ON THE DUAL SPACE OF THE HENSTOCK-KURZWEIL

INTEGRABLE FUNCTIONS IN N DIMENSIONS*

Tepper L. Gill and W. W. Zachary

ABSTRACT

The dual space of the class of Henstock-Kurzweil integrable functions is well known in the one-dimensional case and corresponds to the space of multipliers which, in turn, coincides with the class of functions of bounded essential variation. Comparable results in higher dimensions have been elusive. For cases in which the partitions defining the Henstock-Kurzweil integrals are defined on n-cells (parallelepipeds) with their sides parallel to the coordinate axes (i.e., Cartesian products of compact intervals), we prove that a function is a multiplier of the class of n-dimensional (n ≥ 2) Henstock-Kurzweil integrable functions if and only if it is of strongly bounded essential variation as defined by Kurzweil. This result was proved earlier by T.Y. Lee, T. S. Chew, and P.Y. Lee in the two-dimensional case by a different method. The sufficiency part of our proof makes use of a generalization of a method used earlier by P.Y. Lee in the one-dimensional case.

*2000 Mathematics Subject Classification. Primary 26A39; Secondary 46E99 Key Words and Phrases: Nonabsolute integration, Henstock-Kurzweil integrability, dual spaces, multipliers.

1. INTRODUCTION

In recent years there has been a great deal of interest in theories of integration more general than the well known Lebesgue theory. These theories are nonabsolute in the sense that an integral ∫ f may exist even when the integral ∫ |f| does not, a property that Lebesgue integrals do not have. The best known of these theories bear the names of Denjoy, Perron, Henstock, and Kurzweil (for summaries, see [CD, F, G, H, Ku1, L1, M, Mc, P1, S1]).

In the present work we will be concerned with the Henstock-Kurzweil approach, which is a generalization of Riemann’s original constructive method. This approach has been thoroughly investigated in the onedimensional case, but less extensively studied in higher dimensions. We will restrict ourselves to the discussion of the case of functions defined on particular types of compact sets in R^{n} (n-cells, see below).

In order to simplify our presentation and to make otherwise long complicated formulas comprehensible, we will use vector notation where possible. We will also use it in some nonconventional cases, where the intent should be obvious. In other cases, we will define our meaning. For example, if {a_{i}}_{i=1}^{n}, {b_{i}}_{i=1}^{n} are sequences of real numbers with a_{i} < b_{i}, i = 1, 2, ⋯, n; we denote the corresponding n-cell by

I_{n} ≡ [a_{1}, b_{1}] × [a_{2}, b_{2}] × ⋯ × [a_{n}, b_{n}].  (1.1)

We denote this by I_{n} = [a, b]. Let x = (x_{1} ⋯ x_{n}), and for q ≤ n, let x_{q} = x_{1}, ⋯, x_{q} and x_{n−q} = x_{q+1}, ⋯, x_{n}. Below, we record most of the nonconventional cases:

I′ = i_{1}, ⋯, i_{n}, I′_{q} = i_{1}, ⋯, i_{q}, K′ = k_{1}, ⋯, k_{n}, K′_{q} = k_{1}, ⋯, k_{q},

[a : x_{K′}^{[1,n]}] = a_{1}, x_{k_{1}}^{(1)}; ⋯; a_{n}, x_{k_{n}}^{(n)},

ξ_{I′}^{[1,n]} = ξ_{i_{1}}^{(1)}, ⋯, ξ_{i_{n}}^{(n)}, [ξ_{K′}^{[1,n]} : ξ_{K′+1}^{[1,n]}] = ξ_{k_{1}}^{(1)}, ξ_{k_{1}+1}^{(1)}, ⋯, ξ_{k_{n}}^{(n)}, ξ_{k_{n}+1}^{(n)}.

One says that a real-valued function f defined on I_{n} is integrable in the Henstock-Kurzweil sense or is Henstock-Kurzweil integrable, and denotes its integral ∫_{I_{n}} f(x) dx if, for each ε > 0, there exists a positive function δ : I_{n} → R_{+} (generally called a gauge or gage) such that if P = {J_{i} | i = 1, 2, ⋯, m} is a partition of I_{n} into a set of nonoverlapping n-cells {J_{i}} with union I_{n} = ⋃_{i=1}^{m} J_{i} such that if x^{(i)} ∈ J_{i} ⊂ B(x^{(i)}, δ(x^{(i)})), where B(a, r) denotes the open ball of radius r and center a in R^{n}, then

|∑_{i=1}^{m} f(x^{(i)})μ(J_{i}) − ∫_{I_{n}} f(x)dx| < ε,

where μ(J_{i}) denotes the Lebesgue measure of the n-cell (J_{i}). Such a partition is called a δ-fine partition. Henstock-Kurzweil integrals can be described in terms of primitive functions, thereby providing a formulation of the fundamental theorem of calculus [CD, F, G, Ku1, L1, M, Mc, P1, S1]. In the one-dimensional case for example, if f is Henstock-Kurzweil integrable on the interval [a, b], then there exists a (primitive) function F (x), x ∈ [a, b] , such that

∫_{a}^{b} f(x)dx = F (b) − F (a),  (1.2)

and the derivative of F exists and is equal to f a.e. on [a, b]. Generalizations of this result to higher dimensions are known [CD, H, Ku1, Ku2, M, MO, O2, Mc, P1] and it will be convenient to describe one of these by using a formalism analogous to that discussed by Tolstov [To]. For a real-valued function ϕ defined on I_{n}, consider the first difference:

Δ_{i}(ϕ(x); [a_{i}, b_{i}]) ≡ ϕ(x_{i−1}, b_{i}, x_{n−i−1}) − ϕ(x_{i−1}, a_{i}, x_{n−i−1}). (1.3)

Similarly, second differences are defined as (for i ≠ j ):

Δ_{x_{i}x_{j}}(ϕ; [a_{i}, b_{i}] × [a_{j}, b_{j}])≡ Δ_{x_{j}}(Δ_{x_{i}}(ϕ; [a_{i}, b_{i}]); [a_{j}, b_{j}])
= ϕ(⋯, b_{i}, ⋯, b_{j}, ⋯) − ϕ(⋯, a_{i}, ⋯, b_{j}, ⋯)
− ϕ(⋯, b_{i}, ⋯, a_{j}, ⋯) + ϕ(⋯, a_{i}, ⋯, a_{j}, ⋯).

One easily shows that

Δ_{x_{i}x_{j}}(ϕ; [a_{i}, b_{i}] × [a_{j}, b_{j}]) = Δ_{x_{j}x_{i}}(ϕ; [a_{i}, b_{i}] × [a_{j}, b_{j}]),  (1.4)

and the following convenient formula for n^{th} differences has been given by Tolstov ([To], p. 70):

Δ_{x_{I′}}(ϕ; I_{n}) = ϕ(b) + ∑_{q=1}^{n−1} ∑_{k_{1}=1}^{n} ⋯ ∑_{k_{q}=1}^{n}(−1)^{q}ϕ_{K′_{q}} + (−1)^{n}ϕ(a), (1.5)

where the summation indices k_{1}, ⋯, k_{q} satisfy the constraints k_{1} < k_{2} < ⋯ < k_{q} and the functions ϕ_{K′_{q}} are obtained from ϕ by evaluating that function at the lower limits a_{1}, ⋯, a_{q} at the x_{k_{1}} = a_{k_{1}}, ⋯, x_{k_{q}} = a_{k_{q}}, and at the upper limits x_{m} = b_{m} for the remaining indices. The expressions (1.5) will be very useful to us later in this paper.

It will be convenient for our subsequent considerations to use some notation that is slightly different from that employed by Tolstov. Thus, we will denote a difference by the symbol Δ with a subscript that indicates the order of the difference with particular differences being indicated by a specification of the intervals upon which the function is evaluated and, for purposes of brevity, we will omit the square brackets in the indication of the intervals. The latter simplification should not cause any confusion because we only consider closed intervals. Thus, we will indicate differences such as (1.3) by Δ_{1}(ϕ(x_{i−1}, ·, x_{n−i−1}); a_{i}, b_{i}) and (1.5) by Δ_{n}(ϕ; I_{n}). The symmetry properties (1.4) and higher order versions thereof show that differences are uniquely defined by a specification of their order and of the intervals upon which they act.

The n-dimensional (n ≥ 2) generalizations of (1.2) are

∫_{I_{n}} f(x)dx = Δ_{n}(F; I_{n}),  (1.6)

where F denotes the n-dimensional primitive function of f on the n-cell I_{n}. (See eq. (3.13) in the proof of Theorem 3.1 for the justification of the above equation.) We will denote by HK_{n} the collection of Henstock-Kurzweil integrable functions in n dimensions, where n ≥ 1.

