An analogue of the Denjoy-Perron-Henstock-Kurzweil integral 361
contradicting the choice of x
n
, y
n
. Assume now i = 2. As above we can also assume that x
n
ց x, y
n
ր y. Simple calculations show that |F x, y − F x
n
, y
n
| ≤ µ
g
a, x × [y
n
, y + µ
g
[x
n
, x × c, y
n
≤ µ
h
a, x × [y
n
, y + µ
h
[x
n
, x × c, y
n
= F hx, y − F hx, y
n
− F hx
n
, y + F hx
n
, y
n
. By the continuity of F h, the right hand side of this inequality tends to 0. We
obtain again a contradiction. The cases i = 3, 4 are treated in the same way. We have proved that F is continuous at x, y ∈ Q. The continuity of F on ∂Q is
established similarly.
1.3 Some further properties
The usual restriction and extension properties of the G-integral is given in the following theorem.
Theorem 2 i Let f ∈ GQ and Q
′
= a
′
, b
′
× c
′
, d
′
⊂ Q. Then f|
Q
′
∈ GQ
′
and F f|
Q
′
x, y = F fx, y − F fa
′
, y − F fx, c
′
+ F fa
′
, c
′
, ∀x, y ∈ Q
′
. ii For a ≤ m ≤ b and c ≤ n ≤ d, let
Q
1
= a, m × c, n, Q
2
= m, b × c, n, Q
3
= a, m × n, d, Q
4
= m, b × n, d. Then for each f
1
, f
2
, f
3
, f
4
∈
4
Y
i =1
GQ
i
, there exists a unique f ∈ GQ such that f|
Q
i
= f
i
, 1 ≤ i ≤ 4. Moreover,
Z
Q
f =
4
X
i =1
Z
Q
i
f
i
. iii Let a m m
′
b and f ∈ D
′
Q. Then f ∈ GQ if and only if f|
a,m
′
×c,d
∈ Ga, m
′
× c, d and f|
m,b×c,d
∈ Gm, b × c, d. If Q
1
= a, m
1
× c, n
1
, Q
2
= m
2
, b × c, n
2
, Q
3
= a, m
3
× n
3
, d, Q
4
= m
4
, b × n
4
, d are subrectangles of Q such that
4
\
i =1
Q
i
6= ∅, then for every f ∈ D
′
Q, f ∈ GQ if and only if f|
Q
i
for all i ∈ {1, 2, 3, 4}. We conclude this part with the important fact that the G-integral represents an
extension of the Lebesgue integral. Theorem 3
If we identify f ∈ L
1
Q with the distribution f : g 7→ L
Z
Q
fg, g ∈ DQ, where L
Z
is the Lebesgue integral, then f ∈ GQ and
Z
Q
f = L
Z
Q
f. Moreover, GQ is the completion of L
1
Q or CQ with respect to the norm kfk = sup
L
Z
x a
Z
y c
f : x, y ∈ Q
.
362 D. D. Ang – K. Schmitt – L. K. Vy
Proof. Let f ∈ L
1
Q, and let F x, y = L
Z
x a
Z
y c
f, x, y ∈ Q. Then F ∈ ˆ
CQ. For g ∈ DQ, by Fubini’s theorem and the integration by parts formula, we have
L
Z
Q
fg = −
Z
b a
Z
d c
Z
y c
fx, tdt ∂
2
gx, ydy dx
= −
Z
b a
Z
d c
∂
1
F x, y∂
2
gx, ydx dy
=
Z
d c
Z
b a
F x, y∂
12
gx, ydx dy
= L
Z
Q
F ∂g. We therefore conclude that f = ∂F in D
′
Q. Thus f ∈ GQ and F = F f. In particular,
L
Z
Q
f = F b, d = F fb, d =
Z
Q
f, and L
1
Q ⊂ GQ. Now, let f ∈ GQ. Choose a sequence {F
n
} ⊂ C
2
Q such that F
n
→ F f uniformly on Q. Since F
n
a, · → 0 uniformly on [c, d], F
n
·, c → 0 uniformly on [a, b], by replacing F
n
x, y by F
n
x, y − F
n
a, y − F
n
x, c + F
n
a, c, we can assume that F
n
a, · = 0, F
n
·, c = 0. Thus f
n
= ∂F
n
∈ CQ ⊂ GQ and F f
n
= F
n
, ∀n. We have kf
n
− fk = kF f − F
n
k
∞
→ 0, proving the density of CQ and thus of L
1
Q in GQ.
