Publicacions
Ma emá iques,
Vol
37
(1993),
45-55
.
POINTWISE
CONVERGENCE
OF
THE
FOURIER
TRANSFORM
ON
LOCALLY
COMPACT
ABELIAN
GROUPS
A
bs ac
MARIA
L
.
TORRES
DE
SQUIRE
*
We
ex end
o
locally
compac
abelian g oups,
Feje 's
heo em
on
poin wise
con e gen e
o
he
Fou ie
ans o m
.
We
p o e
ha
lim
<pu
*
(y)
=
(y)
almos
e e ywhe e
o
any
unc ion
in
he
space
(LP,l')(G)
(hence
in
LP(G)),
2
<_
p<
oo,
whe e
{cpp}
is
Simon's
gene aliza ion
o
locally
compac
abelian
g oups
o
he
summabili y
Feje
Ke nel
.
Using
his
esul ,
we
ex end
o locally
compac
abelian
g oups
a
heo em
o
F
.
Holland
on
he
Fou ie
ans o m
o
unbounded
measu es
o
ype
q
.
1
.
No a ion
and
P elimina y
Resul s
Th oughou ,
G
is
a
locally
compac
abelian
g oup,
wi h
dual
g oup
F,
and
Haa
measu e
m
.
By
he
s uc u e
heo em,
G
is
ep esen ed
by
Ra
x
GI,
whe e a
is
a
nonnega i e
in ege
and
Gl
is
a
g oup which
con ains
an open compac
subg oup
H
.
The
se o
basic
neighbou -
hoods
o
xeG
is
deno ed
by
JV,,(G)
.
We
w i e
C,(G),Co(G)
o
he
spaces o
unc ions
on
G
ha
a e
con inuous,
wi h
compac
suppo
and
anish
a
in ini y,
espec i ely
.
We
conside
he
amalgam
spaces
(Lp,lq)(G),(Co,l9)(G)(1
<_
p,
q
<_
oo)
as
de ined
in
[S]
.
The
Fou ie
ans o m
(in e se
Fou ie
ans o m)
o a
measu e
M,
is
deno ed
by
( ,)
.
We
le
A,(G)
be
he
se o
all
unc ions
in
C,(G)
such ha
e
L'
(F)
.
The
cha ac e is ic
unc ion
o a
subse
E
o
G
is
deno ed
by
XE
.
The
conjuga e
p' o a
numbe
p
is
such
ha
1/p
+
1/p'
=
1
.
Fo
each
U
e
N
o
(G),
A
.B
.
Simon
[Si]
de ined
a
unc ion
WU
as
he
p oduc
o
wo
unc ions
aU
and
Q
de ined
on
R'
and on
G1,
espec i ely
.
*1980 Ma hema ics
Subjec
Classi ica ion
(1985
Re ision)
.
P ima y
43A55,
43A25
.
Resea ch
suppo ed
by
NSERC
g an
7914
.
46

M
.
L
.
TORRES
DE
SQUIRE
The
unc ion
/3
U
is
con inuous,
nonnega i e,
wi h
L
l
(G)-no m
equal
o
1,
and
(1)

sup
¡,Qu(x)j
=
BU
<_
2m(U)/1
-
2m(U)

ini e
.
G
Hence
BU
--+
0
as
U
-4
0
.
The
unc ion
aU
is
de ined
as
ollows
.
Le
(-b1,
51)
x
...
x (-b
a
,
S
Q
)
x
UH
be
a
p oduc
neighbou hood
con-
ained
in
U,
whe e
6i
>
0

(i
=
1,
. . . ,
a),
and
UH
is
an
elemen
o
No
(G)
included
in
H
.
Fo
i
=
1,
. . . ,
a

we
se
Ui
=
(
-
bi,
b
i
),
N
i
=
1/bi,
and
de ine
he
unc ion
au
n
on
R
by
1
.1)
Wu
is
con inuous,
nonnega i e
and
bounded
1
.2)
Wu
is
in eg able
and
11
Wu
11
1
=
1
1
.3)
cbu e
C,
(F)
and
11O 11oo
<
1
1
.4)
Wu(x)
=
0(y)y(x)dy
by
1
.3)
1
.5)
Fo
e
>
0 and
U
e
No(G)
gi en,
we can
ind
a
V
such ha
i
V'
<V
and x ~

