Internat. J. Math. & Math. Sci.
(1996) 57-66
VOL. 19 NO.
57
ON A FAMILY OF WIENER TYPE SPACES
R. H. FISCHER, A. T. GORKANU and T. S. LIU
University of Massachusetts
Ondokuz Mayls University
Department of Mathematics
Faculty of Art and Sciences
and Statistic
Department of Mathematics
Amherst, MA.01003, U.S.A
Samsun, Turkey
(Received June 16, 1993 and in revised form June 20, 1994)
ABSTRACT. Research on Wiener type spaces was initiated by N.Wiener in
[15]. A number of authors worked on these spaces or some special cases
of these spaces. A kind of generalization of the Wiener’s definition
was given by H.Feichtinger in [2] as a Banach spaces of functions (or
measures, distributions) on locally compact groups that are defined by
means of the global behaviour of certain local properties of their elements. In the present paper we discussed Wiener type spaces using the
spaces AP’q(G) and FP’q(G) (c.f [8]) as a local component, and Lr(G)as
w,
w,
a global component, where w and v are Beurling weights on G and m is a
Beurling weight on G(c.f.
[13]).
KEY WORDS AND PHRASES. Weighted LP-spaces, Beurling Algebra,
type space, BF-space.
1991 AMS SUBJECT CLASSIFICATION CODES. 43.
Wiener
.
NOTATIONS. Troughout this paper G denotes a locally compact abelian group
(non-compact, non-discrete) with the dual group
Certain well known
terms and their definitions may be found, e.g., in [2], [3], [7], [13],
[14], [9], [8]. In the present paper we have used two kinds of Fourier
Transforms.. The classical Fourier transfor ^) (c.f.e.g[13])and a generalized Fourier transform F (c.f. [4]) discussed below.
Let Q be a fixed open and relatively compact subset of G. We define
SO
(G)--{ f
lf=ELynfn,YneG,fneAQ,n>_l
and
Ell fnllA<=}.
(l)
where
AQ-={ heA (G)
supp (h)
of f
y denotes the translation operator. Any Representation
the form (1) is called an admissible representation. Endowed with
and L
norm
I fllSo ----inf{llf n I A’ f--z,Yn f n
admissible
in
the
R. H. FISCHER, A. T.
58
GDRKANLI
AND T. S. LIU
S (G) is the smallest strongly character invariant Segal algebra on G
O
It is well-known that the Fourier transforms induce isomorphisms betwean the spaces S o(G) and S
The generalized Fourier transform then
o
()
is defined by
<o,f>---<o,>
for feS
O()
oeS’ (G)
0
It is easy to see that above mentioned qeneralized Fourier transform coincides with the Fourier transform in the sense of tempered
distributions for the special case of G=IR
m.
Throughout this work, we also will use Beurling weights, i.e.real
valued, measurable and locally bounded functions w on a locally compact abelian group G which satisfy l<_w(x), w(x+y)<_w(x)w(y) for x,yeG.
For l<p<, we set
LW
p(G)={flf weL p(G)
Under the norm
LI(G)w
llfllp,w-If.wll
p, this is a Banach space. When p--l,
becomes an algebra under convolution, called Beurling algebra, c.f. [13].
In this paper another two imDortant tools are the spaces AP’q(G)
p,’q
II’II P’qm
Fw (G) with the norms
and
w,
spaces and the norms are defined as follows:
and
AP’q (G) ={ f
W,0
feLwP (G)
w,
ll’llFrespectively [8].
and
p
w,60
These
eLq0 {;)
q,
and
FP’q()-w,
(A p’q(G)),
w,
{{{{ F={{f{{P’ q
where
F is the generalized Fourier transform, w and m are
weights on G and
respectively.
Beurling
The main parts of this work deal with certain Wiener-type spaces
[2]: the definition is the following: Let B be a Banach
space. Assume that there exists a homogeneous Banach space
continuously embedded into
which is a Banach algebra
in the sense
(Cb(G),ll.ll.),
under pointwise multiplication and is stable under complex conjugation, such that (B
is continuously embedded into A’ (G)=(A(G)nC (S))’
c
B
c
and also is a Banach module under pointwise multiplication. Here C (G)
c
is the space of continuous functions with compact support, Ac(G)=A(G)
NCc(G) is given the (locally convex) inductive limit topology of its
subspaces
,II A) KcmG compact, and (G) is the topological dual
ll’II
(AK(G) "II
A’c
Ac(G). Let now Bio c be the space of all feA(G) such that hfeB
heAc(G)
this is a locally convex vector space whose topology is
of
fined by the seminorms
fllhfllB,heAc(G).
