517
Internat. J. Math. & Math. Sci.
VOL. 13 NO. 3 (1990) 517-526
ON A FAMILY OF WEIGHTED CONVOLUTION ALGEBRAS
HANS G. FEICHTINGER and A. TURAN
Ondokuz Mayis University
Institut fGr Mathematik
Universi tt Wien
GRKANLI (1)
Faculty of Art and Sciences
()
Strudlhofgasse 4
Department of Mathematics
A-090 Wien, Austria
Samsun,Turkey
(Received March 7, 1988)
Continuing a line of research initiated by Larsen,
ABSTRACT.
Liu and Wang
[13], Gtirkanli IS], and influenced by Reiter’s presen[12], Martin
tation of Beurling and Segal algebras in Reiter [2,10] this paper presents
and Yap
the
study
locally
of
a
compact
LP-conditions
of
of
family
Abelian
their
of
Banach ideals
group.
Fourier
These
Beurling
spaces
transforms.
are
algebras
defined
LI(G},
w
by
G a
weighted
In the first section invariance
properties and asymptotic estimates for the translation and modulation operators are given. Using these it is possible to characterize inclusions in
section 3 and to show that two spaces of this
their parameters are equal.
tities
in these algebras
is
type coincide if and only if
In section 4 the existence of approximate idenfrom which,
established,
among other consequen-
ces, the bijection between the closed ideals of these algebras and those of
the corresponding Beurling algebra is derived.
KEY WORDS AND PHRASES.
Beurling algebra, weighted
LP-spaces,
convolution,
ideal theorem, Banach ideals, factorization
1980 AMS SUBJECT CLASSIFICATION CODES.
43AlS
1Acknowledgement:
Supported by the Scientific and Technical Research
Council of Turkey and the Foundation of the Ondokuz Mayis University
Due to a Max Kade fellowship the first named author spends the academic
year 1989/90 at the University of Maryland at College Park, MD, 20742, which
allowed him to complete the manuscript there.
H. G. FEICHTINGER AND A. T. GURKANLI
518
,
I. NOTATIONS
G
Throughout
denotes a locally compact group
d
and Hast measure dx and
assume
((G)
throughout
compact support. For
A BF-space on G
p
complex-valued functions on G with
we write
(LP(G}, p)
(B,
is a Banach space
Lbc(G),
embedded into
there exists some constant
c
C
K
i.e.such
> 0
with
the Lebesgue spaces
lIB
of (classes of) measurable
K c G
compact set
that for any
fcK
CKIIfll B for
all feB (where
The translation operators L
is the characteristic function of K).
K
In order to avoid trlvialites we
non-compact groups. We write
respectively.
to have non-discrete and
for the space of all continuous,
functions
with dual group
G
are
Y
is defined
M
given by L f(x):= f(x-y), and the multiplication operator
Y
is called [strongly]
x,y G and x cG
as M fix):= X(x)f(x) for
(B,
c B
for all f B
[and
translation invariant if one has
liB
LyfB llfllB
LyB
y
and
G. [Strong] Character invariance is defined in the same way.
B
is called a Banach module over a Banach algebra
A Banach space (B,
A in the algebraic sense for some multiif B is a module over
If B c_ A,
and satisfies
a.b
(a,b)
plication
a-bll B
A
A
and the multiplication in
i.e. if
is continuously embedded into
B
satisfies the above estimate, we call B a Banach ideal in A. A net (e)e
I
in A is called a (bounded) approximate identity for A (shortly BAI), if one
-
A
(A,
has
llallA llbB.
ea-a A ---> 0
lim
any
for
a
A
A Banach
A.B
called essential if the closed linear span of
with B.
If A has a BAI
to assume that one has
w
in
(ideal)
module
(B,[[
lIB
is
coincides
Feichtinger [I])
(e)eI this is equivalent (Braun and
lim0c ecb-b [[B ---)0 for all fB.
on G (Reiter [2]), i.e. a continuous function
w(x+y)-< w(x)w(y) for all x,yeG, we set for
For a Beurling weight w
w(x) >-I
and
satisfying
l-p<m:
LP(G)
w
:= {
-
Recall
f[
It is a Banach space under the natural norm
that
one
::
G
p (G) c p (G)
if
L
L
w1
w2
C (Feichtinger [3])
has
for
all
equivalent,
we
x
convolution.
over
Felchtlnger
It always has a BAI.
