Internat. J. Math. & Math. Sci.
VOL. 13 NO. 2 (1990) 243-246
A NOTE ON SOME SPACES
243
L OF DISTRIBUTIONS WITH LAPLACE TRANSFORM
SALVADOR PEREZ ESTEVA
Instituto de Matemticas
Universidad Nacional
Mxico, D.F.
Autnoma
de Mxico
04510 M6xico
(Received March 2, 1989)
In this paper we calculate the dual of the spaces of distributions
ABSTRACT.
[I ].
introduced in
L
Then we prove that
KEY WORDS AND PHRASES.
is the dual of a subspace of
D’
and denote by
a
L
spaces
Laoy
-Y
PY
S’
and
L
be the classical Schwartz’s
L2OR),
then we define
Laoy
Dp
Since
La
PY
were introduced
as follows:
L
is the subspace of
f
spaces of distributions in
In (Prez-Esteva [I ])
the Laplace transformation.
where
e
[a
DY
py
a
of functions f with supp f C [a,) and
a
is a Hilbert space with the inner product
OR)
loc
-Tx. L oy
-y (x)=e
I
(f,g)
on
Primary 44A35, Secondary 44AI0
INTRODUCTION
Let
e
C OR).
Convolution, Laplace Transform, Strict Inductive Limit.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODE.
I.
L
e_2 fg
dx
where Dp is the distributional derivative of order p.
oY
a
is bijective, we can copy the Hilbert space structure of L
oY
DP[a
We have the continuous inclusions
La
C
Lb
for
a>b
C
a
L qy,
if
p
pY
L
Hence for
p
{0,I,...}
a
PY
<q
the strict inductive limit
L
makes sense.
py’
ind lim
a
PY
La
PY
Then
L
Y
ind lim
p
L
ind lim
PY
p
L-p
PY
is also well defined.
In[l]
it was studied the spaces of distributions
f
is continuous.
f,g: [
L
Y
g
for which the convolution
S.P. ESTEVA
244
of
C OR).
Sy
will denote the norm of
N
(Sy)’=
and conclude that
L2OR),
y
Ly,
S
7
which
will be
will be the set of nonegative integers.
[
THE DUAL OF
DEFINITION I.
such that
which turns out to be a subspace
Y
Then we prove the reflexivity of
ll’II
is the main result of the paper,
2
assumed to be a positive constant, and
2.
L
the strong dual of
Here we describe
Let
a,)e_yg
he the space of all complex measurable functions
L
E
20R
characteristic function of
for every
E
a
[a,).
We provide
pa(g
lla,)e_yg
where
g
in
stands for the
a,)
with the topology given by the
L
seminorms
S
Next we denote by
n
N.
S
and
n
g
LEMMA I.
Ly and Sy are Frechet
Sy, we have that Sy C
L’,
Let
{gp}p 6N
Hence
gp+l
by go
a 6
Fix
Dnf
L
Y
for every
,
n E N
p E N,
e-2y f gpdx,
Dng
there exists
f 6
L
llocR)
gp
L
for any
such that
L
o
satisfies
is determined
PROOF.
a
2
spaces and since
then for every
(DPf)
The sequence
such that
Y
by the system of seminorms
II a,)e_y Dngll
pan(g
It is clear that
L
the subspace of
Define the topology of
m.
a
and
-Dgp + 2ygp
6
Sy.
p 6 N
p 6 N.
6
Then
(2.1)
(Ly)’
and there exists
gpa 6 La
such that
(DPf)
If
a
< b,
we have
[ e-2yf gpa
b
L
C
pY
(DPf)
for
DPf
6
Lbpy’
which
a
L py’
then
I e_2yf
gpb
I
dx
gpa
is the restriction of
belongs to
Let
L
6
PY
e
2yf X
b ,)
X[ b,=o)gpa
gpa
a, =),
to
gp
then
and
(DPf)
D.
Since
gpa dx
shows that
gpb
If
6 ia
DPf
dx,
J e_2yf gpdx
DP+I 6 Lp+ Y
]
L
PY
DPf
E
L
py
we have
Da gpa
is well defined,
SPACES OF DISTRIBUTIONS WITH LAPLACE TRANSFORM
i e_2ygp+l I e_2yDgp
(DP+I)
dx
]{D(e_2y)
<-e_2{Dgp
<-,’>
where
+
2y
e_2 Y}gp
dx
dx
>
+ 2y e_2{g p
D
represents the duality between
gp+l
245
and
D’.