The main object of this paper is to investigate the dual spaces of the class of HK_{n} functions for n = 2, 3, · · ·. As was first done by Alexiewicz [A] in the one-dimensional case, the HK_{n} functions on I_{n} (1.1) can be topologized by using the Alexiewicz norm:

∥f∥_{A} ≡ sup_{x∈I_{n}} |F (x)| ,

where F denotes the indefinite integral of the HK_{n} integrable function f:

F (x) ≡ ∫_{a_{1}}^{x_{1}}⋯ ∫_{a_{n}}^{x_{n}}f(y) dy.

It is known that the space of HK_{n} functions is not complete relative to the topology defined by the Alexiewicz norm, but it can be completed to a space of Schwartz distributions [MO,Ku2]. It is known in dimensions n = 1 [Th] and n = 2 [O2] that the space of HK_{n} functions, topologized by the Alexiewicz norm, is a barreled space but is not a Banach space. In one dimension it is known that there exists a one-to-one correspondence between the continuous linear functionals (relative to the topology defined by the Alexiewicz norm) and the functions of bounded essential variation [O1, S2, MO, L1, CD], and that these functions are multipliers of HK_{1} where, in general, a function g is said to be a multiplier of a space of integrable functions G if hg ∈ G whenever h ∈ G. In particular ([L1], Sec.12), if T denotes a continuous linear functional on HK_{1}([a, b]) for a finite closed interval [a, b], then

T (f) = ∫_{a}^{b} f(x)g(x)dx

for all f ∈ HK_{1}([a, b]) and some g of bounded variation on [a, b]. Moreover, ∥g∥_{B} ≤ 2 ∥T ∥_{A}, where ∥g∥_{B} ≡ inf V (g_{1}), in which V (g) denotes the total variation of g on [a, b] and the infimum is taken over all functions g_{1} of bounded variation on [a, b] which are equal to g a.e. on [a, b]. In this one-dimensional situation, a function has bounded essential variation if and only if its first (Schwartz) distributional derivative is a signed finite Borel measure.

In dimensions n ≥ 2, however, the situation is different because, as we will discuss in more detail below, there are many distinct classes of functions of bounded variation, each of which coincides in the onedimensional case with the class of functions of bounded variation usually considered. Several different classes of functions of bounded variation in two dimensions were listed and their interrelationships discussed by Clarkson and Adams [CA]. We list below a few of these that we will need to use in our subsequent discussion.

DEFINITION 1.1. A function f(x, y) is said to be of bounded variation in the Vitali-Lebesgue-Frechet-de la Vallee Poussin sense on the rectangle I_{2} ≡ [a_{1}, b_{1}] × [a_{2}, b_{2}] if

∑_{i=0}^{q−1} ∑_{j=0}^{p−1} |Δ_{2}(f; [x_{i}^{(1)}, x_{i+1}^{(1)}] × [x_{j}^{(2)}, x_{j+1}^{(2)}])|

is finite for all partitions of I_{2} into nonoverlapping nondegenerate closed rectangles: a_{i} = x_{0}^{(i)} < x_{1}^{(i)} < ⋯ < x_{m_{i}}^{(i)} = b_{i} for i = 1, 2 with m_{1} = q and m_{2} = p .

DEFINITION 1.2. A function f(x, y) is said to be of bounded variation on I_{2} in the Hardy-Krause sense if it is of bounded variation in the sense of Definition 1.1 and, in addition, f(x′, y) is of bounded variation in y for at least one x′ and f(x, y′) is of bounded variation in x for at least one y′.

DEFINITION 1.3. A function f(x, y) is said to be of bounded variation on I_{2} in the Tonelli sense if the one-dimensional total variation of f(x, y) (considered as a function of y) is finite a.e. on (a_{1}, b_{1}) and its Lebesgue integral over (a_{1}, b_{1}) exists (and is finite), with a similar condition on the one-dimensional total variation of f(x, y) considered as a function of x.

Kurzweil ([Ku2], see also [Ku1], p. 95) defined a real-valued function g on I_{2} to be of strongly bounded variation if it is of bounded variation in the sense of Definition 1.1 and if , for every x ∈ [a_{1}, b_{1}], g(x, ·) is of bounded variation on [a_{2}, b_{2}] and, for every y ∈ [a_{2}, b_{2}], g(·, y) is of bounded variation on [a_{1}, b_{1}]. Thus, it is seen that the condition of strongly bounded variation is a special case of the condition of bounded variation in the sense of Definition 1.2. In [Ku2] Kurzweil proved (in the n-dimensional case) that functions of strongly bounded essential variation are multipliers for the class of Perron integrable functions and that these functions correspond to continuous linear functionals of the space of Perron integrable functions. Since the class of Perron integrable functions is equivalent to the class of HK functions, similar results hold for the dual space of the HK functions. Thus, Kurzweil’s result gives a sufficient condition for a function g to be a multiplier of the space HK_{n}. In an important paper [LCL], Lee, Chew, and Lee proved, in the two-dimensional case, that this condition is necessary, as well as sufficient, for a function to be a multiplier of the space HK. These authors claim that their proof can be extended to all higher dimensions, and our main goal in the present paper is to give a different proof of this characterization of the multipliers of the HK_{n} spaces when n ≥ 2. The sufficiency part of our proof makes use of a generalization of a method used by P. Y. Lee ([L1], pp. 72-73) in the one-dimensional case and yields an explicit expression for the integrals by means of integration-by-parts formulae in terms of Riemann-Stieltjes integrals. The existence of these higher-dimensional Riemann-Stieltjes integrals can be proven by using a generalization of Clarkson’s argument [C] for the two-dimensional case. We note that it is sufficient to consider Riemann-Stieltjes integrals rather than the more general Perron-Stieltjes or Ward integrals because the primitives of Henstock-Kurzweil integrals are continuous.

It is known in the two-dimensional case [MO, O2, LCL] that T (f) is a continuous linear functional on the space HK_{2}(I_{2}) (relative to the topology defined by the Alexiewicz norm) if and only if there exists a finite signed Borel measure μ on I_{2} = [a_{1}, b_{1}] × [a_{2}, b_{2}] such that T (f) = ∫_{I_{2}} f(x, y)dμ(x, y) for all f ∈ HK_{2}(I_{2}). In this connection,

Krickeberg [Kr] proved that a real-valued function on R^{n} has first distributional derivatives that are measures if and only if it is of bounded variation in the sense of Tonelli. Ostaszewski [MO,O2] proved that distribution functions of finite Borel measures are of strongly bounded essential variation and are multipliers of HK_{2}(I_{2}), but was not able to prove that these multipliers are associated with the most general form of continuous linear functional on the space of HK_{2} functions. Such a proof was provided later by Lee, Chew, and Lee [LCL]. Multiplier problems have also been investigated for other types of Riemann-type integrals [L2, P1, P2, MP, BS, B].

Our objective in this paper is to give a new proof of the result of Lee, Chew, and Lee on the characterization of the class of multipliers of the space of HK_{n} functions for n = 2, 3, · · ·. We first prove, in Theorem 2.1 in Section 2, that a necessary condition that a function be a multiplier of HK_{n} is that it be of bounded essential variation in the n-dimensional version of the class of functions of bounded variation defined in Definition 1.1. This theorem was proved before we were aware of the work of Lee, Chew, and Lee. After the present work was presented at the conference by the second author, Lee Tuo Yeong informed him of the results in the papers [LCL] and [L2]. It then became clear to us how to modify the proof of Theorem 2.1 so that we obtain a proof of the necessity that functions of strongly bounded essential variation be multipliers for the HK_{n} functions.This is discussed in Section 4. The proof of the sufficiency of this condition in Theorem 3.2 makes use of a generalization to n dimensions (n ≥ 2) of a method used by P.Y. Lee in the one-dimensional case ([L1], Theorem 12.1 and Corollary 12.2, pp. 72-73).

2. Necessary condition for existence of multipliers

In this section we obtain a necessary condition in order that a function g be a multiplier for HK_{n} . The idea of the proof is to reduce from an arbitrary integral dimension (n ≥ 2) to the one-dimensional case and then use similar arguments to those used in earlier proofs for that case ([CD], Theorem 50, p.45; [L1], Theorem 12.9, pp. 78-79; [S2]). As in the Introduction, we restrict ourselves to functions defined on compact n-cells in R^{n}.

THEOREM 2.1. Let f and g denote real-valued functions defined on an n-cell I_{n} (1.1). If fg ∈ HK_{n}(I_{n}) for all f ∈ HK_{n}(I_{n}), then g is equal a. e. to a function of n-dimensional (n ≥ 1) bounded variation in the sense of Vitali et al.

