Internat. J. Math. & Math. Sci.
Vol. i0, No. 3 (1987) 461-472
461
RESONANCE CLASSES OF MEASURES
MARIA TORRES DE SQUIRE
Department of Mathematics and Statistics
University of Regina
Reglna, Saskatchewan
S4S 0A2 Canada
(Received September 30, 1986)
ABSTRACT.
We extend F. Holland’s definition of the space of resonant classes of func-
tions, on the real line, to the space
R(#pq)
sures, on locally compact abelian groups.
(I
) of resonant classes of mea-
p, q
We characterize this space in terms of trans-
formable measures and establish a realatlonship between
definite functions for amalgam spaces.
R(pq)
and the set of positive
As a consequence we answer the conjecture posed
by L. Argabright and J. Gil de Lamadrld in their work on Fourier analysis of unbounded
measures.
KEY WORDS AND PHRASES.
Amalgam spaces, Fourier transform of unbounded measures, positive
definite measures, positive definite quasimeasures, Fourier multipliers.
__
1980 AMS SUBJECT CLASSIFICATION CODE.
INTRODUCT ION.
F. Holland [I] defined the space
43A35, 43A15.
R(#q)
(I
q
) of resonance classes of func-
#ooq,
and proved that
tions, on the real llne, relative to the space of test functions
(2
) iff it is the Fourier transform of an unq
a function belongs to
R(q)
6]. He also pointed out that the set P(C c) of positive
definte functions in Cooper’s sense [2] is included in R(I) [I, I], and proved that
bounded measure
[I,
every function in
Theorem
R(ool
as the functions in
has the same representation
in terms of unbounded measures
8], [3,
Theorems 4.1 and 4.2] (in fact,
[I,
P(C c)
Theorems 7 and
as we will prove here, these representations hold for a larger class of functions and
they are equivalent). These results of Holland together with Bochner’s theorem on poa function is the Fourier transform of a bounded measure
sltlve definite functions [4]
iff it is a linear combination of positive definite functions
that any function in
R(ooq)
lead one to speculate
is a linear combination of positive definite functions.
In the present paper we respond to this conjecture in a more general setting. We define
the space
(I
) of resonance classes of measures (on locally compact
p, q
as a particular case; we
which includes R(#=o
abellan groups) relative to
characterize this space in terms of transformable measures [5], and prove that for
(2
) is a linear combination of positve deany measure in
q
p
s) [6], and for
q < 2, any measure
finite functions for some amagam space (L
(#pq)
,
pq,
(pq)
q)
r,
M.T. DE SQUIRE
462
(pq)
in
can be approximated by linear combinations of positive definite functions
(L r
for some amalgam
s)
From these results we conclude that
P(C c
is dense in
R(I)
and
O(C c )>, the
space generated by the set of positive definite measures as defined in [5, 4], is dense
in the space of transformable measures. This answers the conjecture posed in [5].
Throughout the whole paper
(x) (x
and dual group
m
measure
will be a locally compact abelian group with Haar
G
F. For an element
in
A
G). Given two sets A and B we denote by
E
F
we write
B
the set
[x,]
{x
instead of
ylx
(x)
E
B}.
A, y
f(-x). The
to denote its involution, i.e.
For a function f on G we use
space of contlnuos functions which vanish at infinity, with compact support, will be
with the inductive limit topology, as
denoted by Co, C c, respectively, We endow C
[5] By
in
c
a measure (on G) we will mean an element of the continuous dual of
C (G).
c
be the space of measures on G.
We let
A funetlon
belongs to
q and f
G, belongs to L
set of
and belongs to
f
Loc (I =<
q
=< )q
belongs to
L
if
f
restricted to any compact sub-
(I < q < )
if
f has compact support
L q.
(L p
(Co, q) (I < p, q < ) and the space of measures
M (I < s < ) will be as defined in [7]. We will make constant use of the following
inclusions and inequalities proven in [7].