Fix am open, relatively
and define, for fEB and xeG, the "restriction
pact set
of f over x/Q" to be
for
de-
com-
59
FAMILY OF WIENER TYPE SPACES
[11111
For
fEBloc,
set
hf-" hg for IcA
Ff(x):llfllB(x.O
If now
c
xQ
(t;)
is a solid, translation tn...
variant BFspace on G, one defines the Wiener-type space
W(B,C)--{fEBIoclFfEC}
(B
1
W(B,C)
by
IlfllwB,C)--tlll )-
and
Lastly recall that a Banach convolution triple (BCT) is a triple
3
of BF-spaces such that convolution, given by
,B
B2
fl x f2 (x)=
ff i (x-y) f2 (y)dy
G
for
fieBiNC c (G)
(i:=i,2), extends to a continuous bilinear nap
BIxB2/ B3
(of norm I).
I. THE WIENER-TYPE SPACES
W(AwP’q(G) ,L r)
The construction of the Wiener-type spaces mentioned in the section title requires some preliminary considerations, notably Theorem
1.5 below. First of all we introduce the Banach spaces.
u
1
A (G)= F (Lu
())
where u is an arbitrary weight function on G with the norm
!llluIglll, u
and
F is the classical Fourier transform We set A cu(G)=A u(G)NC
of
the subspaces
equip it with the inductive limit topology T u
Since it is
KeG compact, equipped with their
c(G)
AK(G)=
and
=AU(G),CK(G),
ll.llu-nOrms.
Ac(G),
it is
obvious that T u then is finer than the norm-topology of
the
u(G)
u(G)
pseparates
’----(A c
,T
Hausdorff and hence the dual space A c
u
are closed in A (G)
Note also that the subspaces A U(G)
ints of AU(G)
K
c
to
hEAU(G)
with
respect
to
converges
Indeed, if
u, the
resame
sequence also converges uniformly to h and so supp(h)cK. The
u
u
Consequently,
in A (G)
asoning shows that for F:L, AK(G) is
L
u -closed
also is complete since then it is a
if G is o-compact, then (AU(G)
u)
II’II
(hn)CA(G)
II-II
,Tu)
c
strict inductive limit of Banach spaces
w
satisfies Beurling Domar condition
and
w2
logw 2 (tn[
w2
<, taG), then A c (G)cA (G) and Ac (G)
(shortly (BD) i.e E
n
n>l
wl
is everywhere dense in Ac (G).
1
1
(G) c Lw 1 (G) is methen the inclusion
PROOF. Since
2
w2
w
(G) is everywhere dense in A 1 with respect to the
tlsfied and A
LEMMA I.I. If
wl<w 2
Cc(G)CLw2
Wl<W
norm
II wl
identity
any
Also since w 2 satisfies (BD) then there is an approximate
K
(ee)eiCAw 2 (G),
eAcW I(G)
where
1
(G),supp compact}.
wK2={flfLw
2(G)
function
and e>0. There exists a
w
feA
such that
Take
60
and
ges
.
R. H. FISCHER, A. T.
w
2
eeAc A(G)wl (G).
to g
in
AND T. S. LIU
for all se[. It is easily seen that
c
conver-
I and w 2 be satisfied the hypothesis in lem-
ma I.I. We endow the space
Wl (G).
(.es) me I
This proves our lemma.
COROLLARY 1.2. Let w
pology by A
0RKANLI
AW2(G)is
c
AWl(G)
c
with the induced inductive limit fo-
also everywhere dense in
to induced inductive limit topologv bcause it
s
l
(G) with res,ect
strictiv finer than
w1
Wl (G) which is induced
the original topo]ogv on A
topology bv A (G)
c
COROLLARY 1.3. Again let w.,w be satisfied the hypothesis in lem.w2,.
z.Wl (G) and the Corollary 1.2. one
ma i.I. Using the inclusion A
u;c;%
c
c
wI
w2
Specially
(G )’ is continuously embedded into A c (G)
obtains that A
c
w2
=(A
(G))
if
Wl~W 2 then (A c
c
WI(G))
on G satisPROPOSITION 1.4. If l_<p<_ and the weight function
AP’q(G) is continuously embedded into
fies (BD) condition then
o(A(G)’,--
w,{
A c (S)).
