819
I nternat. J. Math. & Math. Sci.
Vol. 4 No. 4 (1981) 819-822
RESEARCH NOTES
CONTINUITY OF MULTIPLICATION OF DISTRIBUTORS
JAN KUCERA and KELLY MCKENNON
Department of Pure and Applied Mathematics
Washington State University
Pullman, Washington 99164 U.S.A.
(Received February 20, 1981)
In a reference book for distributions [i], it is shown that the multi-
ABSTRACT.
plication (u, f)
uf on C
x
’,
Mx
as well as on
g’,
is hypocontinuous.
We
show here that in both cases it is discontinuous.
KEY WORDS AND PHRASES. Distribution, temperate distribution, dual space, strong
topology, inductive limit.
Primary 46F10, Secondary 46A09.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES.
INTRODUCTION.
i.
The discontinuity of multiplication on C
folklore, but no proof has yet been published.
result is new.
x
N’,
seems to be part of general
As far as we know the second
The presented proofs are simple enough that they can be included
in any future textbook on distributions.
THEOREM i.
Let
B
be the strong and o the weak* topology, both on
and y the usual topology on
(u, f)
C(Rn).
o’
Cy
uf
PROOF.
--
The family
Then the multiplication
is not jointly continuous.
of all increasing sequences of positive integers is
a directed set under the induced product ordering from N
product N
f
m, s
F
m,s
(t)
(mt
and let d
m+l)-I
C
g
if t
’ (Rn),
n/xlx2"’’Xn"
[i, exp(m- s(m))],
s(m)-1/2
d TM g(0)
N
Let F be the directed
For each (m,s)
with f
m, s
(t)
F,
put
0 otherwise, and
J. KUCERA AND K. MCKENNON
820
n
f
If
gm, s
is the Fourier transform of
j=l
t
f
(t)dt < m
m- i,
m, s
i, 2
lldkgm, II < m-n
respectively,
gm, s
lira
(m, s) F
0 in C
s(m) -1/2
+
{gm ,s Fm,s
and
(m, s) F
It remains to show that {F
bounded in zg.
N.
all m
For each such
N and g
for all m
which equals
,
n
R
,
such that s(m) >
s(m)-1/2r(m)
<
Ix12) 1/2,
(1 +
R
Il <_--
N, the space
{u
P,q
with the operator norm
space
@M
C;
f
is Banach.
If
H
uf
H
P
the supremum norm of
N.
The
L(Rn)
For every
continuous} equipped
q
ind lim
q
of rapidly increasing functions equals proj lira
denote by
(d f)(x)
P,q
n
R
and Hilbert
n
space g of rapidly decreasing functions equals the proj lira H
k.
p, q
-2 2
r (m) for
+ }, k
<
r(m)
.
for all g
g
kE Iwk-lelDfl2dx
If
C;
o
Idmg(o) _<
,’ such that
there exists r
Is(m) -1/2 dmg(o) _<
{f
in some
converges to 0 uniformly on every set
m,s
In the sequel, we need a weight function W(x)
H
k
s(m) imply,
where i is the imaginary
Then
IFm, s(g)
spaces
(t)dt
m, s
does not converge to 0 in
(m,s)F
Choose e > 0 and s
l.
f
t
inms(m)n,
(Rn)
For any
y
/_
then the inequalities
(m,s)rlim l(gm’sFm’s)l (m,s)Flim Is(m)-1/2dm gm, s(0)
neighborhood of the origin,
lim
dmgm, s(0)
and
s
Thus
unit.
k
fm’s(Xj)’
P,q
@
q
[4].
then the
Finally,
the dilation operator
and by d
f(gx).
LEMMA i.