The amalgam spaces
(L
(L
(L
p
pl
p,
= (L p,
= (L p2
ql)
q)
q)
[[f[[Pq2
q),
q2)
ql
<
q2
(1.1)
q)
P
>
P2
(1.2)
c L p fl L q
(1.3)
q
p
<
[[f[[ Pql
q
<
<
Ilfll
p
>p
(1.4)
q2
(-
We will assume all the results of duality and convolution product for these spaces,
the Holder and Young’s inequalities, and the Hausdorff-Young theorem for amalgams as
given in
[8, I, 2].
The Fourier transform (inverse of the Fourier transform) of a measure
(on
LI(G)
We let
c
{$U
{ca}
G
the approximate identity
consisting of continuous functions with a fixed support and po-
sitive Fourier transform in
A
). We will denote by
F) will be denoted by
of the algebra
on
LI(F).
U a compact neighbourhood of O}
the space of functions in
C
c
be the family of functions
whose Fourier transform belongs to
L
U
in
with the
following properties
supp
U=U
(1.6)
U SU*U where U L2c
U > 0 and U > 0
lim
uniformly on any
SU
g
{U
is an
(1.8)
compact subset of
approximate identity of
L
g
(1.10)
RESONANCE CLASSES OF MEASURES
B
The duality between a Banach space
by
B’,
F
<f,F>
linear space
there exists a measure
f
C
I
F such that
on
We denote by /_
c
{f*l
generated by the set
B’
and its Banach dual
B. As in [5] we call a measure
f
2(G),
C
463
on G transformable, if the
Cc(G)}, is
f
will be denoted
included in
LI(),
f*(x)d(x)= |I12(-)d()
and
for all
the space of transformable measures.
POSITIVE DEFINITE MEASURES.
2.
[6],
We follow the definition of positive definite measures given by Dupuis in
but
S0(G) which is equivalent to the space of translation bounded
[9]. The advantage is that for
e S0(G)’ its Fourier transform $ bewe have, as proven in [8, 2], that
S0(F)’ [I0] and for f e L
using the Segal algebra
quasimeasures
longs to
$
(*f)^
f
()v
and
(2.1)
(2.2)
S0(G)
We assume all definitions and results about the algebra
given in
[8, 2].
From these it is not difficult to see that the Fourier transform of a transformable
.
(considered as an element of
measure
associated to
As in
S0(G)
[I0],
is positive,
o
positive. In this sense a function
g
o
an element
o(f)
implies
S0(G)’ [10]) corresponds
S0(G)’
in
to the measure
O, if f positive in
S0(G)’ is positive
in
0 almost everywhere. Indeed, let U be the measure gdm and suppose g
g(x)
S0(G)’ positive. For # e C c positive, the function *e d is a positive element
of S0(G) and converges to
in C [5]. So we have that
c
lim <*e,>
lim <*e,g>
O. Hence U is a positive measure and therefore
()
0 almost everywhere [II, Chp. III].
g(x)
iff
in
Let
DEFINITION 2.1.
be a subset of functions of
E
E
itive definite measure for
(DI)
E c L ()
(D2)
<h,>
0
We write (E)
by <(E)>
E
O.
such that
to denote the set of positive definite measures for
to denote the set of measurable functions in
we denote by
is a pos-
if
h e E
for all
S0(G)’. A measure
+ the
E
set of functions in
the linear space
(E). For
E
a set
E, and
P(E)
Definition 2.1
as in
whose Fourier transform is positive, and
generated by (E).
Clearly, Definition 2.1 is equivalent to Dupuis’ definition of positive definite
measures [6]. By [I0, Theorem BI] and [8, (1.9)] the set
hence any L p space [8, (1.4)].
E
can be any amalgam space,
(G), of measures U such
the
with
space of transformable meaconnection
in
C
e c,
Argabright and Gil de Lamadrid have studied the set
<*$,> _>
that
0
for all
sures. We use their results in [5] to prove that
PROPOSITION 2.2.
in
C
A measure
U
(G)
is equal to
belongs to (C C)
iff
<*,U>
(Cc).