PROOF. It is known that
O(A(G) ,Ac(G)) [8].
AP’g(G)
w
is continuously embedded
into
If one uses the above embedding and the Corollary
1.3., easily proves the Proposition.
In order to obtain all the properties of AU(G), etc., required
for the construction of Wiener-type spaces in the sense e.g. of Feichtinger, cf.
[2],
we assume henceforth that the weight function u on
G
is the second weight func-
satisfies (BD) and symmetric ands<u, where
AP’(G)
w
(For example one can take. u=m(x) /(-x))
First of all, AU(G) now satisfies the requirements of [2]: It is
in fact, the emclear that AU(G) is continuously embedded into C b(G)
is
bedding map has norm<l. Moreover, since u satisfies (BD), Au(G)
Banach
a
is
It
is
reqular.
known to be a Wiener space, cf. [13], hence
tion in
algebra under pointwise multiplication and also is translation-invariis character-invariant and that the maps M x are Isometries;
ant: L u
/
moreover, for
Mxg is continuous [8]. By taking Fourier
transforms, we now conclude that AU(G) is translation-invariant, transis continuous from G into
lation maps are isometries and that x
I(6)
geLl(G),x
AU(G)
for each
beAU(G).
Lxh
Banach
under
is closed
In other words: AU(G) is homogeneous
space. Lastly, the symmetry of u implies that AU(G)
complex conjugation.
p’q (G) is a pointwise Banach module over A(G), [8]. Since <_u
Secondly A
w
then AP’q(G) is also a Banach module over Au(G). Consequently one shws
w
) by the Proposiis continuously embedded into (AU(G)
that
c
AwP’q(G)
tlon I. 4.
With this, Feichtinger’s general hypotheses are satisfied and the
construction of the Wiener-type spaces w(AP’q(G),C), C a solid BF-space
w}
proceeds in the standard manner.
Using the arguments in Theorem I.
modification,
t|’e
in
[2]
and doing some
proof of the following theorem is completed.
small
FAMILY OF WIENER TYPE SPACES
THEOREM 1.5.
(i) The Wiener type space
61
W(A’q(G),,o Lur((;))
is a Ba-
nach space under the norm
few(AP’q(G),L
r(G))
v
w,
where
and
F(z)---Fflz)---(]IfllP’q[z/Q),
W
G is any open subset of
z:(; a,d
G with compact closure.
It is also continu-
(A ’q(G)) loc
ously embedded into
(ii) The set
Ao={ feA
(G)
Is continuously embedded into
[suppf
is compact
r
w(AP’q(G)
,Lv (G))
w,
r
q
(iii) W (AP’ (G) ,L u(G)) is left (right) invariant, and
w,
. ’"
where
AP’q(G)
w
W
p’ q
[[L x][I<Lxl[[ W0"
"
P’qw, and
[]]Lx[]]r
r,9
are operator norms on
and L r(G) respectively.
(iv) The translation is sly
r
(G) ,L (G)).
(A’q
,
(V)
r
W(A’q(G),L
, u(G))
n
Wiem
W(A’q(G)w, ,l%r(G)),
t s
is a Banach module over :(A (G) ,L
(G)) with
respect to the pointwise multiplication.
PROPOSITION 1 6
le over W(AI’q(G) W
W
Let p>l
LI(G))
Then
w(AP’q(G)
We
L(G))
is a Banach modu-
with respect to convolution.
It is easy to show that every locally compact lian grip
is a group (i.e. a group having a compact neighbourhood of identity
s a
It is known that AP’q(G)
is Invariant under inner automorphisms)
W)
left(rlght) convolutlon module over Al’q() 8]. Hence since
W
PROOF.
(A’q(GI,AP’q(G)
,A’q(G))
W
(L(G) ,LP(G) ,LP(G))
and
W
W
are a Banach convolution triples on G. Then
(W(A’q(G),LI(G)),W(A’q(G),LP(G)),W(A:q(G)
,
W
W
LP(G)>)
W
is a Banach convolution triples on G and the inequality
<
IIflW(A’q(G) LP(G))11
W
l,q) (G)
and
the Theorem 3. in [2] for el].
4,.P,q(G), (G)). One can easily show the algebraic conditions which
Is Satisfied
"[
are needed to be a module.
feW(Aw,
,L(G)
R. H. FISCHER, A. T.
62
PROPOSITION 1 7
GDRKANLI AND
W(AwP,|(G)
,i)
r
L V(G)) is a BF-space on G.
