left\| f( \theta \, \mathbf {a} + (\mathbf {1} - \theta ) \mathbf {x}_m)\right\| \\= & {} \left\| f( \mathbf {x}_m) + (-1)^{n} f( \mathbf {a})\right\| - \sum \limits _{\begin{array}{c} \mathbf {0} \ne \theta \le \mathbf {1} \\ \theta \ne \mathbf {1} \end{array}} \left\| f_{\mathbf {1} - \theta }^{\mathbf {a}}( \mathbf {x}_m \lfloor (\mathbf {1} - \theta ))\right\| \\\ge & {} m - \sum \limits _{\begin{array}{c} \mathbf {0} \ne \theta \le \mathbf {1} \\ \theta \ne \mathbf {1} \end{array}} M_{\mathbf {1}-\theta }. \end{aligned}$$

The result now follows, as in the \(n=2\) case, from the fact that

$$\begin{aligned} \lim \limits _{m\rightarrow \infty }\Phi \left( m - \sum \limits _{\begin{array}{c} \mathbf {0} \ne \theta \le \mathbf {1} \\ \theta \ne \mathbf {1} \end{array}} M_{\mathbf {1}-\theta }\right) =\infty , \end{aligned}$$

again, by considering the two cases \(\mathrm{Vol}([a, x_m]) \le 1\)   and   \(\mathrm{Vol}([a, x_m]) > 1\). \(\square \)

Theorem 4.2

Let X be a normed space. Let \(\Phi _1, \Phi _2 \in \mathcal {N}\) such that \(\Phi _1(x) \le K \Phi _2(x) \) for all x and some constant K, then \(\mathrm{RV}_{\Phi _2} ([ \mathbf {a}, \mathbf {b}] ; X) \subseteq \mathrm{RV}_{\Phi _1} ([ \mathbf {a}, \mathbf {b}] ; X).\)

Proof

Let \(f \in \mathrm{RV}_{\Phi _2} ([ \mathbf {a}, \mathbf {b}] ; X)\), then for all net partition \(\mathbf {\xi } =\{\mathbf {t}_{{ {\varvec{\alpha }}} }\} \in \Lambda ([ \mathbf {a}, \mathbf {b}])\) we have

$$\begin{aligned} \Phi _1 \left( \dfrac{\Delta _n \left( f, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}] \right) }{\mathrm{Vol} \, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}]} \right) \mathrm{Vol} \, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}]\le & {} K \Phi _2 \left( \dfrac{\Delta _n \left( f, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}] \right) }{\mathrm{Vol} \, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}]} \right) \mathrm{Vol} \, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}], \end{aligned}$$

from which the proposition follows. \(\square \)

Theorem 4.3

Let \(\Phi \in \mathcal {N}\) satisfying condition \(\Delta _2\) and let X be a normed space. Then \(\mathrm{RV}_{\Phi } ([ \mathbf {a}, \mathbf {b}] ; X)\) is a linear space.

Proof

Let K and \(x_0 \in \mathbb {R}\) be as in (2.2) and suppose \(f,g \in \mathrm{RV}_{\Phi } ([ \mathbf {a}, \mathbf {b}] ; X) \). Then, for any net partition \(\mathbf {\xi } =\{\mathbf {t}_{{ {\varvec{\alpha }}} }\} \in \Lambda ([ \mathbf {a}, \mathbf {b}])\)

$$\begin{aligned} \Delta _n \left( f+g, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}] \right)= & {} \left\| \underset{\theta \le \mathbf {1}}{\sum } (-1)^{|\theta |} \, (f+g)( \theta \, \mathbf {x} + (\mathbf {1} - \theta ) \mathbf {y}) \right\| \\\le & {} \left\| \underset{\theta \le \mathbf {1}}{\sum } (-1)^{|\theta |} \, f( \theta \, \mathbf {x} + (\mathbf {1} - \theta ) \mathbf {y}) \right\| \nonumber \\&+ \left\| \underset{\theta \le \mathbf {1}}{\sum } (-1)^{|\theta |} \, g( \theta \, \mathbf {x} + (\mathbf {1} - \theta ) \mathbf {y}) \right\| \\= & {} \Delta _n \left( f, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}] \right) + \Delta _n \left( g, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}] \right) . \end{aligned}$$

Put \( A_n = \dfrac{\Delta _n \left( f, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}] \right) + \Delta _n \left( g, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}] \right) }{\mathrm{Vol} \, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}]}\), then

$$\begin{aligned}&\mathrm{RV}_{\Phi }^{n}(f+g,[ \mathbf {a}, \mathbf {b}],\mathbf {\xi } )\\&\quad = \, \sum \limits _{\mathbf {1} \le {\varvec{\alpha }}\le \mathbf {\kappa } } \, \Phi \left( \dfrac{\Delta _n \left( f+g, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}] \right) }{\mathrm{Vol} \, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}]} \right) \mathrm{Vol} \, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}]\\&\quad \le \sum \limits _{\mathbf {1} \le {\varvec{\alpha }}\le \mathbf {\kappa } } \, \Phi \left( A_n \right) \mathrm{Vol} \, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}]\\&\quad = \sum \limits _{\begin{array}{c} \mathbf {1} \le {\varvec{\alpha }}\le \mathbf {\kappa } \\ A_n < x_0 \end{array}} \, \Phi \left( A_n \right) \mathrm{Vol} \, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}] + \sum \limits _{\begin{array}{c} \mathbf {1} \le {\varvec{\alpha }}\le \mathbf {\kappa } \\ A_n \ge x_0 \end{array}} \, \Phi \left( A_n \right) \mathrm{Vol} \, [ \mathbf {t}_{( {\varvec{\alpha }} - \mathbf {1})}, \mathbf {t}_{ {\varvec{\alpha }}}]\\&\quad \le \, \sum \limits _{\mathbf {1} \le {\varvec{\alpha }}\le \mathbf {\kappa } } \, \Phi \left( x_0 \right) \mathrm{Vol} \, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}]\\&\qquad + \, \sum \limits _{\mathbf {1} \le {\varvec{\alpha }}\le \mathbf {\kappa } } \, \Phi \left( \dfrac{1}{2} \dfrac{2 \Delta _n \left( f, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}] \right) }{\mathrm{Vol} \, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}]} + \dfrac{1}{2} \dfrac{2 \Delta _n \left( g, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}] \right) }{\mathrm{Vol} \, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}]}\right) \mathrm{Vol} \, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}]\\&\quad \le \Phi \left( x_0 \right) \mathrm{Vol} [ \mathbf {a}, \mathbf {b}]\\&\qquad + \,\sum \limits _{\mathbf {1} \le {\varvec{\alpha }}\le \mathbf {\kappa } } \,\dfrac{1}{2} \Phi \left( \dfrac{2 \Delta _n \left( f, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}] \right) }{\mathrm{Vol} \, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}]}\right) + \, \dfrac{1}{2} \Phi \left( \dfrac{2 \Delta _n \left( g, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}] \right) }{\mathrm{Vol} \, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}]}\right) \mathrm{Vol} \, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}]\\&\quad \le \Phi \left( x_0 \right) \mathrm{Vol} [ \mathbf {a}, \mathbf {b}]\\&\qquad + \, \sum \limits _{1 \le {\varvec{\alpha }}\le \mathbf {\kappa } } \,\dfrac{K}{2} \Phi \left( \dfrac{ \Delta _n \left( f, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}] \right) }{\mathrm{Vol} \, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}]}\right) + \, \dfrac{K}{2} \Phi \left( \dfrac{ \Delta _n \left( g, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}] \right) }{\mathrm{Vol} \, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}]}\right) \mathrm{Vol} \, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}]\\&\quad =\, \Phi \left( x_0 \right) \mathrm{Vol} [ \mathbf {a}, \mathbf {b}] + \dfrac{K}{2} \left( \mathrm{RV}_{\Phi }^{n}(f ,[ \mathbf {a}, \mathbf {b}],\mathbf {\xi } ) + \mathrm{RV}_{\Phi }^{n}(g ,[ \mathbf {a}, \mathbf {b}],\mathbf {\xi } ) \right) . \end{aligned}$$