0
for all
c
PROOF. The inclusion
luded in
T
[5,
Theorem
sitive definite function
therefore
(C c
2.2]
c
(G)
is clear. Take
we have that a function
[5, Theorem 4.1],
satisfies condition
(D2).
so
<,>
in
0
by
(G). Since C c is inc+ is a continuous po-
Cc
[5,
Corollary 4.2] and
464
M.T. DE SQUIRE
REMARK 2.3. It is clear that if
P(EI)
c
P(E2)
[5,
so by
Theorem
{[
set
<,o>
such that
M.}
[6,
4.1]
(El, E2 as
C
if
Definition
in
E
is a subset of
C
> 0
in A
for all
C
+,
and characterized it as the
II].
Proposition
As in [6, Proposition II] we use the following le to prove
LE 2.4.
lira f
that
Cc.
{@U = C c,
Proposition
1.8]).
PROOF. If
eorem
2.5.
Es)
(Co,
{f
A
in
+
such
[8,
1.6] that the
Theorem
{f
ea
c)]
lim
P(L p
Eq)
Theorem III
.
p’, Eq’)
Then
I,
f@u*e
is equal to
[7,
D> > O. Therefore
g
P(L
p,
Eq).
(L
is in-
(see also [8
in A
f
as stated in the previous lena, so by
n
lm <f
{fu*ea}
the net
is an element of
f
and
is in-
(fu) (*U)
(fu*e)
belongs to L
{fu*e}
net
p
q
(L p, q)+,
then
Theorem 3.2] we have
The other inclusion follows
[6, Proposition IV].
from
S 2.6.
such that
in
3.
(L
is in
<f,>
then there exists a net
and since
p, q <
Let
THEOREM 2.5.
there is a net
+,
Thus by (2.1) its Fourier transform
Finally as in [6
cluded in A
A
g
we have by
[8, (2.5), (2 6],
is positive
that
f
Eq),
in A.
f
n
(L p,
be any of the amalgam spaces
< s < ). If
PROOF. Since
cluded in
A
Let
,
(I < p, q <
2.1), then
(E)
then
of positive definite quasimeasures to be the set of all
Dupuis defined the set
quasimeasures
El c E2
From (2.2) and
p
Eq
eorem
(L
2.5 we have that if
0
p, q < ), then
(I
is a positive measure
belongs to
P(L p
Eq)
RESONCE CLASSES OF ASUS.
Bertrandias and Dupuis [2] defined the space
Pq
(I
)
p, q
of test func-
tions on locally compact abelian groups based on Holland’s definition of the space
(I
)
q
Let
DEFINITION 3.1.
tions
C (G)
in
The space
i’
p, q
,
p
2
in
g
Pq
in
T
Pq
(G)
r
(R)
Pq
=L
()
Let
C2(G)
and
pq
is included in
C
c
for
(G) (I ! p, q
) is continuous iff there is a
p
q )(F) if p < such that
in (L
,
$(-) d()
Holland’s definition of the space of resonance classes
[I, 5].
if
,
,
pq
if p
[2, 2 c)].
DEFINITION 3.3.
4.2.
for
on
M ,(F)
q
next definition extends
of functions
eorem
Hence the space
! P, q, r
rq
pq
by the Hausdorff-Young theorem.
q
T()
e
[13] [9]
A
@
ii) A linear functional
for
consists of all func-
endowed with the norm
used by Bertrandias and Dupuis for their definition of the Fourier
i) As sets
unique measure
(G)
(C0, Eq)(F)
The space
belongs to
[14]. We will use this
RE 3.2.
!
such that
transfo is equal (as a set) to
I(G)
.
for the real line.
p, q
.
A measure
on
G
is resonant relative to
RESONANCE CLASSES OF MEASURES
(R2)
<,>
The map
is continuous on
465
i.e. there exists a constant
Pq
pq
(
We denote by
and by
R(pq)
(pq).
the space of functions in
(#
included in
the space of resonance classes of measures relative to
Pq
Pq
(pq)
By (I.I) it is clear that
is
< s < q <
if
H. Feichtinger has given a more general definition of resonance classes of func-
B’
tions relative to the space
S0(G)
is a Banach space of functions containing
as a dense subspace (private communication).