Before proceeding to the proof, we discuss n-dimensional variations in the sense of Vitale et al and their connection with the n^{th} order differences (1.5). Given the definition of n-cells I_{n} as in (1.1), consider partitions of these sets of the form

a_{1} = x_{0}^{(1)} < x_{1}^{(1)} < ⋯ < x_{m_{1}}^{(1)} = b_{1}, ; a_{2} = x_{0}^{(2)} < x_{1}^{(2)} < ⋯ < x_{m_{2}}^{(2)} = b_{2}, ; ⋮ ; a_{n} = x_{0}^{(n)} < x_{1}^{(n)} < ⋯ < x_{m_{n}}^{(n)} = b_{n}.  (2.1)

Then, generalizing Definition 1.1 to n ≥ 3 dimensions, we define ndimensional variations in the sense of Vitali et al of real-valued functions ϕ on I_{n} by:

V (ϕ; I_{n}) ≡ ess sup_{P} ∑_{i_{1}=1}^{m_{1}} ⋯ ∑_{i_{n}=1}^{m_{n}} |Δ_{n}(ϕ; [x_{I′−1}^{[1,n]} : x_{I′}^{[1,n]}])| ,  (2.2)

where the n^{th} differences Δ_{n} are defined by (1.5) and the essential supremum is taken over all partitions (2.1).

For the second differences we have

\begin{matrix}Δ_{2}(ϕ; x_{i_{1}−1}^{(1)}, x_{i_{1}}^{(1)}; x_{i_{2}−1}^{(2)}, x_{i_{2}}^{(2)}) = ϕ(x_{i_{1}}^{(1)}, x_{i_{2}}^{(2)}) \\ −ϕ(x_{i_{1}−1}^{(1)}, x_{i_{2}}^{(2)}) − ϕ(x_{i_{1}}^{(1)}, x_{i_{2}−1}^{(2)}) + ϕ(x_{i_{1}−1}^{(1)}, x_{i_{2}−1}^{(2)}), \end{matrix}

so that

\begin{matrix}|Δ_{2}(ϕ; x_{i_{1}−1}^{(1)}, x_{i_{1}}^{(1)}; x_{i_{2}−1}^{(2)}, x_{i_{2}}^{(2)})| ≤ ω(ϕ(·, x_{i_{2}}^{(2)}); x_{i_{1}−1}^{(1)}, x_{i_{1}}^{(1)}) \\ + ω(ϕ(·, x_{i_{2}−1}^{(2)}); x_{i_{1}−1}^{(1)}, x_{i_{1}}^{(1)}), \end{matrix}  (2.3)

where ω denotes one-dimensional essential oscillations:

ω(h; a, b) ≡ ess sup_{x,y∈[a,b]}[h(x) − h(y)]

= ess sup_{x,y∈[a,b]} h(x) − ess inf_{x,y∈[a,b]} h(x). (2.4)

It easily follows from (1.5) that, for example,

\begin{matrix}Δ_{n}(ϕ; [x_{I′−1}^{[1,n]} : x_{I′}^{[1,n]}]) \\ = Δ_{n−1}(ϕ(⋯, x_{i_{n}}^{(n)}); x_{i_{1}−1}^{(1)}, x_{i_{1}}^{(1)}; ⋯; x_{i_{n−1}−1}^{(n−1)}, x_{i_{n−1}}^{(n−1)}) \\ −Δ_{n−1}(ϕ(⋯, x_{i_{n}−1}^{(n)}); x_{i_{1}−1}^{(1)}, x_{i_{1}}^{(1)}; ⋯; x_{i_{n−1}−1}^{(n−1)}, x_{i_{n−1}}^{(n−1)}), \end{matrix} (2.5)

and the expressions on the right-hand side of (2.5) may be replaced by pairs of (n − 1)-differences in which the function ϕ is evaluated at the endpoints of other ”component” intervals of the n-cell I_{n}. Then, from (2.3), (2.5), and induction, we obtain:

\begin{matrix}| & Δ_{n}(ϕ; [x_{I′−1}^{[1,n]} : x_{I′}^{[1,n]}]) & | \\ ≤ ∑_{c_{1}=1}^{2} ⋯ ∑_{c_{n−1}=1}^{2} ω(ϕ(x_{c_{1}i_{1}}^{(1)}, ⋯, x_{c_{n−1}i_{n−1}}^{(n−1)}, ·); x_{i_{n}−1}^{(n)}, x_{i_{n}}^{(n)}) \end{matrix} (2.6)

where, in (2.6), for each c_{j} (j = 1, 2, ⋯, n − 1) the values c_{j} = 1 or c_{j} = 2 are to be chosen corresponding to whether the function ϕ is evaluated at the left or right endpoints, respectively, of the intervals [x_{i_{j}−1}^{(j)}, x_{i_{j}}^{(j)}], (j = 1, 2, ⋯, n − 1).

PROOF OF THEOREM 2.1. For n = 1 the definition of functions of bounded variation in the sense of Vitali et al reduces to the usual concept of functions of bounded variation, so the theorem is true in this case. See the references cited in the first paragraph of this section.

For n ≥ 2, following an argument of P. Y. Lee for n = 1, since fg is HK_{n} integrable for all f that are absolutely HK_{n} integrable (and hence Lebesgue integrable); g must be essentially bounded. To begin the proof for the n ≥ 2 cases, we use an argument analogous to one used in the n = 1 case in the papers cited above. Suppose that g is not equal a. e. to a function of bounded variation in the sense of Vitali et al in n dimensions (n ≥ 2). Thus, for any function g_{1} equal to g a.e., g_{1} is not of bounded variation in the stated sense. Then there exists a point γ ∈ I_{n} such that g_{1} is not of bounded variation in the stated sense on any open n-cell which contains γ. Since the variation of g_{1} in I_{n} is unbounded, the variation of g_{1} in every open subset ϑ of I_{n} completely enclosing γ is also unbounded. Otherwise, one would be able to enclose every point of ϑ in a sub-n-cell of I_{n} in which g_{1} has bounded variation. But it would then follow, by an application of the Heine-Borel theorem, that there exists a finite number of n-cells which cover ϑ on each of which g_{1} has bounded variation. Then g_{1} would be of bounded variation on ϑ, a contradiction. In particular, g_{1} is not of bounded variation in the stated sense on every n-cell with one vertex at γ.

We next use a generalization of the one-dimensional argument of C̆elidze and Dz̆vars̆eĭs̆vili ([CD], Theorem 50, p. 45). We call y ∈ I_{n} ≡ [a , b ] a singular point of g if, for all sub-n-cells Ĩ_{n} ≡ [ã, b̃ ] ⊂ I_{n}, y ∈ Ĩ_{n} and V (g; Ĩ_{n}) = +∞, where V is defined by (2.2). From the assumption that g is not of bounded essential variation, we infer that the set of such singular points is not empty. We may assume, without loss of generality, that the point a ∈ I_{n} is a singular point.

By the definitions (2.2) and inequalities (2.6), there exists a subn-cell of each I_{n}, J_{1}^{(n)}≡ [α^{(1)}, β^{(1)}] ⊂ I_{n} and partitions of J_{1}^{(n)}:

α_{1}^{(1)} = x_{1}^{(1)} < ⋯ < x_{q_{1,1}}^{(1)} = β_{1}^{(1)},
α_{2}^{(1)} = x_{1}^{(2)} < ⋯ < x_{q_{2,1}}^{(2)} = β_{2}^{(1)},
⋮
α_{n}^{(1)} = x_{1}^{(n)} < ⋯ < x_{q_{n,1}}^{(n)} = β_{n}^{(1)},

such that

\begin{matrix}∑_{i_{1}=1}^{q_{1,1}} ⋯ ∑_{i_{n}=1}^{q_{n,1}} ∑_{c_{1}=1}^{2} ⋯ ∑_{c_{n−1}=1}^{2} ω(g(x_{c_{1}i_{1}}^{(1)}, ⋯, x_{c_{n−1}i_{n−1}}^{(n−1)}, ·); x_{i_{n}−1}^{(n)}, x_{i_{n}}^{(n)}) \\ ≥ V (g; J_{1}^{(n)}) ≥ 1. \end{matrix}