Since this holds for all n, it follows that

$$\begin{aligned} \mathrm{TRV}_{\Phi } (f+g,[ \mathbf {a}, \mathbf {b}] ) \le \Phi \left( x_0 \right) \mathrm{Vol} [ \mathbf {a}, \mathbf {b}] + \dfrac{K}{2} \left( \mathrm{TRV}_{\Phi } (f ,[ \mathbf {a}, \mathbf {b}]) + \mathrm{TRV}_{\Phi } (g ,[ \mathbf {a}, \mathbf {b}]) \right) , \end{aligned}$$

from which we conclude that \(f+g \in \mathrm{RV}_{\Phi } ([ \mathbf {a}, \mathbf {b}] ; X)\). On the other hand, if \(\gamma \) is any scalar, then

$$\begin{aligned} \begin{array}{lll} \Delta _n \left( \gamma f, [ \mathbf {t}_{\alpha - \mathbf {1}}, \mathbf {t}_{\alpha }] \right) &{}=&{} \left\| \underset{\theta \le \mathbf {1}}{\sum } (-1)^{|\theta |} \, \gamma f ( \theta \, \mathbf {x} + (\mathbf {1} - \theta ) \mathbf {y}) \right\| \\ &{} = &{} |\gamma |\left\| \underset{\theta \le \mathbf {1}}{\sum } (-1)^{|\theta |} \, f ( \theta \, \mathbf {x} + (\mathbf {1} - \theta ) \mathbf {y}) \right\| = |\gamma |\Delta _n \left( f, [ \mathbf {t}_{\alpha - \mathbf {1}}, \mathbf {t}_{ \alpha }] \right) . \end{array} \end{aligned}$$

Thus,

$$\begin{aligned} \mathrm{RV}_{\Phi }^{n}(\gamma f,[ \mathbf {a}, \mathbf {b}],\mathbf {\xi } )= & {} \sum \limits _{1 \le {\varvec{\alpha }}\le \mathbf {\kappa } } \, \Phi \left( \dfrac{\Delta _n \left( \gamma f, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}] \right) }{\mathrm{Vol} \, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}]} \right) \mathrm{Vol} \, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}]\\= & {} \sum \limits _{1 \le {\varvec{\alpha }}\le \mathbf {\kappa } } \, \Phi \left( \dfrac{|\gamma |\Delta _n \left( f, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}] \right) }{\mathrm{Vol} \, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}]} \right) \mathrm{Vol} \, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}]. \end{aligned}$$

If \(|\gamma | \le 1\), then

$$\begin{aligned} \mathrm{RV}_{\Phi }^{n}(\gamma f,[ \mathbf {a}, \mathbf {b}],\mathbf {\xi } )= & {} \sum \limits _{1 \le {\varvec{\alpha }}\le \mathbf {\kappa } } \, \Phi \left( \dfrac{|\gamma |\Delta _n \left( f, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}] \right) }{\mathrm{Vol} \, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}]} \right) \mathrm{Vol} \, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}]\\\le & {} |\gamma | \sum \limits _{1 \le {\varvec{\alpha }}\le \mathbf {\kappa } } \, \Phi \left( \dfrac{\Delta _n \left( f, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}] \right) }{\mathrm{Vol} \, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}]} \right) \mathrm{Vol} \, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}]\\= & {} |\gamma | \mathrm{RV}_{\Phi }^{n}(f,[ \mathbf {a}, \mathbf {b}],\mathbf {\xi } ). \end{aligned}$$

On the other hand, if \(|\gamma | > 1\) then, again by (2.2), there is a constant \(K'\) and a point \(x_0\) such that

$$\begin{aligned} \mathrm{RV}_{\Phi }^{n}(\gamma f,[ \mathbf {a}, \mathbf {b}],\mathbf {\xi } )= & {} \sum \limits _{1 \le {\varvec{\alpha }}\le \mathbf {\kappa } } \, \Phi \left( \dfrac{|\gamma |\Delta _n \left( f, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}] \right) }{\mathrm{Vol} \, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}]} \right) \mathrm{Vol} \, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}]\\\le & {} \Phi (x_0) \mathrm{Vol} [ \mathbf {a}, \mathbf {b}] + K' \sum \limits _{1 \le {\varvec{\alpha }}\le \mathbf {\kappa } } \, \Phi \left( \dfrac{\Delta _n \left( f, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}] \right) }{\mathrm{Vol} \, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}]} \right) \mathrm{Vol} \, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}]\\\le & {} \Phi (x_0) \mathrm{Vol} [ \mathbf {a}, \mathbf {b}] + K' \, \mathrm{RV}_{\Phi }^{n}(f,[ \mathbf {a}, \mathbf {b}],\mathbf {\xi } ). \end{aligned}$$

It follows that \(\gamma f \in \mathrm{RV}_{\Phi } ([ \mathbf {a}, \mathbf {b}] ; X)\). We conclude that \(\mathrm{RV}_{\Phi } ([ \mathbf {a}, \mathbf {b}] ; X)\) is a linear space. \(\square \)

Lemma 4.4

Let X be a metric semigroup and suppose \(f \in \mathrm{RV}_{\Phi } ([ \mathbf {a}, \mathbf {b}] ; X)\). If \( \mathbf {x}\),\( \mathbf {y} \in [ \mathbf {a}, \mathbf {b}]\) are such that \(x_k = y_k\) for some \(0 \le k \le n\), then

$$\begin{aligned} \underset{\theta \in \mathcal {E}(n)}{\sum } f( \theta \, \mathbf {x} + (\mathbf {1} - \theta ) \mathbf {y}) = \underset{\theta \in \mathcal {O}(n)}{\sum } f( \theta \, \mathbf {x} + (\mathbf {1} - \theta ) \mathbf {y}). \end{aligned}$$