THEOREM 3 4
i) A function
A measure
ii)
e (L
iii) For 2 < q _--<
c (L q
if
(pq)
?(pq)
=< p, q =< =o)
(1
e
and
U
Mq,(F)
(pq)
c
(L
q,
P)
if 2
_<--
p
_<_
=0, and
< p < 2
[I,
PROOF. The proof of i) is similar to the real case
rem
loc"
iff
if p < oo.
we have that
2)
f e L
satisfies (RI) iff
f
belongs to
p’, q’)(F)
o,
if p
B
where
Theorem I] using
[7,
Theo-
3.1].
and set the map T()
<,> on Pq If T is
in
continuous then by Remark 3.2 there exists a measure 9 as stated in the theorem
such that I (-)dg()
/ #(x)d(x). Since C2(G c #=of(G) we conclude that
and
e M ,(r) then for
and
e
e nl(r) [8
Conversely if
q
Theorem 1.4]. Hence by [5 Corollary 3ol] and Youngs inequality we have that
( q ). The proof for
Therefore I
$(-)d()l _<
(x)d(x)l
(#pq)
To prove ii) take
.
,
#T
II
I
eT
IIllq, ll$11q
p finite is the same.
(L p
Part iii) follows from the Hausdorff-Young theorem and the fact that the spaces
q
and M
(2 < q < ) are included in
[15, Remark 6 25]
e
2
=< =< ’
q
LIIoc nT
R(I)
T
[5,
conclude from Theorem 3.4
R(pq) (Opq)
Theorem 2.5] that
=T.
(*i
and
for
p
The following corollary is easily deduced from the previous theorem and the
Hausdorff-Young theorem.
Let
COROLLARY 3.5.
!
f
!
p
,
e M ,(F)
there exists a unique
q
v
2
! q !
if p
,
.
A
function,
p
q
(L
f
R(pq)
belongs to
)(F) if p <
,
iff
such that
=.
Since
rem
q Cc
I] (with p
=>
if q
2, Corollary 3.5 includes the results of Eberlein [16, Theo) and Stewart [7, Theorem 4.4] (with p
o) as special cases.
q
We observe that the measure
in Corollary 3.5 is precisely the Fourier trans-
form of f.
We now establish the relationship between
R(#pq)
(2 < q < )
and the set of posi-
tive definite functions.
PROPOSITION 3.6.
if 2 < p <
oo,
PROOF If
.
Let
q’
of
<P(L
f e
R(=oq ),
p’
e (L
2
_<--q _--<
oo. The space
2)>
if
then
f
0l
92 + i(O
q
if p <
=o, and
(j
P) if 2 p =o,
Young theorem
e (L
.,4). So by Remark 2.6 we conclude that 9.
q’ 2) if < p < 2
2 < p < oo, to P(L
p
’3
q,
R(pq)
<P(L
is a subspace of
p’, q’)>
< p < 2
=< =<
4)
and
(j
9j
9) where
j
> O,
?
[II, Chp. III] By
(L
4)
q, 2)
if
belongs to
j
E
Mq,
if
the Hausdorff-
=< p =<,2
(J,
q
P(L
P
I,.
if
M.T. DE SQUIRE
466
<P(L
COROLLARY 3.7.
2.51
PROOF. By Theorem
p(L
2, I),
[6,
then by
p(L
Theorem II]
R(oo2).
f e
lude that
1)> R(Ooo2 and <p(L2)> R(22).
2, I) c (L 2, 0o) and P(L 2) c L 2,
2
g
so by Remark 2.3 and Theorem 3.4 we
M2,
p(L2),
f e
Similarly if
f is in
hence if
then by Theorem 3,4,
E
conc-_
R(22)
and
the equalities follow from Proposition 3.6.
cases
For the remaining
,
2, 2 < p <
and q
(L
transforms on
(1 _< s < 2)
r,
(L
f
(L q
p’
s)
Eq;)
f e (L
EP
[21,
on
[17, Corollary 6.3]. Indeed
q,
P)
(L
such that
2.41.