Further, on each sub-n-cell K_{1}^{(n)} ≡ [x_{i_{1}−1}^{(1)}, x_{i_{1}}^{(1)}] × ⋯ × [x_{i_{n}−1}^{(n)}, x_{i_{n}}^{(n)}] ⊂ J_{1}^{(n)}, there exist disjoint measurable sets L_{i_{1},···,i_{n}}^{(1)}, H_{i_{1},···,i_{n}}^{(1)} ⊂ K_{1}^{(n)} with μ(L_{i_{1},···,i_{n}}^{(1)}) = μ(H_{i_{1},···,i_{n}}^{(1)}) ≡ η_{1} > 0 (where μ denotes Lebesgue measure on R^{n}) such that

g(z)≥\frac{3}{2}m_{I′}^{∗} + \frac{1}{2}m_{I′}, if z ∈ L_{I′}^{(1)},  (2.7a)

and

g(z)≤\frac{1}{2}m_{I′}^{∗} + \frac{3}{2}m_{I′}, if z ∈ H_{I′}^{(1)},  (2.7b)

where

m_{I′}^{∗} ≡ ∑_{c_{1}=1}^{2} ⋯ ∑_{c_{n−1}=1}^{2} ess sup_{y∈[x_{in−1}^{(n)},x_{in}^{(n)}]} g(x_{c_{1}i_{1}}^{(1)}, ⋯, x_{c_{n−1}i_{n−1}}^{(n−1)}; y), (2.8)

m_{I′} ≡ ∑_{c_{1}=1}^{2} ⋯ ∑_{c_{n−1}=1}^{2} ess inf_{y∈[x_{in−1}^{(n)},x_{in}^{(n)}]} g(x_{c_{1}i_{1}}^{(1)}, ⋯, x_{c_{n−1}i_{n−1}}^{(n−1)}; y), (2.9)

and, in defining (2.8) and (2.9), we have used the fact that two sets of the forms {x_{c_{1}i_{1}}^{(1)}, ⋯, x_{c_{n−1}i_{n−1}}^{(n−1)}}×[x_{i_{n}−1}^{(n)}, x_{i_{n}}^{(n)}] and {x_{c′_{1}i_{1}}^{(1)}, ⋯, x_{c′_{n−1}i_{n−1}}^{(n−1)}}×

[x_{i_{n}−1}^{(n)}, x_{i_{n}}^{(n)}] are disjoint if c_{i} ≠ c′_{i} for at least one value of i ∈ {1, 2, ⋯, n− 1}.

By repetition of the above considerations, we obtain by induction a sequence of disjoint n-cells J_{k}^{(n)} ≡ [α^{(k)}, β⃗^{(k)}] ⊂ I_{n} with β_{i}^{(k)} < α_{i}^{(k−1)}

for i = 1, 2, ⋯, n and k = 2, 3, ⋯ such that lim_{k→+∞} α_{i}^{(k)} = lim_{k→+∞} β_{i}^{(k)} =

a_{i} for each i = 1, 2, ⋯, n. Moreover, on each n-cell J_{k}^{(n)} there exists a partition

α_{1}^{(k)} = x_{q_{1,k−1}}^{(1)} < ⋯ < x_{q_{1,k}}^{(1)} = β_{1}^{(k)}, ; α_{2}^{(k)} = x_{q_{2,k−1}}^{(2)} < ⋯ < x_{q_{2,k}}^{(2)} = β_{2}^{(k)}, ; ⋮ ; α_{n}^{(k)} = x_{q_{n,k−1}}^{(n)} < ⋯ < x_{q_{n,k}}^{(n)} = β_{n}^{(k)},  (2.10)

with

\begin{matrix}∑_{q_{1,k−1}+1}^{q_{1,k}} ⋯ ∑_{q_{n,k−1}+1}^{q_{n,k}} ∑_{c_{1}=1}^{2} ⋯ ∑_{c_{n−1}=1}^{2} ω(g(x_{c_{1}i_{1}}^{(1)}, ⋯, x_{c_{n−1}i_{n−1}}^{(n−1)}, ·); x_{i_{n}−1}^{(n)}, x_{i_{n}}^{(n)}) \\ ≥V (g; J_{k}^{(n)})≥k. \end{matrix}

(2.11)

On each set K_{k}^{(n)} ≡ {x_{c_{1}i_{1}}^{(1)}, ⋯, x_{c_{n−1}i_{n−1}}^{(n−1)}}×[x_{i_{n}−1}^{(n)}, x_{i_{n}}^{(n)}] ⊂ J_{k}^{(n)}, there exist disjoint measurable sets L_{I′}^{(k)}, H_{I′}^{(k)} such that η_{k−1} > μ(L_{I′}^{(k)}) = μ(H_{I′}^{(k)}) ≡ η_{k} > 0 and such that (2.7) holds with the sets L_{I′}^{(1)} and H_{I′}^{(1)} replaced by L_{I′}^{(k)} and H_{I′}^{(k)}, respectively. In addition, lim_{k→+∞} η_{k} = 0.

We now define a real-valued function f on I_{n} by

f(z) = \{ \begin{matrix}p_{k} & if z ∈ L_{I′}^{(k)}, \\ −p_{k} & if z ∈ H_{I′}^{(k)}, \\ 0 & otherwise, \end{matrix}

where

\frac{1}{p_{k}} ≡ η_{k}\sqrt{V (g; J_{k}^{(n)})}  (2.12)

and q_{j,k−1} + 1≤ i_{j}≤ q_{j,k} for j = 1, 2, ⋯, n. Then

∫_{I_{n}} fdz = ∫_{J_{k}^{(n)}} fdz= ∑_{i_{1}=q_{1,k−1}+1}^{q_{1,k}}⋯ ∑_{i_{n}=q_{n,k−1}+1}^{q_{n,k}} [ ∫_{L_{I′}^{(k)}}fdz + ∫_{H_{I′}^{(k)}}fdz] = 0

so f ∈ HK_{n}(I_{n}). Similarly, by (2.7), we obtain

∫_{I_{n}} fgdz = ∑_{i_{1}=q_{1,k−1}+1}^{q_{1,k}} ⋯ ∑_{i_{n}=q_{n,k−1}+1}^{q_{n,k}} p_{k}[ ∫_{L_{I′}^{(k)}} gdz − ∫_{H_{I′}^{(k)}} gdz]
≥ ∑_{i_{1}=q_{1,k−1}+1}^{q_{1,k}} ⋯ ∑_{i_{n}=q_{n,k−1}+1}^{q_{n,k}} p_{k}η_{k}[m_{I′}^{∗} − m_{I′}]
= ∑_{i_{1}=q_{1,k−1}+1}^{q_{1,k}} ⋯ ∑_{i_{n}=q_{n,k−1}+1}^{q_{n,k}} ∑_{c_{1}=1}^{2} ⋯

∑_{c_{n−1}=1}^{2} p_{k}η_{k}ω(g(x_{c_{1}i_{1}}^{(1)}, ⋯, x_{c_{n−1}i_{n−1}}^{(n−1)}, ·); x_{i_{n}−1}^{(n)},x_{i_{n}}^{(n)})

using (2.8), (2.9), and (2.4). Then, from the inequalities (2.11) and the definition of p_{k} (2.12), we have ∫_{I_{n}} fgdz≥\sqrt{V (g; J_{k}^{(n)})}. Since V (g; J_{k}^{(n)}) →

+∞ as k → +∞, it follows that fg ∉ HK_{n}(I_{n}). This contradiction completes the proof of Theorem 2.1.

An example of a function in two dimensions that is of bounded variation in the sense of Vitali et al but not in the sense of Hardy-Krause was given by Clarkson and Adams [CA], which we change in a trivial manner to conform to the notation in the present paper. Thus, we define a real-valued function h on I_{2} = [a_{1}, b_{1}] × [a_{2}, b_{2}] by:

h(x, y) = (x − a_{1}) sin(\frac{1}{x − a_{1}} ) for x ≠ a_{1}; = 0 for x = a_{1}.

One can easily modify this definition to give examples of functions in n dimensions (n ≥ 2) that belong to the n-dimensional class of Vitali et al, but are not of strongly bounded variation. Let us define, for example, h on I_{n} by:

h(x) = (x_{1} − a_{1}) sin(\frac{1}{x_{1} − a_{1}} ) for x ≠ a_{1}; = 0 for x = a_{1}.

Other examples of this type can easily be constructed.

3. Sufficient Conditions for Existence of Multipliers

In this section we give a new proof of the result of Kurzweil [Ku2], in the n-dimensional case with n ≥ 2, that functions of strongly bounded essential variation are multipliers for the class of Henstock-Kurzweil integrable functions. (Actually, Kurzweil considered the equivalent class of Perron integrable functions.) Our proof yields integrationby-parts formulae analogous to the one-dimensional result of P. Y.

Lee ([L1], Theorem 12.1 and Corollary 12.2, pp. 72-73) which the proof generalizes to the n-dimensional case, and which agrees with the explicit result stated by Kurzweil in the n = 2 case.