Proof

Let f, \( \mathbf {x}\) and \( \mathbf {y}\) be as in the statement. If \(\theta = (\theta _1, \theta _2, \ldots , \theta _n) \in \mathcal {E}(n)\) then \(\widetilde{\theta } := (\theta _1, \theta _2, \ldots , 1-\theta _k, \ldots , \theta _n) \in \mathcal {O}(n)\), thus, \(( \theta \, \mathbf {x} + (\mathbf {1} - \theta ) \mathbf {y}) \) has the same entries as \(( \widetilde{\theta } \, \mathbf {x} + (\mathbf {1} - \widetilde{\theta }) \mathbf {y})\) since the kth entry \(\widetilde{\theta }_k \, x_k + (1 - \widetilde{\theta }_k) y_k = (1 - \theta _k) \, x_k + \theta _k y_k = (1 - \theta _k) \, y_k + \theta _k x_k.\) Hence,

$$\begin{aligned} \underset{\theta \in \mathcal {E}(n)}{\sum } f( \theta \, \mathbf {x} + (\mathbf {1} - \theta ) \mathbf {y})= & {} \underset{\theta \in \mathcal {E}(n)}{\sum } f( \widetilde{\theta } \, \mathbf {x} + (\mathbf {1} - \widetilde{\theta }) \mathbf {y}) = \underset{\theta \in \mathcal {O}(n)}{\sum } f( \theta \, \mathbf {x} + (\mathbf {1} - \theta ) \mathbf {y}). \end{aligned}$$

\(\square \)

Lemma 4.5

Let X be a metric semigroup and let \([ \mathbf {x}, \mathbf {y}]\) be an n-dimensional interval. Suppose that \(t^{(j)} \in [x_j, y_j]\) for some \(1 \le j \le n\). Then

$$\begin{aligned} \Delta _n(f, [ \mathbf {x}, \mathbf {y}] ) \le \Delta _n(f, [ \mathbf {x}, \widetilde{\mathbf {y}}] ) + \Delta _n(f, [\widetilde{\mathbf {x}}, \mathbf {y}] ), \end{aligned}$$

where \(\mathbf {\widetilde{x}}:=(\widetilde{x}_1, \ldots , \widetilde{x}_n)\) with \(\widetilde{x}_i = x_i\)   if   \(i \ne j\)  and   \(\widetilde{x}_j = t^{(j)}\),   and   \(\mathbf {\widetilde{y}}:=(\widetilde{y}_1, \ldots , \widetilde{y}_n)\) with \(\widetilde{y}_i = y_i\) if   \(i \ne j\)   and   \(\widetilde{y}_j = t^{(j)}\).

Proof

Note that the interval \([ \mathbf {x}, \mathbf {y}]\) can be divided into two intervals, namely \([ \mathbf {x}, \widetilde{\mathbf {y}}]\) and \([\widetilde{\mathbf {x}}, \mathbf {y}]\), thus, by virtue of property (2.1) and lemma 4.4 we have

$$\begin{aligned}&\Delta _n(f, [ \mathbf {x}, \mathbf {y}] )\\&\quad := d\left( \underset{\theta \in \mathcal {E}(n)}{\sum } f( \theta \, \mathbf {x} + (\mathbf {1} - \theta ) \mathbf {y}), \underset{\theta \in \mathcal {O}(n)}{\sum } f( \theta \, \mathbf {x} + (\mathbf {1} - \theta ) \mathbf {y})\right) \\&\quad = d\left( \underset{\theta \in \mathcal {E}(n)}{\sum } f( \theta \, \mathbf {x} + (\mathbf {1} - \theta ) \mathbf {y}) + \underset{\theta \in \mathcal {E}(n)}{\sum } f( \theta \, \mathbf {\widetilde{x}}+ (\mathbf {1} - \theta ) \mathbf {\widetilde{y}}) ,\right. \\&\qquad \left. \underset{\theta \in \mathcal {O}(n)}{\sum } f( \theta \, \mathbf {x} + (\mathbf {1} - \theta ) \mathbf {y})+ \underset{\theta \in \mathcal {E}(n)}{\sum } f( \theta \, \mathbf {\widetilde{x}}+ (\mathbf {1} - \theta ) \mathbf {\widetilde{y}})\right) \\&\quad = d\left( \underset{\theta \in \mathcal {E}(n)}{\sum } [ f( \theta \, \mathbf {x} + (\mathbf {1} - \theta ) \mathbf {y}) + f( \theta \, \mathbf {\widetilde{x}}+ (\mathbf {1} - \theta ) \mathbf {\widetilde{y}})] ,\right. \\&\qquad \left. \underset{\theta \in \mathcal {O}(n)}{\sum } [f( \theta \, \mathbf {x} + (\mathbf {1} - \theta ) \mathbf {y})+ f( \theta \, \mathbf {\widetilde{x}}+ (\mathbf {1} - \theta ) \mathbf {\widetilde{y}})]\right) \\&\quad = d\left( \underset{\begin{array}{c} \theta \in \mathcal {E}(n) \\ \theta _j = 1 \end{array}}{\sum } [ f( \theta \, \mathbf {x} + (\mathbf {1} - \theta ) \mathbf {y}) \right. \\&\qquad + f( \theta \, \mathbf {\widetilde{x}}+ (\mathbf {1} - \theta ) \mathbf {\widetilde{y}})] + \underset{\begin{array}{c} \theta \in \mathcal {E}(n) \\ \theta _j = 0 \end{array}}{\sum } [ f( \theta \, \mathbf {x} + (\mathbf {1} - \theta ) \mathbf {y}) + f( \theta \, \mathbf {\widetilde{x}}+ (\mathbf {1} - \theta ) \mathbf {\widetilde{y}})] ,\\&\qquad \underset{\begin{array}{c} \theta \in \mathcal {O}(n) \\ \theta _j = 1 \end{array}}{\sum } [f( \theta \, \mathbf {x} + (\mathbf {1} - \theta ) \mathbf {y}) \\&\qquad + \left. f( \theta \, \mathbf {\widetilde{x}}+ (\mathbf {1} - \theta ) \mathbf {\widetilde{y}})] + \underset{\begin{array}{c} \theta \in \mathcal {O}(n) \\ \theta _j = 0 \end{array}}{\sum } [f( \theta \, \mathbf {x} + (\mathbf {1} - \theta ) \mathbf {y})+ f( \theta \, \mathbf {\widetilde{x}}+ (\mathbf {1} - \theta ) \mathbf {\widetilde{y}})]\right) \\&\quad = d\left( \underset{\theta \in \mathcal {E}(n)}{\sum } f( \theta \, \mathbf {x} + (\mathbf {1} - \theta ) \mathbf {\widetilde{y}}) + \underset{\theta \in \mathcal {E}(n)}{\sum } f( \theta \, \mathbf {\widetilde{x}} + (\mathbf {1} - \theta ) \mathbf {y}), \right. \\&\qquad \left. \underset{\theta \in \mathcal {O}(n)}{\sum } f( \theta \, \mathbf {x} + (\mathbf {1} - \theta ) \mathbf {\widetilde{y}}) + \underset{\theta \in \mathcal {O}(n)}{\sum } f( \theta \, \mathbf {\widetilde{x}} + (\mathbf {1} - \theta ) \mathbf {y})\right) \\&\quad \le \Delta _n(f, [ \mathbf {x}, \mathbf {\widetilde{y}}] ) + \Delta _n(f, [\mathbf {\widetilde{x}}, \mathbf {y}] ). \end{aligned}$$