Remark
belongs to
f
otherwise
<_ p < 2, 2 _< q _<
<_ p _< =o, 2 < q _< oo;
=< s =< 2),
< r < 2,
(1
are not onto
then there exists
2
that is, for
the inclusions in Proposition 3.6 are proper because the Fourier
q,
+
g
and clearly to
P(L q
Eq ).
(1
if
2
=< s =< 2),
_<
P
). But
R(pq).
2, s)
hence
l<o,f>l
(L
g
(L
2 < q <
and
p’, q’),
<,,g>
defined by
g
Therefore
and on
_<
p
h e (L
for ali
So the function
P)
(L p
would be in
f
Ms
p
on
Eq ),
The remaining cases are
s imi lar
< q < 2.
(
FOR
pq
We have seen that any measure in
4. THE SPACE
(pq)
(2 < q < oo)
is a linear combination of
< q < 2, any measure in
positive definite functions; we want to prove now that for
( Pq
,
is approximated by linear combinations of positive definite functions.
We endow the spaces
and
Llloc
L
,,
(pq),
R(pq)
and
.,lloc,L1c)
and O(L
as subspaces of
and
respectively.
loc
if p
oI,C c),
with the weak*-topology
respectively [18, Chp. IV]. We consider
-_=<,<
PROPOSITION 4.1. Let
gq )(F)
g (L p
.
oo,
g
then
( Pq
fl
and
==< q
0o,
Conversely, if
PROOF. We prove the proposition for
For
<>
g
lim
the
Cml’
v
< ,*U>
{*$U
net
llm
p
exists a measure
EvMq,(F)
Ifv there measure
and
lim
in
V
(pq),
E
llm
then
*U
0o, the proof for p finite is the same.
in
<*U,> <,> <,>.
,
*U
is a
converges to
is transformable and
that
< 2.
if p < oo, such that
Cc
[5], so
Since
so by Theorem 3.4,
we have that
C2(G) = oo
(q).
we conclude
The converse is
clear.
REMARK 4.2.
If
f
g
R(pq)
f**u(X)
f
because
(I < p < =o,
I *u(X-t)f(t)dt I U()[x’]
IG
(x)f(x)dx
This implies that
f(x)
d()
G
is transformable and
have that
< q < 2), then
lira
LI()
U
lira
I
[5, Corollary 3.1]. Hence
Cc
we
where the limit exists on L
on
IG(X)IF *U()[x’] d()
U()[x,]
d()
e
for
dx"
any compact subset of G (c.f. [I, Theorem 9] and [7, Theorem 4.2]).
THEOREM 4.3.
Let
approximated by elements in
it(Opq),
hence in
q < 2. Every element in
<P(L
If 2 < p < =o, in <p(L
I, P’)>
PROOF. By Proposition 4.1 we only have to prove that for
{*U
belongs to
<P(L
P’)>
By Theorem 3.4, the measure
R(pq),
I, 2)>
IIe (1 ),
if
can be
< p < 2.
the net
is a linear combination of positive measures
llj
RESONANCE CLASSES OF MEASURES
4)
(j
M
in
So by [8
q
(2.5)]
for
[[j][q,[]U[]q[][[
oo
$o
we have that
4)
(j
[8,
L
E
467
2.5]
PJ*U
(j
I..4). Now, if f E LI(G) +, then by [8, P5] and the definition of the Fourier
transform we have that <f,Oj*@U >
<f*@u,Oj> <U,Pj> > O, therefore OJ*U E P(LI)
(j
4), and we conclude that *. belongs to <p(LI)>. If
(nn), then
u
p
’
Eq
g
each
(L
4) belongs to (L
and therefore jU
[19, 7 h)],
j (j
==, EP) if
hence by (2.1) and the Hausdorff-Young theorem we have that
in
is
(L
j*U
+ we have that
2) if < p < 2 So for
2 < p < =o, in (L
A
<, j*U> <U_,Pj> > O, this implies by Theorem 2.5 that j*U is in P(L I, EP’) if
< p _<__ 2.