We begin with a lemma on the existence of certain types of Riemann-Stieltjes integrals, which generalizes the corresponding result of Clarkson [C] when n = 2. Consider partitions of I_{n} of the form (2.1) and let ξ_{i}^{(j)} denote real numbers which satisfy the inequalities:

x_{i−1}^{(j)} ≤ ξ_{i}^{(j)} ≤ x_{i}^{(j)} (i = 1, 2, ⋯, m_{j}; j = 1, 2, ⋯, n)  (3.1)

with

ξ_{1}^{(j)} = x_{0}^{(j)} = a_{j}, ξ_{m_{j}}^{(j)} = x_{m_{j}}^{(j)} = b_{j}.  (3.2)

If the sums

∑_{i_{1}=1}^{m_{1}} ⋯ ∑_{i_{n}=1}^{m_{n}} h(ξ_{I}^{[1,n]})Δ_{n}(ϕ(x_{K′}^{[1,n]})),  (3.3)

where Δ_{n} denotes the n^{th} difference of the function ϕ defined by (1.5), tend to a finite limit as the norm of the subdivisions approaches zero; then the integral with respect to ϕ is said to exist. It will be denoted by

∫_{a_{1}}^{b_{1}}⋯ ∫_{a_{n}}^{b_{n}}h(x)d_{x}ϕ(x),  (3.4)

and will be called a Riemann-Stieltjes integral. We have:

LEMMA 3.1. A necessary and sufficient condition that the integrals (3.4) exist for each real-valued continuous function h defined on I_{n} is that ϕ belongs to the n-dimensional class of functions of bounded variation in the sense of Vitali et al (Definition 1.1).

REMARK. For n = 2 Clarkson [C] considered, in addition to the convergence of (3.3) to (3.4) which he called the restricted integral, the

convergence of the sums ∑_{i=1}^{m_{1}} ∑_{j=1}^{m_{2}} h(ξ_{ij}^{(1)}, ξ_{ij}^{(2)})Δ_{2}(ϕ(x_{i}^{(1)}, x_{j}^{(2)})), wherein

the numbers {ξ_{i}^{(j)} : i = 1, ⋯, m_{j}; j = 1, 2} in (3.3) with n = 2 have been replaced by the numbers {ξ_{ik}^{(j)} : i = 1, ⋯, m_{1}; k = 1, ⋯, m_{2}; j = 1, 2}. The limit of these latter sums, when they exist, was called the unrestricted integral. We will not need these integrals, and so will not consider their n-dimensional generalizations.

The proof of Lemma 3.1 follows from a straightforward extension of Clarkson’s argument in the n = 2 case.

We now have:

THEOREM 3.2. Let f and g denote real - valued functions defined on I_{n}. Suppose that f ∈ HK_{n}(I_{n}) (n ≥ 1) and let g be of strongly bounded essential variation. Then fg ∈ HK_{n}(I_{n}) and we have the following integration-by-parts formulae in terms of Riemann-Stieltjes integrals:

∫_{a_{1}}^{b_{1}} ⋯ ∫_{a_{n}}^{b_{n}} f(x)g(x)dx = Δ_{n}(F g; a , b ) + (−1)^{n}B_{n}^{(0)}

+ ∑_{q=1}^{n−1}(−1)^{q} ∫_{a_{1}}^{b_{1}} ⋯ ∫_{a_{q}}^{b_{q}} Δ_{n−q}(F (x_{q}; ⋯)d_{x_{q}}g(x_{q}; ⋯); a_{n−q}, b_{n−q}),

(3.5)

where F denotes the n-dimensional primitive of f,

B_{n}^{(0)} ≡ ∫_{a_{1}}^{b_{1}}⋯ ∫_{a_{n}}^{b_{n}}F (x) d_{x}g(x),  (3.6)

and the (n − q)-differences in (3.5) are defined by (1.5) with the function ϕ replaced by the product of the functions F and g with the differentials d_{x_{1}} ⋯ d_{x_{q}} inserted between them.

REMARK. For n = 1, (3.5) reduces to the result

∫_{a}^{b} fgdx = Δ_{1}(F g; a, b) − ∫_{a}^{b} F (x)dg(x),  (3.7)

discussed by P. Y. Lee ([L1], Theorem 12.1 and Corollary 12.2, pp.72-73), while for n = 2 it becomes

\begin{matrix}∫_{a_{1}}^{b_{1}} ∫_{a_{2}}^{b_{2}} f(x_{1}, x_{2})g(x_{1}, x_{2})dx_{1}dx_{2} = Δ_{2}(F g; a_{1}, b_{1}; a_{2}, b_{2}) + B_{2}^{(0)} \\ − ∫_{a_{1}}^{b_{1}} Δ_{1}(F (x_{1}, ·)d_{x_{1}}g(x_{1}, ·); a_{2}, b_{2}) − ∫_{a_{2}}^{b_{2}} Δ_{1}(F (·, x_{2})d_{x_{2}}g(·, x_{2}); a_{1}, b_{1}), \end{matrix}

in agreement with the result given by Kurzweil [Ku2].

Proof of THEOREM 3.2. For n = 1, g is of bounded variation in the usual sense, fg ∈ HK_{1}, and the integral of fg satisfies (3.7) to which (3.5) reduces in this case.

For the case n ≥ 2, in terms of the δ-fine partitions (2.1), (3.1), (3.2), define

J_{n} ≡ ∑_{i_{1}=1}^{m_{1}} ⋯ ∑_{i_{n}=1}^{m_{n}} f(ξ_{I′}^{[1,n]})A_{I′}g(ξ_{I′}^{[1,n]}),  (3.8)

where A_{I′} ≡ ∏_{j=1}^{n} (x_{i_{j}}^{(j)} − x_{i_{j}−1}^{(j)}). We write

J_{n} = T_{n} + (J_{n} − T_{n}),  (3.9)

T_{n} ≡ ∑_{i_{1}=1}^{m_{1}−1} ⋯ ∑_{i_{n}=1}^{m_{n}−1} f(ξ_{I′}^{[1,n]})A_{I′}g(ξ_{I′}^{[1,n]}),

and an easy calculation shows that the terms in J_{n} − T_{n} involve products of at most n − 1 summations. Using the finite additivity of the differences Δ_{n} (see, e.g., [To], p. 68), we decompose T_{n} in the form

T_{n} = T_{n}^{(1)} + T_{n}^{(2)} + T_{n}^{(3)},  (3.10)

where

T_{n}^{(1)} ≡ ∑_{k_{1}=1}^{m_{1}−1} ⋯ ∑_{k_{n}=1}^{m_{n}−1}

{ ∑_{i_{1}=1}^{k_{1}} ⋯ ∑_{i_{n}=1}^{k_{n}} f(ξ_{I}^{[1,n]})A_{I} − Δ_{n}(F; [a : x_{K′}])}

× (−1)^{n}Δ_{n}(g; [ξ_{K′}^{[1,n]} : ξ_{K′+1}^{[1,n]}]),

T_{n}^{(2)} ≡ ∑_{k_{1}=1}^{m_{1}−1} ⋯ ∑_{k_{n}=1}^{m_{n}−1} Δ_{n}(F; [a : x_{K′}])[(−1)^{n}Δ_{n}(g; [ξ_{K′}^{[1,n]} : ξ_{K′+1}^{[1,n]}])],

(3.11)

and

T_{n}^{(3)} ≡ ∑_{i_{1}=1}^{m_{1}−1} ⋯ ∑_{i_{n}=1}^{m_{n}−1} f(ξ_{I′}^{[1,n]})A_{I′}[g(ξ_{I′}^{[1,n]}) − Δ_{n}(g; [ξ_{I′}^{[1,n]} : b])]. (3.12)

First consider T_{n}^{(1)}. Since f ∈ HK_{n}(I_{n}), for every ε > 0 there exists δ(ξ_{I′}^{[1,n]}) > 0 such that for any δ-fine partition (2.1), (3.1), (3.2) (i. e., there exists a positive function δ such that each n-cell ∏_{j=1}^{n} [x_{i_{j}−1}^{(j)}, x_{i_{j}}^{(j)}]⊂ I_{n}

is contained in the open ball in I_{n} with center (ξ_{I′}^{[1,n]}) and radius δ) one has

| ∑_{i_{1}=1}^{k_{1}} ⋯ ∑_{i_{n}=1}^{k_{n}} f(ξ_{I′}^{[1,n]})A_{I′} − Δ_{n}(F; [a : x_{K′}])| < ε,  (3.13)

where F denotes the n-dimensional primitive of f. (This is the justification for writing eq.(1.6).) From (3.13) and the definition (2.2) of n-dimensional variations in the sense of Vitali et al, it follows that

|T_{n}^{(1)}| < εV (g; I_{n}).  (3.14)