U,
hen
cp (x)
<
e
and
G_V
cp
,(x)dx
<
e
.
1
.6)
1
.7)
aU¡
( )
-
1
-
cos(Ni )
7 Ni
Fo
=

(
i
,
. . . ,
a
)

in
R
a
,

he
unc ion
aU
is
gi en
by
au( )

=
Há
l
au
i
( i)
.
Clea ly
aU
is
con inuous,
nonnega i e,
and
i s
L
1
(R
a)-no m
is
equal
o
1
.
Each
Wu
has he
ollowing p ope ies
.
Fo
a
p oo
see
[Si]
.
lime
0u
(^Y)
=
1
.
he
amily
{WU1U
e
No(G)}
is
an
app oxima e
iden i y
in
L
1
(G)
.
We
add
o
his
lis
he
ac
ha
each
W
u
belongs
o
he
Wiene
algeb a
(C0,
11
0)
P oposi ion
1
.1
.
Fo
each
U
in
No(U),
he
unc ion
aU
belongs
o
(Co,11)(R")
.
P oo
..
Since
au
n
(i
=
1,
. . .
,
a)
is
an
e en
unc ion
we
ha e
o
n
in
Z-
{0,
-1}
ha
sup
au
n
(
+
n)
=
sup
nu,
(
-
(1
+
n))
<_

2

1
-
.
E[o,1]

E[o,1]

Ni7
n
2
I
n
e {0,
-1}, hen
he e
exis s
a
cons an
Ci
such
ha
sup
au
n
(
+
n)
<
Ni
-
NiCi
E
[o,1l
CONVERGENCE
OF
THE
FOURIER
TRANSFORM
ON
GROUPS

47
because
he
limi
lim
_
_n
1
-
cos
Ni(n
+
)
(N
i
(n
+
))2
exis s
.
The e o e
o
all
i
=
1,
. . . ,
a
and
all
in ege
n
we
ha e
ha
whe e
sup

lceu
a
(
+n)(
<
can
E
[0,1]
c
=
max
(2/(Niz),
NiCi/7 )
1<i<a
and an
is
equal
o
l/n
2
i
n
E
Z-
{0,
-1},
and
o
1
i
nE
{0,
-1}
.
Finally,
o
i
=
1,
. . . ,
a
we
ha e
ha
Z
F om
he
de ini ion
o
he
no m 11111,
i is
easy o see ha
IIaullool
=Ilá
1liauilloo1
"
Co olla y
1
.2
.
Fo
each
U
in
N0(G),
he
unc ion
WU
belongs
o
(COI1
1
)(G)
.
P oo
.
By
(1)
we
ha e
o
all
( ,
s)
in
G
ha
hence
lau
;
11
.1
=
1
:

SUP

l
aui
( )
l
Z
E[n,n+1]
_
E
sup
lau
;
(
+
n)
Z
E[0,1]
<
C
L
:
an
<
00
.
Wu( ,
s)
=
au( )Ou(s)
<
Buau( ),
llWu1jool
<_
Bu
i

sup

Ia ( )1
=
Bullaul
i
.1
.
neZa
+nE[0,1]a
Fo he
es
o
his
pape
Wu,'au,
and
Ru
a e
as
indica ed
in
his
sec ion
.
48

M
.
L
.
TORRES
DE
SQUIRE
2
.
Main
Theo em
In
his
second
sec ion
we
wan
o
p o e
ha
(2)

limo
Wu
*
(y)
.
=
(y)

almos
e e ywhe e
o
all
in
(LP,
lw)(G)(2
<
p
<
oo)
.
Fi s ,
we
p o e
wo
lemmas
.
Lemma
2
.1
.
Le
V
and
K
be
wo
elemen s
o
NO(G)
o he o
V
=
(
-
s1,
61)
x
. . .
x
(
-
sa
Sa
)
x
VH
and
K
=
[--yl,
`y1]
x
.
. .
x
[
-
ya,
a]
x
KH,
whe e
Si
>
0,
-y2
>
0
(i
=
1,
. .
. ,
a),
VH
and
KH
a e
elemen s
o
No
(G)
con ained
in
H,
and
KH
is
compac
.
Fo
1<_
p
<
oo,
we
de ine
li
=
min(S?
P
,
~
yi)