2 Fubini theorems for
G
-integrable distributions
In this section, we consider some Fubini type theorems for the G-integral. We next apply these results to some initial value problems for the two-dimensional wave
equation with nonsmooth initial data.
2.1 Fubini theorems
We first make some remarks on traces of integrals of G-integrable distributions. For f ∈ CQ and x ∈ [a, b], the function
Z
x a
fs, ·ds : [c, d] → R, y 7→
Z
x a
fs, yds clearly satisfies
Z
x a
fs, ·ds = [F fx, ·]
′
on [c, d].
An analogue of the Denjoy-Perron-Henstock-Kurzweil integral 363
Generalizing to the case f ∈ GQ, we define, for f ∈ GQ, x ∈ [a, b], y ∈ [c, d]:
Z
x a
fs, ·ds = [F fx, ·]
′
in D
′
c, d,
Z
y c
f·, tdt = [F f·, t]
′
in D
′
a, b. It is clear that
Z
x a
fs, ·ds ∈ Gc, d,
Z
y c
f·, tdt ∈ Ga, b, where Ga, b and Gc, d are respectively the spaces of G-integrable distributions
on a, b and c, d, i.e. Ga, b = {g
′
∈ D
′
a, b : g ∈ C[a, b]}, where g
′
is the distributional derivative of g, etc. The consistency of the above definition can also be seen by remarking that if
f ∈ GQ, {f
n
} ⊂ CQ, f
n
→ f in GQ, then
Z
x a
f
n
s, ·ds →
Z
x a
fs, ·ds in Gc, d, and
Z
y c
f
n
·, tdt →
Z
y c
f·, tdt in Ga, b, x ∈ [a, b], y ∈ [c, d]. This means that the mapping GQ → Gc, d, f 7→
Z
x a
fs, ·ds is the unique extension of the mapping
CQ → Gc, d, f 7→
Z
x a
fs, ·ds. We are now in a position to prove a simple Fubini type theorem for GQ.
Theorem 4 For f ∈ GQ,
Z
Q
f =
Z
b a
Z
d c
f·, tdt =
Z
d c
Z
b a
fs, ·ds .
Proof. We first remark that the above repeated integrals exist in the sense of the one-dimensional G-integral. Since F f·, da = F fa, d = 0, one has
F f·, d = F
Z
d c
f·, tdt and
Z
b a
Z
d c
f·, tdt = F fb, d =
Z
Q
f. This proves the first equality. The second is proved in a similar way.
We next derive another form of Fubini’s theorem for some subclasses of GQ. A fundamental property of those classes is that one can define traces of their elements
on segments parallel to the sides of Q. We put G
i
Q = {∂
i
F : F ∈ CQ}, i = 1, 2, and
G
∗ 1
Q = {∂
1
F : F ∈ L
1
Q, F ·, y ∈ C[a, b], a.e. y ∈ [c, d], and ∃g = gF ∈ L
1
c, d such that |F x, ·| ≤ g, ∀x ∈ [a, b]}, and
G
∗ 2
Q = {∂
2
F : F ∈ L
1
Q, F x, · ∈ C[c, d], a.e. x ∈ [a, b] and ∃g = gF ∈ L
1
a, b such that |F ·, y| ≤ g, ∀y ∈ [c, d]}. Some elementary properties of these classes are given in the following proposition.
364 D. D. Ang – K. Schmitt – L. K. Vy
Proposition 1 i G
i
Q ⊂ G
∗ i
Q, i = 1, 2. ii L
1
Q ⊂ G
∗ 1
Q ∩ G
∗ 2
Q, and G
1
Q ∪ G
2
Q ⊂ GQ. iii If f ∈ G
i
Q then ∂f
j
∈ GQj ∈ {1, 2} and {i} = {1, 2} \ {j} For the definition of traces, we need the following lemma.
Lemma 4 If F ∈ L
1
Q and if ∂
1
F = 0 in D
′
Q, then there exists g ∈ L
1
c, d such that F x, y = gy a.e. x ∈ a, b, y ∈ c, d.