For each k
N and multi-index
llg[Iwk(x)D exp(_l_xl2 )I I D exp(-Jx] 2 )Iloo
0+
PROOF.
lim llglIwk(x)D exp(-_x 2)i,o
lim
g
n,
N
where
xl
2=x2+x 2+... +xn2
2
c 0+
lira
0+
IIWk(x)
d
D
exp(-
Ixl 2 )IIoo=
lira
0+
IIWk(x)D exp(-Ixl 2 )Iloo llDexp (-
x
2) Iloo
821
CONTINUITY OF MULTIPLICATION OF DISTRIBUTIONS
sequence {f
lim
exp(-Ixl2)llq+r
PROOF.
By Prop 8 of
D exp(-Ix 2)
A
supl h
Then S
m
m
llp, q
m
fm P,q _<
m
and
S-2m-2q
ll!q+r
S-2m-2q(l + m2)
In
+ m.
<
m+m
lim sup S
-nI l<+r
2m-2q(l
gqA
< lim sup
+
m
Y.
N
m
If we put f
(i + m
2)
Let
B
n
-i
S
m
+
exp(-ll 2) Ioo
R
x
h
m
N, then
m
m
2) lxl 2) 2
dx
+ m2) 2L)D
n
II
q
+
r.
Then
/Rn
ID
exp(-Ixl 2)
lim sup m
-2q
(i + m
12dx
2 q+r-
-
n
+
m.
be the strong and o the weak* topology on
the multiplication (u,f)
exp(-Ixl21dx.
>
m/m
THEOREM 2.
and
R
2-n+q+r
2dx
then there is a
IIWq-pll (x)D
II_
such that
m/oo
<
I I_<
0+
2"
exp(-]xl2)ll
q+r
llfm (x)
II
m p,q-
2
lm(X) exp(-xm)lq+rIWq+r-ll(x)D exp(-(l
Take a multi-index
lim sup
If
[ n ],
and Lemma i, there exists A > 0 such that
exp(-l 2)
-q
(x)
Define h
sup
.
x 2
I q exp(-ll
)llp,q
[4]
+
p, and r
in g (Rn) such that sup
m
m
llfm(X)
lira sup
0+
_<
If p, q 6 N, 0 < q
2
(u f)
g
OM
go
g’(Rn).
Then
is not jointly
continuous.
PROOF.
of 0 in
of
O, U
The polar P of the singleton
g’.
c
@M
{exp(-Ixl2)}
<
g is a
o-neighborhood
If the multiplication was continuous, there would be neighborhoods
and V <
g,
such that UV
P.
For some q
N, there exists a
822
J. KUCERA AND K. MCKENNON
a neighborhood G of 0 in @
q
of radius g about the origin in @
q,P
lira
For any g
fm(X)
m
of V.
(fm g)
{fm
+ oo,
Since
M
in B(e)
+
where r
exp(-ix]2))l i’-<
Since V
is bounded in
g, too; i.e., sup
is bounded in
c U, and there exists a ball B(g)
g
@
M’
such that
[n ].
UV c p, which implies
V, we have f g
Ig(fm exp(-,x12))l
in the polar V
exp(-Ixl2)llq+r
M
such that B(e) c G.
Lemma 2 implies existence of a sequence
m+
@
such that G
m
fm exp(-Ixl2)is contained
sequence {f exp(-Ixl2)}
m
Hence
g, the
llf m exp(-Ixl2)ll
q+r
<
+ oo,
which is a
contradiction.
REFERENCES
i.
SCHWARTZ, L., Thorie des distributions, Nouvelle Edition, Hermann, Paris,
(1966).
2.
SCHAEFER, H. H., Topological Vector Spathe, s., 3rd printiug, Springer-Verlag,
New York, Heidelberg, Berlin (1971).
3.
RUDIN, W., Functional Analysis, McGraw-Hill, New York (1973).
4.
KUERA,
5.
TROVES,
J., On Multipliers of Temperate Distributions, Czech. Math. J.
21(96) (1971), pp. 610-618.
F., Topological Vector Spaces
Academic Press, New York (1967).
Diistlributions and
Kernels,