\(\square \)

Theorem 4.6

Let X be a metric semigroup. If \(f:[ \mathbf {a}, \mathbf {b}] \rightarrow X\) is a function, then

$$\begin{aligned} \mathrm{RV}^{n}_{\Phi }(f,[ \mathbf {a}, \mathbf {b}], \mathbf {\xi } _1 \times \cdots \times \{\mathbf {\xi } _{z}\cup \{t^{(z)} \} \} \times \cdots \times \mathbf {\xi } _n )\ge \mathrm{RV}^{n}_{\Phi } \left( f,[ \mathbf {a}, \mathbf {b}], \mathbf {\xi } \right) \end{aligned}$$

for any net partition \(\mathbf {\xi } =\prod \nolimits _{i=1}^{n} \mathbf {\xi } _i\) of \([ \mathbf {a}, \mathbf {b}]\).

Proof

Suppose \(\mathbf {\xi } = \prod \nolimits _{i=1}^{n} \mathbf {\xi } _i\) where \(\mathbf {\xi } _i = \{a_i= t_0^{(i)}, t_1^{(i)}, \ldots , t_{k_i}^{(i)} = b_i\}\). Assume \(z \in \{1, \ldots , n\}\) and let \( t^{(z)} \) be such that \(t_0^{(z)}< t_1^{(z)}<\dots< t_{r-1}^{(z)}< t^{(z)}< t_r^{(z)}< \dots < t_{k_j}^{(z)}.\) Put \(\varrho := \prod \nolimits _{i=1}^{n} \varrho _i\) where \(\varrho _i = \{a_i= s_0^{(i)}, s_2^{(i)}, \ldots , s_{\widehat{k}_i}^{(i)} = b_i\}\), with

$$\begin{aligned} s_{j}^{(l)} = t_{j}^{(l)}, \, \text { for } \, l \ne z&\text { and }&\left\{ \begin{array}{l@{\quad }ll} s_{l}^{(z)}= t_{l}^{(z)} &{} \text {if} &{} 0 \le l \le r-1 \\ s_{r}^{(z)}= t^{(z)} &{} &{} \\ s_{l}^{(z)} = t_{l-1}^{(z)} &{} \text {if} &{} l \ge r+1. \end{array}\right. \end{aligned}$$

Then,

$$\begin{aligned} \sum \limits _{\begin{array}{c} 1 \le {\varvec{\alpha }}\le \widehat{ \mathbf {\kappa }} \\ \alpha _z< r \end{array}} \, \Phi \left( \dfrac{\Delta _n \left( f, [ \mathbf {s}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {s}_{ {\varvec{\alpha }}}] \right) }{\mathrm{Vol} \,[ \mathbf {s}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {s}_{ {\varvec{\alpha }}}]}\right)= & {} \sum \limits _{\begin{array}{c} 1 \le {\varvec{\alpha }}\le \mathbf {\kappa } \\ \alpha _z < r \end{array}} \, \Phi \left( \dfrac{\Delta _n \left( f, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}] \right) }{\mathrm{Vol} \, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}]}\right) \mathrm{and} \\ \sum \limits _{\begin{array}{c} 1 \le {\varvec{\alpha }}\le \widehat{ \mathbf {\kappa }} \\ \alpha _z> r+1 \end{array}} \, \Phi \left( \dfrac{\Delta _n \left( f, [ \mathbf {s}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {s}_{ {\varvec{\alpha }}}] \right) }{\mathrm{Vol} \,[ \mathbf {s}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {s}_{ {\varvec{\alpha }}}]}\right)= & {} \sum \limits _{\begin{array}{c} 1 \le {\varvec{\alpha }}\le \mathbf {\kappa } \\ \alpha _z > r \end{array}} \, \Phi \left( \dfrac{\Delta _n \left( f, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}] \right) }{\mathrm{Vol} \, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}]}\right) . \end{aligned}$$

Hence,

$$\begin{aligned}&\sum \limits _{\mathbf {1} \le {\varvec{\alpha }}\le \widehat{ \mathbf {\kappa }} } \, \Phi \left( \dfrac{\Delta _n \left( f, [ \mathbf {s}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {s}_{ {\varvec{\alpha }}}] \right) }{\mathrm{Vol} \,[ \mathbf {s}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {s}_{ {\varvec{\alpha }}}]}\right) [ \mathbf {s}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {s}_{ {\varvec{\alpha }}}]\\&\qquad - \sum \limits _{\mathbf {1} \le {\varvec{\alpha }}\le \mathbf {\kappa } } \, \Phi \left( \dfrac{\Delta _n \left( f, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}] \right) }{\mathrm{Vol} \, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}]}\right) \mathrm{Vol} \, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}]\\&\quad = \sum \limits _{\begin{array}{c} \mathbf {1} \le {\varvec{\alpha }}\le \widehat{ \mathbf {\kappa }} \\ \alpha _z = r \end{array}} \, \Phi \left( \frac{\Delta _n \left( f, [ \mathbf {s}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {s}_{ {\varvec{\alpha }}}] \right) }{\mathrm{Vol}\,[ \mathbf {s}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {s}_{ {\varvec{\alpha }}}]}\right) \mathrm{Vol}\,[ \mathbf {s}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {s}_{ {\varvec{\alpha }}}] \\&\qquad - \sum \limits _{\begin{array}{c} \mathbf {1} \le {\varvec{\alpha }}\le \mathbf {\kappa } \\ \alpha _z =r \end{array}} \, \Phi \left( \dfrac{\Delta _n \left( f, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}] \right) }{\mathrm{Vol}\,[ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}] }\right) \mathrm{Vol}\,[ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}] \\&\qquad +\sum \limits _{\begin{array}{c} \mathbf {1} \le {\varvec{\alpha }}\le \widehat{ \mathbf {\kappa }} \\ \alpha _z = r + 1 \end{array}} \, \Phi \left( \dfrac{ \Delta _n \left( f, [ \mathbf {s}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {s}_{ {\varvec{\alpha }}}] \right) }{\mathrm{Vol}\,[ \mathbf {s}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {s}_{ {\varvec{\alpha }}}] }\right) \mathrm{Vol}\,[ \mathbf {s}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {s}_{ {\varvec{\alpha }}}]. \end{aligned}$$