To evaluate T_{n}^{(2)}, we have by (1.5):

\begin{matrix}Δ_{n}(F; [a : x_{K′}]) = F (x_{K′}^{[1,n]}) \\ + ∑_{q=1}^{n−1} ∑_{k_{1}=1}^{n} ⋯ ∑_{k_{q}=1}^{n} (−1)^{q}F_{K′_{q}} + (−1)^{n}F (a), \end{matrix}  (3.15)

where k_{1} < ⋯ < k_{q} and (see the discussion following eq. (1.5)) the functions F_{K′_{q}} are obtained from F by evaluation at the lower limits a_{i} for the indices K′_{q} and evaluation at the upper limits x_{k_{i}}^{(i)} for the remaining indices. By Lemma 3.1 the contributions to T_{n}^{(2)} obtained by replacing Δ_{n}(F; ⋯) in (3.11) by the first term in (3.15) converge to the Riemann-Stieltjes integrals (3.6). Similarly, for the contributions to T_{n}^{(2)} obtained from the terms in (3.15) involving the functions F_{K′_{q}}, we use repeatedly the decompositions (2.5) of Δ_{n}(F; ⋯) into differences of lower order to prove, again using Lemma 3.1, that these contributions to T_{n}^{(2)} converge to the following Riemann-Stieltjes integrals:

∑_{k_{1}=1}^{n} ⋯ ∑_{k_{q}=1}^{n} (−1)^{q} ∫_{a_{q+1}}^{b_{q+1}} ⋯ ∫_{a_{n}}^{b_{n}} {F (a_{K′_{q}}; x_{n−q})} ×
{d_{x_{n−q}}Δ_{q}(g(⋯; x_{n−q}); [a_{K′_{q}} : b_{K′_{q}}])}

≡ ∑_{j_{1}=1}^{n} ⋯ ∑_{j_{q}=1}^{n} (−1)^{q}B_{n}^{(j_{1},···,j_{q})},

and, as usual, we have the constraints j_{1} < ⋯ < j_{q} on the summation indices. Finally, for the contributions to T_{n}^{(2)} arising from the remaining term in (3.15), we obtain

(−1)^{n}F (a) ∑_{k_{1}=1}^{m_{1}−1} ⋯ ∑_{k_{n}=1}^{m_{n}−1} Δ_{n}(g; ξ_{I}^{[1,n]}) = F (a)Δ_{n}(g; [a : b])

using the additivity of the Δ_{n} and the identifications (3.2). Thus, upon collecting the above results, we obtain:

T_{n}^{(2)} = T̃_{n}^{(2)} + F (a)Δ_{n}(g; [a : b])  (3.16)

with, for every ε > 0, |T̃_{n}^{(2)} − B_{n}| < ε, where

B_{n} ≡ (−1)^{n}B_{n}^{(0)} + ∑_{q=1}^{n−1} ∑_{j_{1}=1}^{n} ⋯ ∑_{j_{q}=1}^{n} (−1)^{q}B_{n}^{(j_{1},···,j_{q})},  (3.17)

where j_{1} < ⋯ < j_{q}.

We now consider T_{n}^{(3)} as given by (3.12). Again using formula (1.5), we have:

Δ_{n}(g; [ξ_{I′}^{[1,n]} : b]) = g(b) + ∑_{q=1}^{n−1} ∑_{j_{1}=1}^{n} ⋯ ∑_{j_{q}=1}^{n}(−1)^{q}g_{j_{1},···,j_{q}} + (−1)^{n}g(ξ_{I′}^{[1,n]}),

(3.18)

where the subscripts j_{1}, ⋯, j_{q} are subject to the constraints j_{1} < ⋯ < j_{q} and the functions g_{J′_{q}} are derived from g in a similar manner as described in the discussions following (1.5) and (3.15). Substitution of (3.18) into (3.12) gives:

T_{n}^{(3)} = (−1)^{n−1} ∑_{i_{1}=1}^{m_{1}−1} ⋯ ∑_{i_{n}=1}^{m_{n}−1} f(ξ_{I}^{[1,n]})A_{I′}[g(b) + ∑_{q=1}^{n−1} ∑_{i_{1}=1}^{n} ⋯ ∑_{i_{q}=1}^{n} (−1)^{q}g_{I′_{q}}]

(3.19)

where, as in (3.18), i_{1} < ⋯ < i_{q}.

We now transform the summations ∑_{i_{j}=1}^{m_{j}−1} (j = 1, 2, ⋯, n) in (3.19)

to summations of the form ∑_{i=1}^{m} and call the resulting expressions T_{n}^{(4)} :

T_{n}^{(4)} = (−1)^{n−1} ∑_{i_{1}=1}^{m_{1}} ⋯ ∑_{i_{n}=1}^{m_{n}} f(ξ_{I′}^{[1,n]})A_{I′}[g(b) + ∑_{q=1}^{n−1} ∑_{i_{1}=1}^{n} ⋯ ∑_{i_{q}=1}^{n}(−1)^{q}g_{I′_{q}}],

(3.20)

with i_{1} < ⋯ < i_{q}. The difference between T_{n}^{(3)} and T_{n}^{(4)} involves terms in which the number of summations is ≤ n − 1 and one easily shows by straightforward calculations that these terms cancel with some of the terms in J_{n} − T_{n}. We note that the summations for given values of q in (3.20) are special cases of the summations (3.8) because, in the functions g_{J′_{q}}, some of the arguments are the variables ξ_{i_{j}}^{(j)} and some are upper limits. Thus, the terms in (3.20) can be evaluated by using the same procedure that we are using to evaluate J_{n}; viz., by using analogues of the decompositions (3.10), (3.16), and the transformation from (3.19) to (3.20).

For the terms in (3.19) with a given value of q in the set {1, 2, ⋯, n− 1} we have, by analogy with the decompositions (3.10):

\begin{matrix}T_{n,q}^{(4)} ≡ (−1)^{n+q−1} ∑_{α_{1}=1}^{n} ⋯ ∑_{α_{q}=1}^{n} \\ ∑_{i_{1}=1}^{m_{1}} ⋯ ∑_{i_{n}=1}^{m_{n}} f(ξ_{I′}^{[1,n]})A_{I′}g(ξ_{α_{1}}^{(1)}, ⋯, ξ_{α_{q}}^{(q)}; b_{n−q}) \\ = T_{n,q}^{(4,1)} + T_{n,q}^{(4,2)} + T_{n,q}^{(4,3)}, \end{matrix}  (3.21)

where α_{1} < ⋯ < α_{q} and

T_{n,q}^{(4,1)} ≡ (−1)^{n+q−1} ∑_{α_{1}=1}^{n} ⋯ ∑_{α_{q}=1}^{n} ∑_{k_{1}=1}^{m_{1}−1} ⋯ ∑_{k_{q}=1}^{m_{q}−1}

{ ∑_{i_{1}=1}^{k_{1}} ⋯ ∑_{i_{q}=1}^{k_{q}} ∑_{i_{q+1}=1}^{m_{q+1}} ⋯ ∑_{i_{n}=1}^{m_{n}} f(ξ_{I′}^{[1,n]})A_{I′}

−Δ_{n}(F; [a_{q} : x_{K′_{q}}^{[1,q]}]; [a_{n−q} : b_{n−q}])}

×(−1)^{q}Δ_{q}(g(⋯; b_{n−q}); [ξ_{K′_{q}}^{[1,q]} : ξ_{K′_{q}+1}^{[1,q]}]),

T_{n,q}^{(4,2)} ≡ (−1)^{n+q−1} ∑_{α_{1}=1}^{n} ⋯ ∑_{α_{q}=1}^{n} ∑_{k_{1}=1}^{m_{1}−1} ⋯ ∑_{k_{q}=1}^{m_{q}−1}

{Δ_{n}(F; [a_{q} : x_{K′_{q}}^{[1,q]}]; [a_{n−q} : b_{n−q}])} ×
{(−1)^{q}Δ_{q}(g(⋯; b_{n−q}); [ξ_{K′_{q}}^{[1,q]} : ξ_{K′_{q}+1}^{[1,q]}])} ,

\begin{matrix}T_{n,q}^{(4,3)} ≡ (−1)^{n+q−1} ∑_{k_{1}=1}^{n} ⋯ ∑_{k_{q}=1}^{n} ∑_{i_{1}=1}^{m_{1}} ⋯ ∑_{i_{n}=1}^{m_{n}} & \{f(ξ_{I′}^{[1,n]})A_{I′}\} \\ × [g(ξ_{K′_{q}}^{[1,q]}; b_{n−q}) + (−1)^{q−1}Δ_{q}(g(⋯; b_{n−q}); [ξ_{K′_{q}}^{[1,q]}, b_{q}])] , \end{matrix}

where we again have the constraints k_{1} < ⋯ < k_{q} on the summation indices.