(i
=
1,
.
.
. ,
a),
and we
le
WH
be
he
in e io
o
KH
.
Then
he se
W
=
[
-
171, ll,
]
x
. .
.
x
[-77a,
l
a
]
x
WH
belongs
o
Vo(G)
and
o
a ixed
y
=
(yo,
so)
=
(y,,
...
,
ya,
so) in
G,
he
elemen
W
y
=
y
+
W
o
N
y
(G)
has
he
ollowing
p ope ies
:
2
.1)
W
y
C
y
+KH
2
.2)
I
II
.
=
[
-
l1
+
y1,
971
+
y1]
x
.
.
.
x
[-?1
a
+
yaga
+
ya]i
hen
1
1/P
a
[ Ia nu(yo
-
x)Pdx
]
=
Q(IIi-1bi)
I
2
.3)
Ra
-
II
a
C
U
I
n
,
whe e
{I
,} is
a
coun able
amily o
compac
subse s
o
Ra,
and
1/P
I
aa(yo
-
x)Pdx
J

=
O(1i
i
1si)*
In
2
.4)
The e
exis s
a
cons an
C
such
ha
SUPN
C(I
n
)
<_
C,
whe e
C(In)
is
he
ca dinali y
o
he
se
{j e
Z
a
1
(j
+
[0,1]
a
)
n
in
7~
`N}
.
P oo
.
Se e al cons an s
will
appea
du ing
he
p oo
and
since hei
speci ic
alue
is
i ele an
o
ou
needs
we
jus
w i e C1,
C2
. . . .
C
Q
.
Pa
2
.1)
is
clea
.
Se
Ji
=
[
-
7%i
+
yi,
77i
+y¡](¡=
1,
. . . ,
a)
.
Pa
2
.2)
ollows
om
he
con inui y
o
au,
because
(3)
11/P
~ i

11/P
au
n
(yi
-
x)Pdx
J
=


aun
(x)Pdx
J

<
C
1
N
i
7l
lp
<
C28i
.
espec i ely
.
Then
and
whe e
Since

añ/p
con e ges
we
conclude
ha
Simila ly
Clea ly
supN
C
(L(n,
i))
and sup
N
C
(R(n,
i))
a e
less
han
o
equal
o
2,
hence
o
i
=
1,
. . . ,
a,
he
se
R
-
Ji
is
equal o
UIn
,
whe e
each
I
n
is
compac ,
supC
(I
n
)
<
2
and
CONVERGENCE
OF
THE
FOURIER
TRANSFORM
ON
GROUPS

49
Now,
o
each
i
=
1,
. . . ,
a,
le
L(n,
i)
and
R(n,
i)
(nEN)
be
he
in e als
[
-
n
-
1
-
?7i+yi,
-n-
li+yi]
and
[n+i7i+yi,n+1+77i+yi]
R
-
Ji
=
(
-
oo,
-
7i
+
yi)
U
(?7i
+
yi,
w)
C
U
L(i,
n)
U
U
R(i,
n),
N N
cxu
;
(yi
-
x)Pdx
<
C36pan
L(n,i)
1

1
an
_
(ni
+
n)2P
-1

(ni
+
n
+
1)2p-1
.
1/P
aun
(yi
-
x)Pdx
J
G
C46i
.
N
(ni)
,

l1/P
//

au i
(yi
-
x)Pdx
J
=
Cssi
.
N

J
,(n,i)
1/P
aun
(yi
-
x)Pdx
J
=
0(6i)
.
N
I
n
Since
R
=
(R
-J
a
)
UJ
a
,
and
J
a
is
compac ,
by
(3)
and
(4)
we
see
ha
R
=
UK
n
,
wi h each
K
n
compac ,
supC(K
n
)
<
C
6
,
and
1/P
aua
(ya
-
x)Pdx
l
=
O(6a)
.
N
K
n

50

M
.
L
.
TORRES
DESQUIRE
We
p o e
p ope ies
2
.3)
and
2
.4)
by
induc ion
on
a
.
The
case
a
=
1
ollows
om
(3)
.
Suppose
ha
2
.3)
and
2
.4)
hold
o
a
-
1
.
Tha
is,
Ra
-1
-IIa-1
C_
UI
n
,
each
I
n
a
compac
subse
o
Ra-1,
supC(In)
<
C7,
and
By
(4)
wi h
i
=
a,
we
ha e
ha
R
-J
a
<_
UIj,
each
Ij
a
compac
subse
o
R,
sup
C
(Ij)
<
2
and
Then
1/P
Ha-1
a ~
(y¡
-
xi)Pdx1