W(AwP’(G)
r(G)) is continuously
(i),
(thus a seminorm
That means given any heA
c
such
there exists a constant number
OC
By the Theorem 1 5
PROOF
’q
e,edded into
Ph(f)=h.fP’
W,U
(A ,u(G))loc"
(A,)..
on
T. S. LIU
Dh>0
that
{{hf{P’Dh{{f{W(h’q(G) L r(G))
for all f
W(AP’q(G),LP(G))
w,
(I)
Hence one can write
{{h.f{{p<D
h_ "{{f{w(AP’q(G)w, LrV(G)){{"
lso
(G)
*s a regular Banach algebra with respect to pointwise ul-
tiplicat*on, then one may choice a function
and h(x)=l
0h<l
hence
(x) lf(x)l<(x)lf(x)l
DKo>0
h((G)=-((G)[ICc(G)
fo all xK. We let suph
fyin
exists
(2)
fo
;
Since L
xG
Eo"
L
Then
K(x)h(x),
o’
then there
such that
o
{h{x)f(x)IdxD K
o
.llh.fllp.
(3)
Also one has
K
o
The proof is completed combining the formHlas (2), (3) and (4).
W(AwP’q(G), ,Lu(G)r
is a BaPROPOSITION 1.8. The Wiener type space
nach convolution module (left and right because G is an Abelian grou
1
over some Beurllng algebra LW (G).
o
W(A.P.’(G),L,r.(G))
is a
PROOF. We proved in Proposition 1.7 that
1
Lr(G)) +
By the Theorem 1.5., this
BF-space, thus
space is left invarlant and translation operator in W(AP’q(G),Lr(G)is
W
continuous. NOW a simple application of vector valued inteqral shows
1
that W(AP’q(G),L r(G)) is a Banach module over LW (G), where
%
W
O
W(AwP’q(G)
,
Lloc(G).
o(x)=max(l IIILxlW(AwP’q(G)
W
L
r
,L r(G)) is a left (right) Banach module
COROLLARY 1 9
(
1
satisfying v(x)>Wo(X) for all xSG,where
a
is
weight
v(x)
if
Lv(G)
w o (x) is defined as in Proposition 1.8.
W(AwP’q(G)
over
REMARK. By the Theorem 1.5 (iii), one writes
p’ q"
IlLx III r,u’
IlLx III < III Lx III w,
where
p’q
III-III III. III W)
AP’q(G) and Lr(G)
W
and" IIILxlIIP’q<w(x)
III .III.,
are operator norms on
is also known that
It
respectively.
[8]. Then we have
and
(l)
W(AwP:q(G),L(G)),
<v(x)
63
FAMILY OF WIENER TYPE SPACES
W,v are
weight functions, then the function w.9
bounded. Using the inequality(2) it is easily seen that
for all xeG. Since
is
locally
x-
IIILIII
i
so
.oo oune.
LP(G)
W
Given a weighted space
is denoted by
r_{
w
Pw
(a)
i
the associated weighted sequence space
and defined
ie %eq
r
(aiW(i) iele r}
where the discrete weioht W given by W(i)--W(x i)
for iel
r
known that
is a Banach space with respect to the norm
w
I[zll r w---(
where z----(a i)
It
is
1
laiw(i)Ir)
il
E
ieI"
Zwr,
LEMMA I.i0. For any
llLpZllr,
z0 the function z-+
w is equivalent to the weight function w, i.e there is a constant c>0 such that
one has
c
-Iw (P) <ll LpZll r, w-<c. W (p)
PROOF. Result can be obtained by a slight modifications of the
proof of Lemma 2.2. in [7].
It is also easy to prove the followi Imam using the closed graph theonaa.
r
LEMMA I.II. If
I c r2
wI
w
then there is some constant c>0such that
2
I II r2 ,w2<_C-II zll r I W l
for all
r1
z=lai)ieieWl
if we use the Lemma I.I0 and Lemma I.II easily prove the following lemma
LEMMA 1.12.
w w
r
r c
1
if and only if
2
w2<w I.