Each expression T_{n,q}^{(4,1)} can be estimated in a similar manner as T_{n}^{(1)} in (3.14), and a decomposition analogous to (3.16) for T_{n}^{(2)} can be obtained for T_{n,q}^{(4,2)} with associated terms T̃_{n,q}^{(4,2)} converging to appropriate Riemann-Stieltjes integrals analogous to (3.17). For T_{n,q}^{(4,3)}, we apply formula (1.5) to Δ_{q} and obtain expressions analogous to (3.19):

\begin{matrix}T_{n,q}^{(4,3)} = (−1)^{n} ∑_{i_{1}=1}^{m_{1}−1} ⋯ ∑_{i_{n}=1}^{m_{n}−1} ∑_{k_{1}=1}^{n} ⋯ ∑_{k_{q}=1}^{n} [f(ξ_{I′}^{[1,n]})A_{I′}] × [g(b)+ \\ ∑_{q′=1}^{q−1} ∑_{β_{1}=1}^{q} ⋯ ∑_{β_{q′}=1}^{q} (−1)^{q′}g_{β_{1},···,β_{q′}}([ξ_{K′_{q′}}, b_{K′_{q′}}]; b_{n−q′−1})], \end{matrix}

(3.22)

with the constraints k_{1} < ⋯ < k_{q} and β_{1} < ⋯ < β_{q′} on the summation indices.

We now transform the summations ∑_{i_{j}=1}^{m_{j}−1} (j = 1, 2, ⋯, n) in (3.22)

to summations of the form ∑_{i=1}^{m} to obtain new expressions, say T_{n,q}^{(5)},

as we did in passing from T_{n}^{(3)} to T_{n}^{(4)}. Just as in the earlier case, the difference between (3.22) and T_{n,q}^{(5)} involves terms in which the number of summations is ≤ n − 1 and these terms cancel some of the terms in J_{n} − T_{n}. We note that the terms in the summation over q in (3.20) contain expressions that combine with the first term in (3.20), and the combination of these quantities can be estimated by using an argument similar to that used in (3.13). Finally, by combining the evaluations of (3.20) with the results (3.16), we obtain the formula (3.5), thereby completing the proof of Theorem 3.2.

REMARK. We note that, for dimensions n ≥ 3, the contributions to (3.21) for q ≥ 2 contain terms that cancel some of the contributions to (3.21) with smaller values of q. This has the effect of changing the sign of the contributions of the latter terms to (3.20).

4. Characterization of multipliers

We now combine Theorem 3.2 with a modified version of Theorem 2.1 to obtain the following characterization of multipliers for the class of HK_{n} functions.

THEOREM 4.1. Let f and g denote real-valued functions defined on I_{n}. Then, for a given positive integer n ≥ 1, fg ∈ HK_{n}(I_{n}) for all f ∈ HK_{n}(I_{n}) if and only if g is of strongly bounded essential variation on I_{n}.

Before discussing the proof of this theorem, we need to modify the definition of the n-dimensional variations (2.2) to obtain quantities more appropriate for functions of strongly bounded variation. A definition appropriate for this case has been given by Kurzweil in [Ku2]. For partitions of the form (2.1), in addition to considering essential suprema of n^{th} differences as in (2.2), one must also consider essential suprema of all lower order differences involving the ”component intervals” of I_{n}. For example, in the case n = 2 we consider:

\begin{matrix}V_{SBV} (ϕ; I_{2}) ≡ ess sup_{P} ∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}}[|Δ_{2}(ϕ; x_{i_{1}−1}^{(1)}, x_{i_{1}}^{(1)}; x_{i_{2}−1}^{(2)}, x_{i_{2}}^{(2)})| + \\ ∑_{c_{1}=1}^{2} |Δ_{1}(ϕ(x_{c_{1}i_{1}}^{(1)}, ·); x_{i_{2}−1}^{(2)}, x_{i_{2}}^{(2)})| + ∑_{c_{2}=1}^{2} |Δ_{1}(ϕ(·, x_{c_{2}i_{2}}^{(2)}); x_{i_{1}−1}^{(1)}, x_{i_{1}}^{(1)})|], \end{matrix}

(4.1)

where, as in (2.6), the values c_{j} = 1, 2 for j = 1, 2 correspond to the evaluation of the function ϕ at the left or right endpoint, respectively, of the intervals [x_{i_{j}−1}^{(j)}, x_{i_{j}}^{(j)}]. Then, by analogy with the derivation of the inequalities (2.6), we obtain upper bounds for the n-dimensional (n = 2, 3, ⋯) versions of (4.1) in terms of the one-dimensional essential oscillations ω defined by (2.4):

\begin{matrix}V_{SBV} (ϕ; I_{n}) ≤ n ess sup_{P} ∑_{i_{1}=1}^{m_{1}} ⋯ ∑_{i_{n}=1}^{m_{n}} ∑_{ν=1}^{n} \\ [ ∑_{c_{j_{1}}=1}^{2} ⋯ ∑_{c_{j_{n−1}}=1}^{2} ω(ϕ(x_{c_{j_{1}} i_{j_{1}}}^{(j_{1})} , ⋯, x_{c_{j_{n−1}} i_{j_{n−1}}}^{(j_{n−1})} , ·); x_{i_{ν}−1}^{(ν)}, x_{i_{ν}}^{(ν)})], \end{matrix} (4.2)

where j_{p} < j_{p+1}(p = 1, 2, ⋯, n − 2) and ν has the value in the set {1, 2, ⋯, n} that is distinct from the values {j_{1}, j_{2}, ⋯, j_{n−1}}. We see that, whereas the variations (2.2) can be bounded by one-dimensional essential oscillations over any of the one-dimensional ”component intervals” of I_{n} that we wish, as in (2.6), we note from (4.2) that the upper bounds of the variations V_{SBV} (ϕ; I_{n}) involve one-dimensional essential oscillations over all of the one-dimensional component intervals. Thus, it will be convenient in the proof of Theorem 4.1, to be discussed below, to decompose the variations V_{SBV} (ϕ; I_{n}) in the following manner:

V_{SBV} (ϕ; I_{n}) = ∑_{ν=1}^{n} V_{SBV}^{(ν)} (ϕ; I_{n}) where, for each ν ∈ {1, 2, ⋯, n} as

defined following (4.2), with j_{p} < j_{p+1} for p = 1, 2, ⋯, n − 2:

\begin{matrix}V_{SBV}^{(ν)} (ϕ; I_{n}) ≤ n ess sup_{P} ∑_{i_{1}=1}^{m_{1}} ⋯ ∑_{i_{n}=1}^{m_{n}} ∑_{c_{j_{1}} =1}^{2} ⋯ \\ ∑_{c_{j_{n−1}}=1}^{2} ω(ϕ(x_{c_{j_{1}} i_{j_{1}}}^{(j_{1})} , ⋯, x_{c_{j_{n−1}} i_{j_{n−1}}}^{(j_{n−1})} , ·); x_{i_{ν}−1}^{(ν)}, x_{i_{ν}}^{(ν)}). \end{matrix} (4.3)

PROOF OF THEOREM 4.1. It follows from Theorem 3.2 that the condition that g is of strongly bounded essential variation is sufficient to have fg ∈ HK_{n}(I_{n}) for all f ∈ HK_{n}(I_{n}) with n = 1, 2, ⋯, so our task is to show that the proof of Theorem 2.1 can be modified to provide a proof of the necessity of this condition. For n = 1 the definition of functions of strongly bounded variation reduces to the usual concept of functions of bounded variation, so the result is true in this case.

For n ≥ 2, following the argument in the proof of Theorem 2.1, we may again reduce to the case in which g is essentially bounded. Now, suppose that g is not equal a.e. to a function of strongly bounded variation. Then, modifying the argument in the proof of Theorem 2.1, for any function g_{2} = g a.e., g_{2} is not of strongly bounded variation and there exists a point κ ∈ I_{n} such that g_{2} is not of strongly bounded variation on any open n-cell which contains κ. In particular, we may suppose that κ is a singular point of g where, however, we now define the concept of singular point in a different manner than in the proof of Theorem 2.1 by using the definition of variation given following the statement of the present theorem (i.e., the n-dimensional analogues of (4.1)). We may assume, without loss in generality, that the point (a_{1}, ⋯, a_{n}) ∈ I_{n} is a singular point in this modified sense.