Now if \(\widetilde{t}_{\alpha _i}^{(i)}=t_{\alpha _i}^{(i)}\) for \(i\ne z\) and \(\widetilde{t}_{\alpha _z}^{(z)}=t^{(z)}\), by Lemma 4.5, we obtain

$$\begin{aligned}&= \sum \limits _{\begin{array}{c} \mathbf {1} \le {\varvec{\alpha }}\le \widehat{ \mathbf {\kappa }} \\ \alpha _z = r \end{array}} \, \Phi \left( \dfrac{\Delta _n \left( f, [ \mathbf {s}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {s}_{ {\varvec{\alpha }}}] \right) }{\mathrm{Vol}\,[ \mathbf {s}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {s}_{ {\varvec{\alpha }}}] }\right) \mathrm{Vol}\,[ \mathbf {s}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {s}_{ {\varvec{\alpha }}}] \\&\quad - \sum \limits _{\begin{array}{c} \mathbf {1} \le {\varvec{\alpha }}\le \mathbf {\kappa } \\ \alpha _z =r \end{array}} \, \Phi \left( \frac{\Delta _n \left( f, [\widetilde{\mathbf {t}}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}] \right) }{\mathrm{Vol}\,[ \mathbf {t}_{ {\varvec{\alpha }} - 1}, \mathbf {t}_{ {\varvec{\alpha }}}]} + \frac{\Delta _n \left( f, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \widetilde{\mathbf {t}}_{ {\varvec{\alpha }}}] \right) }{\mathrm{Vol}\,[ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}]}\right) \mathrm{Vol}\,[ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}]\\&\quad +\sum \limits _{\begin{array}{c} \mathbf {1} \le {\varvec{\alpha }}\le \widehat{ \mathbf {\kappa }} \\ \alpha _z = r + 1 \end{array}}\, \Phi \left( \dfrac{\Delta _n \left( f, [ \mathbf {s}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {s}_{ {\varvec{\alpha }}}] \right) }{\mathrm{Vol}\,[ \mathbf {s}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {s}_{ {\varvec{\alpha }}}] }\right) \mathrm{Vol}\,[ \mathbf {s}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {s}_{ {\varvec{\alpha }}}] \\&\mathbf {=} \sum \limits _{\begin{array}{c} \mathbf {1} \le {\varvec{\alpha }}\le \widehat{ \mathbf {\kappa }} \\ \alpha _z = r \end{array}} \, \Phi \left( \dfrac{\Delta _n \left( f, [ \mathbf {s}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {s}_{ {\varvec{\alpha }}}] \right) }{\mathrm{Vol}\,[ \mathbf {s}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {s}_{ {\varvec{\alpha }}}] }\right) \mathrm{Vol}\,[ \mathbf {s}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {s}_{ {\varvec{\alpha }}}] \\&\quad - \sum \limits _{\begin{array}{c} \mathbf {1} \le {\varvec{\alpha }}\le \mathbf {\kappa } \\ \alpha _z =r \end{array}} \, \Phi \left( \frac{\Delta _n \left( f, [\widetilde{\mathbf {t}}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}] \right) }{\mathrm{Vol}\,[\widetilde{\mathbf {t}}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}]}\dfrac{\mathrm{Vol}\,[\widetilde{\mathbf {t}}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}]}{\mathrm{Vol}\,[ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}]}\right. \\&\quad + \left. \frac{\Delta _n \left( f, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \widetilde{\mathbf {t}}_{ {\varvec{\alpha }}}] \right) }{\mathrm{Vol}\,[ \mathbf {t}_{{\varvec{\alpha }} - \mathbf {1}}, \widetilde{\mathbf {t}}_{ {\varvec{\alpha }}}]}\dfrac{\mathrm{Vol}\,[ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \widetilde{\mathbf {t}}_{ {\varvec{\alpha }}}]}{\mathrm{Vol}\,[ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}]}\right) \mathrm{Vol}\,[ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}]\\&\quad +\sum \limits _{\begin{array}{c} \mathbf {1} \le {\varvec{\alpha }}\le \widehat{ \mathbf {\kappa }} \\ \alpha _z = r + 1 \end{array}} \, \Phi \left( \dfrac{\Delta _n \left( f, [ \mathbf {s}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {s}_{ {\varvec{\alpha }}}] \right) }{\mathrm{Vol}\,[ \mathbf {s}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {s}_{ {\varvec{\alpha }}}] }\right) \mathrm{Vol}\,[ \mathbf {s}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {s}_{ {\varvec{\alpha }}}] \end{aligned}$$

and, by the convexity of \(\Phi \) we get:

$$\begin{aligned}\ge & {} \sum \limits _{\begin{array}{c} \mathbf {1} \le {\varvec{\alpha }}\le \widehat{ \mathbf {\kappa }} \\ alpha_z = r \end{array}} \, \Phi \left( \dfrac{\Delta _n \left( f, [ \mathbf {s}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {s}_{ {\varvec{\alpha }}}] \right) }{\mathrm{Vol}\,[ \mathbf {s}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {s}_{ {\varvec{\alpha }}}] }\right) \mathrm{Vol}\,[ \mathbf {s}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {s}_{ {\varvec{\alpha }}}] \\&- \sum \limits _{\begin{array}{c} \mathbf {1} \le {\varvec{\alpha }}\le \mathbf {\kappa } \\ \alpha _z =r \end{array}} \, \dfrac{\mathrm{Vol}\,[\widetilde{\mathbf {t}}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}]}{\mathrm{Vol}\,[ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}]} \Phi \left( \frac{\Delta _n \left( f, [\widetilde{\mathbf {t}}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}] \right) }{\mathrm{Vol}\,[\widetilde{\mathbf {t}}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}]}\right) \mathrm{Vol}\,[ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}]\\&- \dfrac{\mathrm{Vol}\,[ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \widetilde{\mathbf {t}}_{ {\varvec{\alpha }}}]}{\mathrm{Vol}\,[ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}]} \Phi \left( \frac{\Delta _n \left( f, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \widetilde{\mathbf {t}}_{ {\varvec{\alpha }}}] \right) }{\mathrm{Vol}\,[ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \widetilde{\mathbf {t}}_{ {\varvec{\alpha }}}]}\right) \mathrm{Vol}\,[ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}]\\&+\sum \limits _{\begin{array}{c} \mathbf {1} \le {\varvec{\alpha }}\le \widehat{ \mathbf {\kappa }} \\ \alpha _z = r + 1 \end{array}} \, \Phi \left( \dfrac{\Delta _n \left( f, [ \mathbf {s}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {s}_{ {\varvec{\alpha }}}] \right) }{\mathrm{Vol}\,[ \mathbf {s}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {s}_{ {\varvec{\alpha }}}] }\right) \mathrm{Vol}\,[ \mathbf {s}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {s}_{ {\varvec{\alpha }}}]\\= & {} \sum \limits _{\begin{array}{c} \mathbf {1} \le {\varvec{\alpha }}\le \widehat{ \mathbf {\kappa }} \\ \alpha _z = r \end{array}} \, \Phi \left( \dfrac{\Delta _n \left( f, [ \mathbf {s}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {s}_{ {\varvec{\alpha }}}] \right) }{\mathrm{Vol}\,[ \mathbf {s}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {s}_{ {\varvec{\alpha }}}] }\right) \mathrm{Vol}\,[ \mathbf {s}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {s}_{ {\varvec{\alpha }}}] \\&- \sum \limits _{\begin{array}{c} \mathbf {1} \le {\varvec{\alpha }}\le \mathbf {\kappa } \\ \alpha _z = r + 1 \end{array}} \, \Phi \left( \frac{\Delta _n \left( f, [ \mathbf {s}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {s}_{ {\varvec{\alpha }}}] \right) }{\mathrm{Vol}\,[ \mathbf {s}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {s}_{ {\varvec{\alpha }}}]}\right) \mathrm{Vol}\,[ \mathbf {s}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {s}_{ {\varvec{\alpha }}}] \\&- \sum \limits _{\begin{array}{c} \mathbf {1 } \le {\varvec{\alpha }}\le \mathbf {\kappa } \\ \alpha _z = r \end{array}} \Phi \left( \frac{\Delta _n \left( f, \mathrm{Vol}\,[ \mathbf {s}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {s}_{ {\varvec{\alpha }}}] \right) }{\mathrm{Vol}\,[ \mathbf {s}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {s}_{ {\varvec{\alpha }}}]}\right) \mathrm{Vol}\,[ \mathbf {s}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {s}_{ {\varvec{\alpha }}}]\\&+\sum \limits _{\begin{array}{c} \mathbf {1} \le {\varvec{\alpha }}\le \widehat{ \mathbf {\kappa }} \\ \alpha _z = r + 1 \end{array}} \, \Phi \left( \dfrac{\Delta _n \left( f, [ \mathbf {s}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {s}_{ {\varvec{\alpha }}}] \right) }{\mathrm{Vol}\,[ \mathbf {s}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {s}_{ {\varvec{\alpha }}}] }\right) \mathrm{Vol}\,[ \mathbf {s}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {s}_{ {\varvec{\alpha }}}] = 0. \end{aligned}$$

It follows that

$$\begin{aligned}&\sum \limits _{\mathbf {1} \le {\varvec{\alpha }}\le \widehat{ \mathbf {\kappa }} } \, \Phi \left( \Delta _n \left( f, [ \mathbf {s}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {s}_{ {\varvec{\alpha }}}] \right) \right) \, \mathrm{Vol} \, [ \mathbf {s}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {s}_{ {\varvec{\alpha }}}]\\&\quad - \sum \limits _{\mathbf {1} \le {\varvec{\alpha }} \le \mathbf {\kappa } } \, \Phi \left( \Delta _n \left( f, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}] \right) \right) \, \mathrm{Vol} \, [ \mathbf {t}_{ {\varvec{\alpha }} - \mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}] \ge 0, \end{aligned}$$

and therefore, \( \mathrm{RV}^{n}_{\Phi }\left( f,[ \mathbf {a}, \mathbf {b}], \mathbf {\xi }_1 \times \cdots \times \mathbf {\xi } _{z}\cup \{t^{(z)}\} \times \cdots \times \mathbf {\xi } _n \right) \ge \mathrm{RV}^{n}_{\Phi } (f,[ \mathbf {a}, \mathbf {b}], \prod \nolimits _{i=1}^{n} \mathbf {\xi } _i ). \) \(\square \)

Corollary 4.7

Let X be a metric semigroup. If \(f:[ \mathbf {a}, \mathbf {b}] \rightarrow X\), \(\mathbf {\xi } \) and \(\delta \) are any net partitions of \([ \mathbf {a}, \mathbf {b}]\), such that \(\mathbf {\xi } \subseteq \delta , \) then \( V^{n}(f,[ \mathbf {a}, \mathbf {b}], \mathbf {\xi } ) \le V^{n}\left( f,[ \mathbf {a}, \mathbf {b}], \delta \right) . \)

Proof

It suffices to apply Theorem 4.6 finitely many times. \(\square \)

5 A representation theorem

In our next result, we present a counterpart of the classical Riesz’s lemma (cf., [3, 24]) in the n-dimensional case. It will be assumed that \(X=\mathbb {R}\) and the following well-known notation will be used: given any multi-index \(\beta = (\beta _1, \ldots , \beta _k)\) we define

$$\begin{aligned} D^{\beta } := \dfrac{\partial ^{\beta _1 + \cdots + \beta _k}}{\partial x_1^{\beta _1}\cdots \partial x_k^{\beta _k}}. \end{aligned}$$

Theorem 5.1

Let \(\, \Phi \in \mathcal {N}.\, \) If \( \, f \in \mathrm{RV}_{\Phi } ([ \mathbf {a}, \mathbf {b}] ; \mathbb {R})\) and is of class \(C^{n}\), then

$$\begin{aligned} \sum \limits _{\mathbf { 0}\ne \eta \le \mathbf {1}}(n-|\eta |)! \int _{[ \mathbf {a}, \mathbf {b}]\lfloor \eta } \Phi \left( \left| D^{\eta } f^{ \mathbf {a}}_{\eta }( \mathbf {x} \lfloor \eta ) \right| \right) d( \mathbf {x}\lfloor \eta ) = \mathrm{TRV}_{\Phi } (f,[ \mathbf {a}, \mathbf {b}] ). \end{aligned}$$
(5.1)