THEOREM 1.13. Let U
U be the weight functions in construc1 and 2
rI
r
tion of Wiener type spaces
and
,L%)
,L%)
weights on
respectively. Also assume that Wl,W2,91,u
i,2 weights
2
<
u
u
on G and l_<p,q,rl,r2 <. If
r
and
1
2 4_r I
2
2
W(AwP?I(G)
I(G))
W(AwP’?m
; 2(G)
22(G))
UI~U
A p’q
w2, 2
then
r
-
(G)
p’q
(O)
= AWlm
I
r
W(AwPq’t2 (C) ,Lv2(G)) W(AwP[?tal (G) ,Lo 1I(G))
2
(I)
GORKANLI
R. H. FISCHER, Ao T.
64
PROOF. Since
AI
’q
2
’2
((;)
A
C
I rll wp’ q,m
I
feAw2q
for all
AcUllG)-Auc 21G)
f’
u
and (A
1
(G))’
c
(G)
tllen tl,ere exists c>0 such that
’ml
p’ q
<c.ll ell w2,
(2)
I-
Wl<W 2, u ml<,2 [8].
(G) and
2
’q
1
AND T. S. LIU
(A 2 (G))
c
2
Also since
UI-.U 2
by the Lemma l.l
then
llence a simple
calculation shows that
(A p’q
(APw2’q’"2 (G)) oc
(G))
Wl
Now using the definition of Wiener-type space,
r
W(A
r
p’qw2,,n2 (G) ,Lv 2 (G))
c_
r
c_
(2) and (3) we have
2
w(AP,qwl’ml (G) ,L92 (G))
Also because tile Proposition 3.7.
W(A 1’q’I (G) ,I,v 22 (G))
(3)
oc"
in
[6]
we write
r
w(AP,q
Wl ’ml
(G)
,Lvl1 (G))
(4)
if and only if
rl
r
c
(gu2(G))d
(LullS))d’
2
r
where (L u
r
and
r
2
o2
Lu
r
2
r
G))
2
(G)
1
I
d
and
151
r
1
(gl(G))d
are the discretes of the spaces
respectively. Since r 2
r1
and
Ul<
u
2
2
Lu2(G)
then
This completes the proof.
p’g
and
(G) cA P’q m (G) if and only f Wl<W 2
It is known that Aw
Wl,
2 ,n 2
proves
the
ml<,,2 [8]. If one uses Theorem 1.13 and Lemm 1.12 easily
following corollary.
COROLLARY 1.14. Let
Ul~U 2, Wl<W 2
r
W(A p’q
(G) ,L (G))
u2
W2’u12
if
and
c., W(A p,q
Wl’l
ml<m 2.
Then
r
(G)
,LullS))
and only if u l<u 2
COROLLARY 1.15. If
W(A
q
’2
Wl~W 2, ml~m2 Ul~U 2 Ul~U 2
r
(G),L u (G)):-W(A
2
1
’ml
(G) ,L u (G))
1
and
rlr 2
then
65
FAMILY OF WIENER TYPE SPACES
2.
]
D,qtG) ,L r(,))
FW,
THE WIENER-TYPE SPACES W
Let u be a weight function on G. Proceedin(! as we did in Section
Au(,)-AU(G)I|C (G) euuipl,ed with their nawe set Ac(G)--A(G)C
c
(,)
c
c
AU(,)
rural inductive limit topologies, thus the topology e.q of
is the
c
u
inductive limit
of subspaces u
Kc(;
=A
compact, and
their
Recall that T is (strictly) finer than the toA
pology T c induced by the usual inductive limit topology of
AK(G
TA
ll.llA-tOpologies.
(G)flKK(G),
Cc(,).
We assume that the weight function u satisfies(Bl)) and symmetric
and the first weight w in AP’q(G) satisfies the condition w<u.
Since W is symmetric then
FP’q()=Aq’P(,),
w,(
,w
(,)
,
(see
18])
Now all conditions are satisfied required for the construction of
Wiener-type space W( F P’(’’ L r (G)=-W( Aq’P(, ),L ())
I f one uses
the
wb}
0),w
^v
,
properties of the space
type space
Aq’P(G)
W(FP’q(G)
,Lr(6))
9
w,
obtains all properties of the
like to that of
W(AP’g(G),L(G))
Wiener-
in Sec-
tion I.
ACKNOWLEDGEMENT. The authors want to thank It.G. Feichtinger for his
significant suggestions regarding this paper. The first named author
wants to thank R.ll. Fischer and T.S. Liu and University of Massachusetts
for their hospitality.
i,
2,
5,
6,
7
9,
10,.
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1980.
FEICIITINGER, II.G. On a new Segal algebra, Mh. Math. 92. (1981),
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FEICIITINGER, II.G. Un espace de Banach distributions temperees sur
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