The proof now proceeds in a similar manner to that of Theorem 2.1. We obtain by induction a sequence of disjoint n-cells J_{k}^{(n)} ≡ [α_{1}^{(k)}, β_{1}^{(k)}] × ⋯ × [α_{n}^{(k)}, β_{n}^{(k)}] ⊂ I_{n} with β_{i}^{(k)} < α_{i}^{(k−1)} for i = 1, 2, ⋯, n and k = 2, 3, ⋯ such that lim_{k→+∞} α_{i}^{(k)} = lim_{k→+∞} β_{i}^{(k)} = a_{i} for each i ∈

{1, 2, ⋯, n}. (For convenience, we use the same notation for these ncells as for the n-cells discussed in the proof of Theorem 2.1.) On each of these n-cells J_{k}^{(n)}, there exists a partition of the form (2.10) such that inequalities analogous to (2.11) hold for each value of ν defined following (4.2):

\begin{matrix}∑_{i_{1}=q_{1,k−1}+1}^{q_{1,k}} & ⋯ & ∑_{i_{n}=q_{n,k−1}+1}^{q_{n,k}} \\ ∑_{c_{j_{1}} =1}^{2} ⋯ ∑_{c_{j_{n−1}} =1}^{2} ω(g(x_{c_{j_{1}} i_{j_{1}}}^{(j_{1})} , ⋯, x_{c_{j_{n−1}} i_{j_{n−1}}}^{(j_{n−1})} , ·); x_{i_{ν}−1}^{(ν)}, x_{i_{ν}}^{(ν)}) \\ ≥ \frac{1}{n}V_{SBV}^{(ν)} ≥ \frac{k}{n}. \end{matrix}  (4.4)

Then, on each set K_{k}^{(n,ν)} ≡ {x_{c_{j_{1}} i_{j_{1}}}^{(j_{1})} , ⋯, x_{c_{j_{n−1}} i_{j_{n−1}}}^{(j_{n−1})} }×[x_{i_{ν}−1}^{(ν)}, x_{i_{ν}}^{(ν)}] ⊂

J_{k}^{(n)}, there exist disjoint measurable sets L_{i_{j_{1}} ,···,i_{j_{n−1}} ,i_{ν}}^{(k,ν)}, H_{i_{j_{1}} ,···,i_{j_{n−1}} ,i_{ν}}^{(k,ν)}

such that η_{k−1,ν} > μ(L_{i_{j_{1}} ,···,i_{j_{n−1}} ,i_{ν}}^{(k,ν)}) = μ(H_{i_{j_{1}} ,···,i_{j_{n−1}} ,i_{ν}}^{(k,ν)}) ≡ η_{k,ν} > 0 and such that analogues of (2.7) hold:

z ∈ L_{i_{j_{1}} ,···,i_{j_{n−1}} ,i_{ν}}^{(k,ν)} ⇒ g(z) ≥ \frac{3}{2}M_{i_{j_{1}} ,···,i_{j_{n−1} ,iν}}^{(ν)} + \frac{1}{2}m_{i_{j_{1}} ,···,i_{j_{n−1}} ,i_{ν}}^{(ν)}, (4.5a)

while

z ∈ H_{i_{j_{1}} ,···,i_{j_{n−1}} ,i_{ν}}^{(ν)} ⇒ g(z) ≤ \frac{1}{2}M_{i_{j_{1}} ,···,i_{j_{n−1}} ,i_{ν}}^{(ν)} + \frac{3}{2}m_{i_{j_{1}} ,···,i_{j_{n−1}} ,i_{ν}}^{(ν)}, (4.5b)

where

M_{i_{j_{1}} ,···,i_{j_{n−1}} i_{ν}}^{(ν)} ≡ ∑_{c_{j_{1}} =1}^{2} ⋯ ∑_{c_{j_{n−1}} =1}^{2} ess sup_{y∈[x_{i_{ν} −1}^{(ν)},x_{i_{ν}}^{(ν)}]}g(x_{c_{j_{1}} i_{j_{1}}}^{(j_{1})} , ⋯, x_{c_{j_{n−1}} i_{j_{n−1}}}^{(j_{n−1})} ; y),

(4.6)

m_{i_{j_{1}} ,···,i_{j_{n−1}} ,i_{ν}}^{(ν)} ≡ ∑_{c_{j_{1}} =1}^{2} ⋯ ∑_{c_{j_{n−1}} =1}^{2} ess inf_{y∈[x_{iν −1}^{(ν)},x_{iν}^{(ν)}]}g(x_{c_{j_{1}} i_{j_{1}}}^{(j_{1})} , ⋯, x_{c_{j_{n−1}} i_{j_{n−1}}}^{(j_{n−1})} ; y),

(4.7)

and, in defining (4.6) and (4.7), we have used results analogous to the one stated following (2.9). In addition, lim_{k→+∞} η_{k,ν} = 0 for each value

of ν defined following (4.2).

For a given z in I_{n}, we now define a real-valued function f(z) by

f(z) = \{ \begin{matrix}p_{k,ν} & if z ∈ L_{i_{j_{1}} ,···,i_{j_{n−1}} ,i_{ν}}^{(k,ν)}, \\ −p_{k,ν} & if z ∈ H_{i_{j_{1}} ,···,i_{j_{n−1}} ,i_{ν}}^{(k,ν)}, ν = 1, 2, ⋯, n, \\ 0 & otherwise, \end{matrix}

where

\frac{1}{p_{k,ν}} ≡ η_{k,ν}\sqrt{V_{SBV}^{(ν)} (g; J_{k}^{(n)})},  (4.8)

and q_{j,k−1} + 1≤i_{j}≤q_{j,k} for j = 1, 2, ⋯, n. Then

\begin{matrix}∫_{I_{n}} fdz = ∫_{J_{k}^{(n)}} fdz = ∑_{i_{1}=q_{1,k−1}+1}^{q_{1,k}} ⋯ ∑_{i_{n}=q_{n,k−1}}^{q_{n,k}} ∑_{c_{j_{1}} =1}^{2} ⋯ ∑_{c_{j_{n−1}} =1}^{2} \\ ∑_{ν=1}^{n}[  ∫_{L_{i_{j_{1}} ,···,i_{j_{n−1} ,iν}}^{(k,ν)}} fdz + ∫_{H_{i_{j_{1}} ,···,i_{j_{n−1} ,iν}}^{(k,ν)}} fdz] = 0, \end{matrix}

so that f ∈ HK_{n}(I_{n}). Similarly, by (4.5) and using (4.6), (4.7), and (2.4), we have

∫_{I_{n}} fgdz ≥ ∑_{i_{1}=q_{1,k−1}+1}^{q_{1,k}} ⋯ ∑_{i_{n}=q_{n,k−1}+1}^{q_{n,k}} ∑_{c_{j_{1}} =1}^{2} ⋯ ∑_{c_{j_{n−1}} =1}^{2}

∑_{ν=1}^{n} p_{k,ν}η_{k,ν}[M_{i_{j_{1}} ,···,i_{j_{n−1}} ,i_{ν}}^{(ν)} − m_{i_{j_{1}} ,···,i_{j_{n−1}} ,i_{ν}}^{(ν)}]

= ∑_{i_{1}=q_{1,k−1}+1}^{q_{1,k}} ⋯ ∑_{i_{n}=q_{n,k−1}+1}^{q_{n,k}} ∑_{c_{j_{1}} =1}^{2} ⋯ ∑_{c_{j_{n−1}} =1}^{2}

∑_{ν=1}^{n} p_{k,ν}η_{k,ν}ω(g(x_{c_{j_{1}} i_{j_{1}}}^{(j_{1})} , ⋯, x_{c_{j_{n−1}} i_{j_{n−1}}}^{(j_{n−1})} , ·); x_{i_{ν}−1}^{(ν)}, x_{i_{ν}}^{(ν)}).

Then, using the inequalities (4.3) and the definition (4.8) of p_{k,ν}, we have

∫_{I_{n}} fgdz≥ ∑_{ν=1}^{n} \sqrt{V_{SBV}^{(ν)} (g; J_{k}^{(n)})}.

Since V_{SBV}^{(ν)} (g; J_{k}^{(n)}) → +∞ as k → +∞ for ν ∈ {1, 2, ⋯, n} by the inequality on the right-hand side of (4.4), it follows that fg ∉ HK_{n}(I_{n}). This contradiction completes the proof of Theorem 4.1.

Acknowlegments

One of us (WWZ) wishes to thank Lee Tuo Yeong for reprints of the papers [LCL] and [L2], and to also thank him, as well as Erik Talvila and Luisa Di Piazza, for discussions concerning the work presented herein.

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TEPPER L. GILL^{1,2,3} W. W. ZACHARY^{1,4} tgill@howard.edu wwzachary@earthlink.net

(1) DEPARTMENT OF ELECTRICAL ENGINEERING, HOWARD UNIVERSITY WASHINGTON, DC 20059

(2) DEPARTMENT OF MATHEMATICS, HOWARD UNIVERSITY, WASHINGTON, DC 20059

(3) DEPARTMENT OF PHYSICS, UNIVERSITY OF MICHIGAN, ANN ARBOR, MI, 48109

(4) DEPARTMENT OF MATHEMATICS AND STATISTICS, UNI-VERSITY OF MARYLAND UNIVERSITY COLLEGE, COLLEGE PARK, MD 20742