2 < p < o, in P(L I, E2) if
Since
is dense in
Proposition
we conclude that
E
L
’
REMARK 4.4. By Remark 2.3, Theorem 4.3 and Proposition 2.2 we conclude that the
)> is dense in/ (c.f. [5, 4]) and <P(Cc)> is dense in L
space
T
PROPOSITION 4.5. i) Let
are dense in
q ___< 2. The spaces <p(LI)> and
floe T"
9(Cc
R(q)
,q
< q < 2. The spaces
ii) Let
<P(L
2)>
R(O2q)
and
<p(L
I, 2) R(O2q).
2) = ’T
P(L
to
Let
f e P (L
Mq,,
hence
First of all we recaii (Remark 2.3) that P(L
q,
R(oo),
Llloc"
f
since
If
p(L
f
I, 2),
l’) and by the Hausforff-Young theorem,
We
finally
(pq),
in
as
p < oo, the space
2
point out that for
because
<P(L
[21
[20, Theorem 5.5.1] using
in
2, 1 q’),
I, P’)>
(
is not a
P’))
(L
E
Also
measure
if
in
(L p’
.
.
,
and [15,
1 p)
(L
defined by
I }. So for each integer
n
q
(L p
Indeed, let
such that
n
we define the
to be the product of n times the characteristic function of the set J
f
function
[n,n+l)Ix
belongs
hence
in
P’) and g Is not a measure so
I, 2)> is not included in (0 Pq ).
inner measure there exists I = [n,n+l)
n
a,
f
f
belongs to P(L
< p < q < 2, then <p(L
p’/q’ + p’. Since m is a
{x e
m(Jn)
(I/n)
where J
n
If
g
hence the function
f
is not included
Theorem 5]
Theorems 5.6 and 15.9], we can prove that there exists a function
such that
and
then again by Theo-
(L
R(2q).
f g
1) ’#T
and
By Theorem 2.5 and [6, Theorem II],its Fourier transform
f
rem 2.5, f e (L
(2q)
are dense in
<p(L1)> = R(q)
PROOF. In view of Theorem 4.3, we have to prove that
f
then
q’).
lfllp,q,-v
fn
Hausdorff-Young theorem
fni I0o2 ;< 7.
f
Since each
Illnllo2
<
7.
n
(np (l/n)
converges
cllfnl121
I/n, therefore f
I) n
2) to
in (L
7. c n2(1/n) Of’=
belongs to
(L
2
c (L
2)
f’, because
a function
c(1/n)
0-2
<
and
is not in
<,>
<$,g>
I, 2)
q’)
p’,
(L
l<$,f>l
I
because for
<
I<,>I
E
S0(R)
thls shows that
> 3.
since
By the Lebesgue Convergence theorem and the fact that So (R)
v
v
f’> hence f’
<,fn>
<*,f n >
S0(R) we have <*,f> <$ f>
defined by <,g>
be the function on (L
Clearly g is in
in
is not
so by the
f. Let
e(e
g
I, 2)
we have that
c (L
p’
q’)
iff
f E
(LP’,q’).
M.T. DE SQUIRE
468
Therefore
(
g
Pq
).
G
can be extended to
f
The construction of the function
using the partition of
disjoint relatively compact subsets as in [7, 3]. Probably the same is true for
< q < p < 2, but we were unable to decide the matter.
5.
REPRESENTATION THEOREMS.
R(
We will prove in this section (Theorem 5.4) that the representation theorems for
and P(C
and 8] and [3 Theorems 4.1 and 4.2] respectively
in [I Theorems
C
L
hold for the space
=< q
(I
R(pq)
loc
REMARK 5.1.
i) If
g
x e G
all
Lq+/-oc R(pq)
f e
Let
LI(G),
*f
then
and they are equlvalent. We
Theorem 3.3].
[5,
first give a remark easily deduce from
=< p _< oo),
< 2,
=< ^P’
(I
exists and
___<
q
e
=o).A
LI().
Therefore for locally almost
we have that
I
f*(x)
I
f(y)(x- y)dy
G
r
()[x,] d().
ii) If the integral on the left is a continuous function of
[
/
e L q’
for all
=<
Let
THEOREM 5.2.
=< .