=
O(II°=i
Sz)
N
I
11/P
a (yo
-
x)Pdx
J

=
I
n
xK

,
1/P
[ ,,
auo
(ya
-
x)Pdx1
=
O(6a)-
R'
-
Ha
=
(R
a-1
x R)
-
(II(a
-
1)
x
Ja)
=
(R
a-1
-
II(a
-
1))
x
(R
U
II(a
-
1))
x(R
-
Ja)
<
U
(I
n
x
K
m
)
U(II(a
-
1)
x
I
j)
.
n,m
N
The
se s
I
n
x
K
m
,
and
II(a
-
1)
x I
j
a e
compac
subse s
o
R,
o
all
n,
m,
j
.
Hence
supC(I,
x
K

,)
<_
Cs
and
supC(II(a
-
1)
x
Ij)
<
C9
.
The e o e
2
.4)
holds
wi h
C
=
max(C8,
C9)
.
Finally,
by
(5)
and
(6)
we
ha e
ha
11/P

11/P
-

11~
II
a_1
a ~
(y2
-
x2)Pdx
l
[L

aUa
(ya
-
x)Pdx
]
=
I

K

y
=
O(IIa
6i)
We
conclude
om
(3)
and
(7)
ha
1/P
a (yo
-
x)Pdx
J
=
N
[ a-IxIj
1/P

1/P
=
IIa=i

aun
(y2
-
x)
P
dx~

a d
(ya
-
x)
P
dx

=
h

N
h
=0
(IIQ
1S2)
.
CONVERGENCE
OF
THE
FOURIER
TRANSFORM
ON
GROUPS

51
Lemma
2
.2
.
Fo
each
V
y
in
N
y
(G)

(yEG)
lim
u_o
o
all
in
(LP,
l')
(G)
(1
<
p
<
oo)
.
G
_
U
cpu(y
-
x) (x)dx
=
0
P oo
.
Le
y
=
(y,,
.
.
. ,
ya,,
so)
=
(yo,
so)
be an
elemen
o
Ra
x
Gl
.
We
choose
wo
elemen s
V
and
K
o
No
(G)
wi h
he
same
o m
as in
Lemma
2
.2,
such
ha
y
+
K
C
V
y
and
V
C
U
.
Following
he
no a ion
o
Lemma
2
.2,
we
se
97z
=
min(S?
P
,
-y2)
(i
=
1,
. . .
,
a),
and
WH
he
in e io
o
KH
.
Then
he
se
W
=
[-77
1
,
77
1
]
x
.
.
.
x
[
-
la, la]
x
WH
sa is ies
he p ope ies
lis ed
in
Lemma
2
.2
.
Hence
by
p ope y
2
.1)
i is
enough
o
p o e
ha
lim

Wu(y
-
x) (x)dx
=
0
.
U-0
c-w
Since
G
-
W
y
=
(R
a
-
Il
a
)
x
G
l
UH
a
x
(Gl
-
(so
+
WH)),
we
ha e
by
he
de ini ion
o
he
unc ion
cpu,
ha
wu(y
-
x)
=
au(yo
-
)
PU
(s0
-
s)
=
0
i
so
-
s
,
and x
=
( ,
s) in
G
.
Hence
Wu(y
-
x) (x)dx
=
J

Wu(y
-
x) (x)dx
G-W
y
(Ra-IIa)
x
(so
+H)
+
ax(so+(H-W,»
WU(y
-
x) (x)dx
.
II
Le
{I,} be
he coun able
amily
o se s
gi en
by
p ope y
2
.3)
.
Fo
each
I
n
,
we
ha e
by
he
Hólde
inequali y
and
(1)
I,x(so+H)
1
Wu
(y
-
x)
(x)
(
dx
<-
Il
.
XI
.
x
(so+x)
I
pBU
<-
<

au(yo
-
x)P
dx
5 2

M
.
L
.
TORRES
DE
SQUIRE
By
p ope y
2
.4)
SUP
N
¡S(I"
x
(so
+
H))1
<
C,
whe e
C
is
a
cons an ,
and
1S(I,
x
(so
+
H))I
is
he
numbe
o
K
a
's
(as
de ined
in
[S])
such
ha
I,
n
x
(so
+
H)
1
K
a
:,A
0
.
This
implies
ha
o
all
n
s
N
II XInx(s,,+H)IIP
<-
IS(In
x
(so+H))I
II lip
.
:5
CIi lipoo
.
Thus,
we
conclude
om
2
.2)
ha