Proof

Let \(\mathbf {\xi } _i := \{t_1^{(i)},t_2^{(i)}, \ldots , t_{k_i}^{(i)}\}\), \(i=1,\ldots ,n\) and let \(\mathbf {\xi } = \prod \nolimits _{i=1}^{n} \mathbf {\xi } _i= \{\mathbf {t}_{\alpha }\}\) be a net partition of \([ \mathbf {a}, \mathbf {b}]\) with \( \mathbf {\kappa } = (k_1, k_2, \ldots , k_n)\). Then, for all \( {\varvec{\alpha }} \le \mathbf {\kappa }\) we have

$$\begin{aligned}&\underset{\theta \le \mathbf {1}}{\sum } (-1)^{|\theta |} \, f( \theta \mathbf {t}_{ \alpha -\mathbf {1} } + (\mathbf {1}- \theta ) \mathbf {t}_{ {\varvec{\alpha }}} ) = \,\, \underset{\begin{array}{c} \theta \le \mathbf {1} \\ \theta _i = 1 \end{array}}{\sum } (-1)^{|\theta |} \, f( \theta \mathbf {t}_{ {\varvec{\alpha }}-\mathbf {1}} + (\mathbf {1} - \theta ) \mathbf {t}_{ {\varvec{\alpha }}} ) \\&\quad - \sum \limits _{\begin{array}{c} 1 \le {\varvec{\alpha }}\le \mathbf {\kappa } \\ \alpha _z = r \end{array}} \underset{\begin{array}{c} \theta \le \mathbf {1} \\ \theta _i = 0 \end{array}}{\sum } (-1)^{|\theta |+1} \, f( \theta \mathbf {t}_{ {\varvec{\alpha }}-\mathbf {1} } + (\mathbf {1} - \theta ) \mathbf {t}_{ {\varvec{\alpha }}} ), \end{aligned}$$

where \( z \le k_z\).

Let \( \mathbf {x} = (x_1, x_2, \ldots , x_n)\) , \( \mathbf {v}= \theta \mathbf {t}_{( {\varvec{\alpha }}-1)} + (\mathbf {1} - \theta ) \mathbf {t}_{ {\varvec{\alpha }}}\) and define the function

$$\begin{aligned} g_1(x_1) := \underset{\begin{array}{c} \theta \le \mathbf {1} \\ \theta _1 = 1 \end{array}}{\sum } (-1)^{|\theta |} \, f ( \mathbf {e}_1 \mathbf {x} + (1 - \mathbf {e}_1) \mathbf {v} ) \end{aligned}$$

(\(\mathbf {e}_i\) denotes the canonical unit vectors of \(\mathbb {R}^{n}\)). Then, since f is differentiable, \(g_1: [t_{\alpha _1-1}^{1}, t_{\alpha _1}^{1}] \rightarrow \mathbb {R}\), satisfies the conditions of the ordinary mean value theorem and thus, there is an \(x_{\alpha _1}^{1} \in (t_{\alpha _1-1}^{1}, t_{\alpha _1}^{1}) \) such that,

$$\begin{aligned} g_1'(x_{\alpha _1}^{1})= & {} \dfrac{ g_1(t_{\alpha _1}^{1}) - g_1(t_{\alpha _1-1}^{1})}{ t_{\alpha _1}^{1}- t_{\alpha _1-1}^{1} }. \end{aligned}$$

That is,

$$\begin{aligned} \underset{\begin{array}{c} \theta \le \mathbf {1} \\ \theta _1 = 1 \end{array}}{\sum } (-1)^{|\theta |} \, D^{\mathbf {e}_1} f_{1- \mathbf {e}_1}^{x_{\alpha _1}^{1} \mathbf {e}_1} (\mathbf {v}\lfloor (\mathbf {1}-\mathbf {e}_1 ) )= & {} \dfrac{\sum _{\begin{array}{c} \theta \le \mathbf {1} \\ \theta _1 = 1 \end{array}} (-1)^{|\theta |} \, f( \mathbf {v} ) - \sum _{\begin{array}{c} \theta \le \mathbf {1} \\ \theta _1 = 0 \end{array}} (-1)^{|\theta |+1} \, f( \mathbf {v} ) }{ t_{\alpha _1}^{1}- t_{\alpha _1-1}^{1} }. \end{aligned}$$

Now define \(g_2: [t_{\alpha _2-1}^{2}, t_{\alpha _2}^{2}] \rightarrow \mathbb {R}\), by

$$\begin{aligned} g_2(x_2):= & {} \underset{\begin{array}{c} \theta \le \mathbf {1} \\ \theta _1 = \theta _2 = 1 \end{array}}{\sum } (-1)^{|\theta |} \, D^{\mathbf {e}_1} f_{1- \mathbf {e}_1}^{x_{\alpha _1}^{1} \mathbf {e}_{_1} + x_{_2} \mathbf {e}_{_2}}( \mathbf {v}\lfloor (1-\mathbf {e}_{_1} ) ). \end{aligned}$$

Then, as before, since \(g_2\) depends only in the second variable, \(x_2\), an application of the mean value theorem implies that there is \(x_{\alpha _2}^{2} \in (t_{\alpha _2-1}^{2}, t_{\alpha _2}^{2})\) such that

$$\begin{aligned} g_2^{\prime }(x_{\alpha _2}^{2})= & {} \dfrac{g_2(t_{\alpha _2}^{2}) - g_2(t_{\alpha _2-1}^{2})}{t_{\alpha _2}^{2} - t_{\alpha _2-1}^{2}}. \end{aligned}$$

Thus,

$$\begin{aligned}&\underset{\begin{array}{c} \theta \le \mathbf {1} \\ \theta _1 = \theta _2 = 1 \end{array}}{\sum } (-1)^{|\theta |} \, D^{\mathbf {e}_2} D^{\mathbf {e}_1} f_{1- \mathbf {e}_1 - \mathbf {e}_2}^{x_{\alpha _1}^{1} \mathbf {e}_1 + x_{\alpha _2}^{2} \mathbf {e}_2}( \mathbf {v}\lfloor (1- \mathbf {e}_1 - \mathbf {e}_2) ) \nonumber \\&\quad = \dfrac{ \dfrac{ \sum _{\theta \le \mathbf {1}} (-1)^{|\theta |} \, f( \mathbf {v} ) }{ t_{\alpha _1}^{1}- t_{\alpha _1-1}^{1} }}{t_{\alpha _2}^{2} - t_{\alpha _2-1}^{2}} = \dfrac{ \sum _{\theta \le \mathbf {1}} (-1)^{|\theta |} \, f( \mathbf {v} ) }{(t_{\alpha _1}^{1}- t_{\alpha _1-1}^{1})(t_{\alpha _2}^{2} - t_{\alpha _2-1}^{2}) }. \end{aligned}$$