[I,
p, q
I
q)
c
PROOF. It is clear that the convolution
If
$
(L
is in either
,
support of
x(y)
(x
and
p,
q)
f*
we have to prove that
or
(C0,
e L
if p
q’
L
exists for
then
q)
(Co
I().
q
f e L,
oc
So by our previous remark
and
c
U
of 0. Let
E
be the
x
x
< q < 0o, then the map
where
G. So given e > 0 there exists a nelghbourhood
V
y), is continuous on
is the characteristic function of
Therefore
*f
q),
II, x
of 0 such that for all x e V we have that
[f*(x)
$
if p <
is continuous on a nelghbourhood
U. If
s g
then
(-) d()
r
e (L p
L’qloc R(pq),
f
If
f(x)(x)dx
$
3] as a particular case.
Theorem
G
such that
in a nelghbourhood
[r ()d().
f(y)(-y)dy
G
The next theorem includes
x
0. Hence under this hypothesis
of 0, then the formula in i) is valid for x
f*(s)[ <_
f*
]U-E
If(y)
*yl lq,
fXu_ E
E. So for x e U 8 V
U
lx(y)
s(y)
dy <
XU_E
q
we have that
IIfXu_EIIqlIx yllq,
< e.
is continuous at s.
I, then the map x
is continuous on G, and as before,
E x
there exists a neighbourhood of zero V such that for x in U N V we have that
If q
f*(x)
f*(s)
<
fXu
II lll(fU_E) x (fU_E)sl 11
<
.
We need now to introduce Simon’s generalization of
compact abelian groups
[22].
This ends the proof.
Castro
summabillty on locally
This consists of a family of functions
compact neighbourhood of O) in
(C0,
I)
{U
(U being a
with the following properties
u-->’ II%llx__<
{U
U
is an approximate identity for
Cc
and
lira
U
U()
for all
(5.2)
L
F.
The following representation theorem is an extension of
(5.3)
[I, Theorem 7] (c.f. [3,
469
RESONANCE CLASSES OF MEASURES
Theorem 4.1]).
(x) f(x)
(C.I)
=< q
Let
THEOREM 5.3.
dx
for all
G
(L p
h
q)(F)
Furthermore if
q, q
for alI
if p <
r, P’)
in (L
I)
and
f e
If
Lq
R(#pq),
1oc
then
d?()
h(-)
dx
F
q)(F)
h e (Co
2q/(2q
r
.
u(x)h(x)f(x)
lira
G
=< p =<
< 2,
if p
2 < p < oo, then
The double integrai exists not necessarily as a Lebesgue in-
tegral but as the sum of the convergent series
I Iv8
a
Va,
where
V
Y)(x)*(Y) dx dY
(L p,
p < oo. If h e
< p < 2, or (L
q
in [7, 3].
definedv
then h belongs to either
L,
are finite union of the sets
8
PROOF. Suppose
if
f(x
q)(F),
V
i2
{#U}
c C (G), we have that
(G) if p > 2. Since
C
that
conclude
we
5.2
Theorem
So
by
(hU)
VV
h*u
and therefore
G
u(x)h(x)f(x)
e (L
Since
as
p’, q’)
F
and
h*u
lim
in (L
h
f h() d()
gral on the right converges to
,P’)(G)
VV
Lq
hu
C
d(.).
h**u(-)
dx-
(L q
p,
q) [8,
Proposition 1.8] the inte-
and this proves the first part of the
theorem
Let
q
qb,
in L
r
hence I/2 < I/r < 3/4
(Co, q)
(*)^
II
Set
IB(b,)l
2q/(2q
r
Since
c
and therefore
r’
because
and
< q < 2, we have that l/r
r
e (Co,
then
e L
[12, 7]. So by
2q
r’
C’
(5.4)
to be equal to the left side of (5.4). So by
B(,@)
111 Ip,q, 1151 [pq
<
ll2q,
and
Theorem 5.2
/ ()()d().
f(x- y)(x)*(y)dx dy
<
1)
< r < 2. If
I111 p,q, !1511 pr’ I111 pr’
Holder’s inequality
therefore
=< C ]]@[[rp,[[][rp, if _<_ p < 2 and [B(,)] __< C [][]r2[[][r2 if p > 2,
where C is a constant depending on f, p and q.