The proof of the lemma relies on arguments similar to those used in the proof of Lemma 1, and is thus omitted. Now suppose f ∈ G
∗ 1
Q and that F, F
1
∈ L
1
Q are as in the definition of G
∗ 1
Q, i.e. f = ∂
1
F = ∂
1
F
1
. It follows from Lemma 4 that [F ·, y]
′
= [F
1
·, y]
′
in D
′
a, b, which proves the consistency of the following definition.
Definition 2 Let f ∈ G
∗ 1
Q. We put, for almost all y ∈ [c, d], f·, y = [F ·, y]
′
in D
′
a, b, where F ∈ L
1
Q, ∂
1
F = f as in the definition of G
∗ 1
Q. We remark that this definition generalizes the one usually given for traces of
continuous functions. Indeed, suppose f ∈ CQ. Consider F x, y =
Z
x a
fs, yds, x, y ∈ Q. Since fx, y =
dF x, y dx
, x, y ∈ Q, one has ∂
1
F = f in D
′
Q, and f·, y = [F ·, y]
′
in D
′
a, b. We have a similar definition for fx, · if f ∈ G
∗ 2
Q. We see that if f ∈ G
∗ 1
Q then f·, y ∈ Ga, b for a.e. y ∈ [c, d]. Hence the integral
Z
b a
f·, y exists as a one-dimensional G-integral. Summarizing we have the following theorem.
Theorem 5 If f ∈ GQ ∩ G
∗ 1
Q, then the function y 7→
Z
b a
f·, y, y ∈ [c, d] is Lebesgue integrable on [c, d], and
Z
Q
f =
Z
b a
Z
d c
f·, ydy. Hence, for all f ∈ GQ ∩ G
∗ 1
Q ∩ G
∗ 2
Q, we have
Z
Q
f =
Z
d c
Z
b a
f·, y dy =
Z
b a
Z
d c
fx, · dx
Proof. Let F ∈ L
1
Q be such that ∂
1
F = f and that F satisfies the conditions in the definition of G
∗ 1
Q. A direct proof shows that for a.e. y ∈ [a, b],
Z
b a
f·, y = F b, y − F a, y. Since |F x, ·| ≤ g, x ∈ [a, b] with g ∈ L
1
c, d, the mapping Gx, y =
Z
y c
F x, tdt is defined for every x, y ∈ Q. Let x
n
, y
n
→ x, y in Q. We have χ
c,y
n
t →
An analogue of the Denjoy-Perron-Henstock-Kurzweil integral 365
χ
c,y
t, t ∈ [c, d]\{y}, and by the continuity of F ·, t on [a, b], F x
n
, t → F x, t, a.e. t ∈ [c, d]. Hence
F x
n
, tχ
c,y
n
t → F x, tχ
c,y
t for a.e. t ∈ [c, d]. Moreover, |F x
n
, tχ
c,y
n
t| ≤ |F x
n
, t| ≤ gt, ∀n ∈ N, a.e. t∈ [c, d]. By the dominated convergence theorem
Gx
n
, y
n
=
Z
d c
F x
n
, tχ
c,y
n
tdt →
Z
d c
F x, tc, ytdt = Gx, y as n → ∞. We have proved that G ∈ CQ. On the other hand, since F x, · ∈
L
1
c, d, we have ∂Gx, y
∂y = F x, y for a.e. y ∈ [c, d]. Integrating by parts, we have
∂
2
G = F in D
′
Q. Hence ∂G = ∂
1
F = f in D
′
Q. Thus G ∈ If. By definition,
Z
Q
f = Gb, d − Ga, d. Moreover,
Z
d c
Z
b a
f·, y =
Z
d c
|F b, y − F a, y|dy ≤ 2
Z
d c
gydy ∞,
i.e. the function y →
Z
b a
f·, y is in L
1
c, d. Furthermore,
Z
d c
Z
b a
f·, y dy =
Z
d c
F b, ydy −
Z
d c
F a, ydy = Gb, d − Ga, d. We have thus proved the first equality of Theorem 5. The second inequality may be
proved similarly.
Remark 1 From Proposition 1, we see that the first equality holds for all f ∈ G
1
Q and the second holds for f ∈ G
1
Q ∩ G
2
Q. In view of Theorem 2 and Proposition 4, we see that Theorem 5 is valid for all F ∈ L
1
Q. It is thus a generalization of the classical Fubini theorem for Lebesgue integrable functions on Q.
2.2 An application to differential equations