By repeating this procedure n times, we obtain that there is \( \mathbf {x}_{\alpha } \in [ \mathbf {t}_{\alpha -1}, \mathbf {t}_{\alpha }]\) such that

$$\begin{aligned} D^{\mathbf {e}_1+\mathbf {e}_2+ \cdots + \mathbf {e}_n} \, f ( \mathbf {x}_{\alpha }) = \dfrac{ \sum _{\theta \le \mathbf {1}} (-1)^{|\theta |} \, f( \mathbf {v} ) }{\mathrm{Vol} \, [ \mathbf {t}_{( {\varvec{\alpha }}-1)}, \mathbf {t}_{ {\varvec{\alpha }}} ]}, \end{aligned}$$

and hence

$$\begin{aligned} \Phi \left( \left| D^{\mathbf {e}_1+\mathbf {e}_2+ \cdots + \mathbf {e}_n} \, f ( \mathbf {x}_{\alpha }) \right| \right) \mathrm{Vol} \, [ \mathbf {t}_{ \mathbf {\alpha }-\mathbf {1} }, \mathbf {t}_{ \mathbf {\alpha }} ]= & {} \Phi \left( \dfrac{\left| \sum _{\theta \le \mathbf {1}} (-1)^{|\theta |} \, f( \mathbf {v} ) \right| }{\mathrm{Vol} \, [ \mathbf {t}_{ \mathbf {\alpha }-\mathbf {1} }, \mathbf {t}_{ \mathbf {\alpha }} ]}\right) \mathrm{Vol} \, [ \mathbf {t}_{ \mathbf {\alpha } -\mathbf {1}}, \mathbf {t}_{ \mathbf {\alpha }} ].\nonumber \\ \end{aligned}$$
(5.2)

Since this holds for each \(\mathbf {t}_{\alpha }\), \( {\varvec{\alpha }}\le \mathbf {\kappa }\), of any net partition \(\mathbf {\xi } \) of \([ \mathbf {a}, \mathbf {b}]\), we must have

$$\begin{aligned}&\sum \limits _{1 \le {\varvec{\alpha }}\le \mathbf {\kappa } } \Phi \left( \left| D^{\mathbf {e}_1+\mathbf {e}_2+\cdots + \mathbf {e}_n} f ( \mathbf {x}_{\alpha }) \right| \right) \mathrm{Vol} [ \mathbf {t}_{ {\varvec{\alpha }}-\mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}} ]\\&\quad = \sum \limits _{1 \le {{\varvec{\alpha }}}\le \mathbf {\kappa } } \, \Phi \left( \dfrac{\left| \sum _{\theta \le \mathbf {1}} (-1)^{|\theta |} f( \mathbf {v} ) \right| }{\mathrm{Vol} \, [ \mathbf {t}_{{\varvec{\alpha }}-\mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}} ]}\right) \mathrm{Vol} [ \mathbf {t}_{( {\varvec{\alpha }}-1)}, \mathbf {t}_{ {\varvec{\alpha }}} ]. \end{aligned}$$

Now define,

$$\begin{aligned} m_{\alpha }:= & {} \inf \left\{ \Phi \left( \left| D^{\mathbf {e}_1+\mathbf {e}_2+\cdots + \mathbf {e}_n} f ( \mathbf {x}) \right| \right) : \mathbf {x} \in [ \mathbf {t}_{\alpha -\mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}]\right\} \, \hbox { and } \\ M_{\alpha }:= & {} \sup \left\{ \Phi \left( \left| D^{\mathbf {e}_1+\mathbf {e}_2+\cdots + \mathbf {e}_n} f ( \mathbf {x}) \right| \right) : \mathbf {x} \in [ \mathbf {t}_{\alpha -\mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}}]\right\} , \end{aligned}$$

then

$$\begin{aligned} \underline{S}(f,\mathbf {\xi } ):= & {} \sum \limits _{ 1 \le {\varvec{\alpha }}\le \mathbf {\kappa }} m_{\alpha } \, \mathrm{Vol} [ \mathbf {t}_{ {\varvec{\alpha }}-\mathbf {1} }, \mathbf {t}_{ {\varvec{\alpha }}} ] \\\le & {} \sum \limits _{1 \le {\varvec{\alpha }}\le \mathbf {\kappa } } \Phi \left( \left| D^{\mathbf {e}_1+\mathbf {e}_2+\cdots + \mathbf {e}_n} f ( \mathbf {x}_{\alpha }) \right| \right) \mathrm{Vol} [ \mathbf {t}_{( {\varvec{\alpha }}-1)}, \mathbf {t}_{ {\varvec{\alpha }}} ] \\= & {} \mathrm{RV}_{\Phi }^{n}(f,[ \mathbf {a}, \mathbf {b}],\mathbf {\xi } )\\\le & {} \sum \limits _{ 1 \le {\varvec{\alpha }}\le \mathbf {\kappa }} M_{\alpha } \, \mathrm{Vol} [ \mathbf {t}_{ {\varvec{\alpha }}-\mathbf {1} }, \mathbf {t}_{ {\varvec{\alpha }}} ]\\=: & {} \overline{S}(f,\mathbf {\xi } ). \end{aligned}$$

Notice that the lower sum, \(\underline{S}\), and \(\mathrm{RV}_{\Phi }^{n}(f,[ \mathbf {a}, \mathbf {b}],\mathbf {\xi } )\) are increasing with respect to refinements of the partition \(\mathbf {\xi } \) while the upper sums, \( \overline{S}\), are decreasing. This means that if \(k_i \rightarrow \infty \) then the upper sums decrease to the limit

$$\begin{aligned} { \mathbf {\int _{[ \mathbf {a}, \mathbf {b}]} } }\Phi \left( \left| D^{\mathbf {e}_1+\mathbf {e}_2+ \cdots + \mathbf {e}_n} f ( \mathbf {x}) \right| \right) \mathrm{Vol} [ \mathbf {t}_{ {\varvec{\alpha }}-\mathbf {1} }, \mathbf {t}_{ {\varvec{\alpha }}} ] \mathrm{d} \mathbf {x}, \end{aligned}$$

and the lower sums increase to the limit

$$\begin{aligned} \underline{\int _{[ \mathbf {a}, \mathbf {b}]}} \Phi \left( \left| D^{\mathbf {e}_1+\mathbf {e}_2+ \cdots + \mathbf {e}_n} f ( \mathbf {x}) \right| \right) \mathrm{Vol} [ \mathbf {t}_{{\varvec{\alpha }}-\mathbf {1}}, \mathbf {t}_{ {\varvec{\alpha }}} ] \mathrm{d} \mathbf {x}, \end{aligned}$$

whereas \(\mathrm{RV}_{\Phi }^{n}(f,[ \mathbf {a}, \mathbf {b}],\mathbf {\xi } )\) increases to the limit \(\mathrm{RV}_{\Phi }^{n}(f,[ \mathbf {a}, \mathbf {b}])\), and consequently

$$\begin{aligned} \mathrm{RV}_{\Phi }^{n}(f,[ \mathbf {a}, \mathbf {b}]) = \int _{[ \mathbf {a}, \mathbf {b}]} \Phi \left( \left| D^{\mathbf {e}_1+\mathbf {e}_2+ \cdots + \mathbf {e}_n} \, f ( \mathbf {x}) \right| \right) \mathrm{d} \mathbf {x}. \end{aligned}$$

Now, since this holds for any function of n variables, with \(n\ge 1\), in particular it holds for any truncated function \(f_{\eta }^{ \mathbf {a}}\), where \(\eta \le \mathbf {1}\), which yields (5.1), since there are \((n-|\eta |)!\) truncations. \(\square \)