(L r, P’) then Jig[ [rp’
If g
[Iga[ lp’)r lip’ where ga gXLo" So for #,
in (e r, zP’) we have that [B(,)[ < C l[a[ [rp, [@8[ [rp’" So I
B(e,@8) is absolutely convergent and the left side of (5.4) exists as stated in the theorem.
Finally, since I ga converges in the norm of (L r, P’) to g we have that
]B(,)]
[
.
.
B(a,8)
f $()() d().
THEOREM 5.4.
If
=<
The proof for 2 < p <
q < 2,
=<
p <
f e
and
Lloc,q
is similar.
then the following are equi-
valent
i) f e
R(pq)
,
ii) There exists a unique
r
in (L
that for all
U
P’)(G)
in
m, in (L p
2q/(2q- I)
M ,(F) if p
q
where
r
q
the double integral exists as in Theorem 5.3.
iii) There exists a unique measure
as in part ii) such that
)(F) if p < o, such
M.T. DE SQUIRE
470
f(x)
li,
F
qu(.)[x,]
du(.)
where the limit exists as in Remark 4.2.
PROOF. By Theorem 5.3 part i) implies part ii) and by Propsttton 4.1 part tit) tpiles part ).
U 8U*SU
Suppose that ii) holds. Since
2
8U e L c
and
for all t in G we have
The left side is equal to
the right side is
f(x)8U*U(t- x)dx
equal to I U()[t,]
of part iii). Hence we conclude that
6.
f(X),u(tdu(),
u*f
and
f**u(t),
x)dx
converges to f in the sense
ii) implies iii).
RTHER RESETS.
In this last section we want to give a characterization of the set of Fourier mul-
and
(I
tipliers from the space
) to L
q
Dupuis’ characterization theorem [6, Theorem IIl].
#q
M I. This will allow us to extend
M() ((q)) the space
M(#) ((#q)) is the space of
is in L1 (M I) for all
q(G)
Folling the notation in [23] we denote by
multipliers from
Since
for
!
L
to
tions (measures)
f
q Cc
2.
q
on
F
that is
=< q =< ,
for 2
EOREM 6.1. Let
(MI).
{(q)
for f e C
l<f,$
C
(n
and suppose that
q’)
(L
S(#,)
2. Then
q
l<f,$ n
c
<
$’1
lira
Ilfllll$n
+
[<f’$n
$
IIq
and
I, q’) c M(q)
n
in
we have that
h’[
and
#
C
clude by the Closed Graph theorem that the map
longs to
e
Mq’ (F) and therefore
From Theorem 6.1 we see that
F*q
We write
O(Fq) Mq,
g
M
P.
O
Take
$*e_
g
(L
g
A
q
and
(F_).
q
E-’
and
By [6. Proposition
c
M(Cc
(L
n
h
lira
M(q),__
f
2)
in
MI,
so
is dense in C, we con-
from
I. q’)(F).
2. Then
q
P(C0, q)
P(Fq)
PROOF.
If
).
#q
to
then the measure
as proven in [23]
and
M
is con-
If[din
be-
(C
M2
to denote the set
EOREH 6.2. Let
Hence
Mq,(F).
f e (L
(q)__ Mq,
M_,c(#
and
where C is a constant depending on the support of f. Since C
tinuous. Hence by Remark 3.2,
all func-
we consider the characterization of these spaces
PROOF. By the Young’s inequality
Let
f
such that
of Fourier
N
$
and
(’e_) v
Since
(L
= P(Fq).
that of Theorem 2.5.
g
(L
(mq) (C0, q).
IV] and eorem 6.1, P(Fmq)
(Fq) +.
> 0
Eq
P(Fq)
(L
I, q)
and
Since
$,
we have that
is included in
Mq’
lim
[8, (1.9)]
$*ea,
> O. erefore
we conclude that
The last equality follow from Remark 2.3 and an argument like
RESONANCE CLASSES OF MEASURES
471
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