It follows that
-Dgp + 2ygp
or
-D(e-2y gp)
e-2y gp+l
gp
Hence, every
LEMMA 2.
H
+ 2yl.
-D
g 6 S
Let
is well defined in
PROOF.
{f
and
Y
H be the differential operator defined by
Then the functional
I
(DPf)
sequence
S.
belongs to
Let
n n6 N
L
f 6
and is continuous.
Y
f 6
C p
H(P)gdx’
e-2y f
Laoy
be such that
converging to
f
f
Dh
b
h 6
with
L oy if
in
b
Loy.
There exists a
< a.
Let
n (x)
b
ioy’
n
uous, we have that
Then
D’n
f
and
h)
f
{n}n6N
n
fn dy
and since the inclusion
f
h
converges to
in
f e_2yh
H(g)dx
Lim
I e_2yf
g dx
Limf e_27fngdx
f e_2yn
Lo.
Lb
o
C Lb
is contin-
If follows that
H(g) dx
(2.2)
(2.3)
On the other hand
e_2.(on
n D(e-2g)
tt(g)dx
-n(B)e-2y(B)g(B)
+
fn
e-2yg
(2.4)
dx
b
But we have the estimate
]g(x) < Ig(b)] +
1/2
ey(x) ]lb,oo)e_y(Dg Yg)](x-b)
Hence
e_2yn
dx
H(g)dx
e_2yfn
g dx
From (2.2) and (2.3) it follows that
f e_2yfgdx I e_2,h
H(g)dx
for
x
>b
S.P. ESTEVA
246
By induction we obtain
;
if
DPh
f
f e_2yh H(P)(g)
e_2yfgdx
f,h E
and
L oy
DPf
Finally, if
(2.6)
dx
Dqh
f,h E L
with
and
q
p,
f
then
Dq-Ph,
hence
by (2.6) we have
e_2yf
H (P) (g)dx
e_2yh
H
(q)
(g)dx
is well defined and it is clearly continuous.
Thus
THEOREM I.
PROOF.
_
is
and 2 we know that
By lemmas
L, Ly)
strong topology
Ly
The strong dual of
Sy.
Ly’ Sy.
It remains to prove that the
coincides with the topology
T
of
First notice that
Sy.
is defined by the system of seminorms
I x[ ,)_ "(P) (g)ll 2’
Let V
{g E Sy: qap(g)
qap (g)
a l
a
Fix
in
L oy’
and
p
then the set
DPu
II
B CE
Hence
{g
p
DPu
Sy: q_p
g
B
,
I}.
pY
N
p
l,
U
Denote by
L
and hence in
the unit ball
If
g
B
U we have
f
I<DPf,
(p)
(g
a,m) H
)112 <
c
B(L’,, Ly).
C V
Ly.
Now, let B be a bounded set in
and i
-p
and is bounded there (see Kucera, McKennon 2 ).
B C L
PY
for some g >0
where U is the unit ball in L -p. Let
oY
V implies for f E EU that
then g
(g) <
P
B
It follows that
Then for some
L
is bounded in
I e-2yf H(P)(g)dxl
Thus
Then
B
B), then for every
(the polar of
V
N.
a
N,
E-I},
<DPf, g>=
so we proved that
COROLLARY I.
L
.
If e_2yf li(p)(g)dx] <
V C B
This completes the proof.
is the strong dual of
S
Y
Y
PROOF. By (Kucera, McKennon 2 ], Theorem 4)
Hence the corollary follows from Theorem I.
we know that
L
is reflexive.
REFERENCES
S. Prez-Esteva, "Convolution Operators for the one-sided Laplace TransformatioH’
Casopis pro pestovn matematiky, Vol. 110(1985), 69-76.
[2
J. Kucera, K. McKennon, "Dieudonn-Scwartz theorem on bounded sets in inductive
limits", Proc. Amer, Math. Soc. 78(1980), 366-368.
[3
[4
J. Kucera, "Multiple Laplace Integral", Czech. Math. J. 19(1969), 181-189.
L. Schwartz,
Theory of Distributions, Hermann, Paris, 1966.