~ou
(y
-
x)
1
(x)
1
dx
<
(g,a-II
a
)xGi
11/P
,
<_
CI
i
I
¡
.Bu

~~
au(yo
-
x)P'dx
J
-
I
n
Applying
Hdlde 's
inequali y
we
ge
Lx(80+W-WH»
Wu
(y
-
x)
I
(x)
I
dx
<
1P
l
<
Bu
1l
1
¡P
.I
S(Ha
x
(so
+(H-
WH))
I
[ ila
au(yo
-
x)P
dx
]
IIa
No e
ha
lla
x
(s
o
+(H
-WH))
is
compac
(H
is
compac
and
H-WH
is
closed),
and
because
Bu
->
0

as
U
-> 0
Now,
since lla
->
y

as
U
-
0

and
so
+
(H-
WH)
<_
so
+
H

is
independen
o
U,
we
ha e
ha
1
S(IIa
x
(so
+(H-
WH)
->
0 as
U
-~
0
.
The e o e
by
p ope y
2
.2)
(10)
J

cp
u
(y-x)I (x)Idx,0
as
U,O
.
IIa
x(so+(H-WH
))
The
esul
ollows
om
(8),
(9),
and
(10)
.
Theo em
2
.3
.
Fo
all
in
(LP,
l')(G),
2
<p<
oo,
almos
e e ywhe e
.
ló
c
Wu(y
-
x) (x)dx
=
(y)
_
O(IIa
1biBU)
"
P oo
..
Le
V
y
be
in
N
y
compac
.
We
ha e
o
show,
by
Lemma
2
.2,
ha
ló
~y
~ou
(y
-
x)
(x)dx
Fu he
CONVERGENCE
OF
THE
FOURIER
TRANSFORM
ON
GROUPS

53
con e ges
o
(x)
almos
e e ywhe e
.
Since he
unc ion
belongs
o
(Lp,
l°°)
C_
(L
2
,
l
°°
),
he
unc ion
g
=
XV
y
belongs
o
L
2
(G),
and
by
Co olla y
1
.2,
1
.1),
and
1
.4)
each
WU
also
belongs
o
L
2
(G)
.
Hence
by
he Pa se al
iden i y,
we
ha e
ha
Wu(y
-
x)
(x)dx
=
Wu(y
-
x)g(x)dx
IV"

G
almos
anywhe e
.
=
~bu(x)9(-x)[y,x]dx
By
he
Lebesque
Domina ed
Con e gen e
heo em
(see
p ope ies
1 .3
and
1
.6)
we
ha e
ha
(x)9(
-
x)
[y,
¡]di
=
9(-x)[y,
x]dx
=
g(y)
3
.
Fou ie
Uans o m
o
Unbounded
Measu es
The
space
M
9
(G)
(1
<_
p
<
oo)
o
unbounded
measu es
o
ype
q
[S],
consis s
o
Radon
measu es
M
wi h
ini e
no m
(lp
,
il
q
gi en
by
[j
:
j
¡p¡(Kc i)
q
]
1lq
.
I
G
=
R,
hen
he amily
{K
a
}
can
be
aken
as
{[n,
n
+
1]
InEZ}
.
In
his
sec ion
we
gene alize
o
locally
compac
abelian g oups,
he
ollowing
heo em
due
o
F
.
Holland
[H]
.
Theo em
3
.1
.
Le
1
<
q
<
2
anda
e
M
q
(R)
.
Then
as
N
-
co
1
N
e
-¡x
d[ ( )
727
-
N
con e ges
in he
no m
o
(L
q
~,
l°°) o
a
unc ion
i
and
h(x)i(x)dx
=J
h(x)dp(x)

(he(Lq,
l
l
)(R))
.
27
(x)
=
(C
.1)
1
e-ix dp( )