LI(C)
w
[I]
only
Two
if
2
if
positive
w2lw T
i.e.
functions
are
Lwl
to
actually,
it
w
LP(c)
w
Moreover
respect
with
Prop.l,
and
(C) Is
w
and w w
2.
2
because it is a Banach algebra with respect to
w
w
write
called a Beurling algebra,
module
[ lf(x)lPwP(x)dx
llflp w
w2(x) Cwl(x)
caIled
LP(G) }
fw
convolution
follows
by
an
is
essential
p <
for
writing
Banach
(Braun
and
convolution
as
vector-valued integral).
For functions in
by
LI(G)
the
Fourier
transform
is
denoted
or f (if we want to stress the Fourier transform
,
mapping). Given a normed space (B,[[
as the image of B under
For any subspace
LI(G)
B)
LI(G)
we consider
endowed with its natural norm
we define the set
cosp(I} := (
cosp(1) by
X, (x}=O for
all fI
).
alternatively
f-B
f as
always
519
FAMILY OF WEIGHTED CONVOLUTION ALGEBRAS
-
2. BASIC RESULTS AND INVARIANCE PROPERTIES.
Let
w,
respectively. For
be weight functions on G and
^PW,; cc:
:= {
1
L’IcG)
W
f
LPwC) )
e
P
Ilfllw,,,,
an.d
p <
we set
,
Ilfll w+ll:rll p,,,.,
Among the most natural examples for spaces of this type (describing
both decay properties of its elements plus a certain amount of smoothness)
L1n2(
m)
w s
are the spaces
for
2(m)
s
s>0, where
denotes the Bessel poten-
spaces of order s (Stein [4]). In fact, these spaces aise as spaces
s/2. For special cases see Grkanli [5].
A p (A m) for re(y)
tial
:=(I+Iy12)
w,
The basic observation concerning A
THEOREM
(APw,mCG)[[ liP m)
2.1.
W,
p
w,0
a
is
-spaces is the following one:
Banach
L WI(G),
in
ideal
hence
Banach algebra with respect to convolution.
PROOF. The result can be obtained either by slight modifications of the
proof of Thm.3.1 in Feichtinger [6] or of Theorems 1,2 in Grkanli [5].
LEMMA 2.2.
For any
_ _
f
eLP(G)w
equivalent to the weight function
that one has
c-’x
r
o,
,ULxII p,w c.x
o
PROOF. For the second estimate we write for
IlLxfllp,w
(J’[fCz-x)lPwPCz)dz) l/p=
p
is
such
xG.
-
LP(G)
w
f
G with
llf.cKllp,w > 0
1/p
(fKlfCu) lPwP(x+u)du)l/P
[[exf[[p, wa (fx+Klf(z-x)[PwP(z)dz)
a
is locally bounded (cf.
w
Since
f.cK[[pW(X)/SUPueKW(-U).
proof is finished by setting C-= max ([[fllp,w,WCx)/SUPuKW(-U) ).
lea
C>0
(J’lfCu)lPwPCx+u)du) 1/p wCx)llfllp,w.
For the first estimate choose a compact set K
LEMMA 2.3. For any
IlLxfllp,w
x-
the function
i.e. there is a constant
w
(G), f
Then
[6])
the
x--->UMzflJ:,"""-,,w
O, the function
is
equivalent to the weight function
PROOF. The proof is simila to that of Lemma 2.2 and is left to the
interested reader.
A
THEOREM 2.4.
over, the functions
weight functions
w
x
ht
o
th
(G)
and
PROOF. Combining
vn
-
p
is
LxlliPw,,.,,
translation
and
character
IIIMxlll,",
invariant.
More-
are equivalent to the
respectively.
Lemma 2.2 and
owe
Z-
and
ete
Lemma 2.3
gives the result, by obser-
n xpon
....llllw,-
wh
wh
is
appropriate.
3. INCLUSIONS AND CONSEQUENCES.
two
We start with the observation that the intersection of
LP-spaces L pw (G) and Lp (G) is just LwP(G),
w2
p2
pl
w
max(wl,w2).. Hence we have A Wl,l n AW2,02
and
max(w1,m2)._ Next we look for inclusions.
where
A p
W,
w
with
weighted
may be taken as
w
maxCw1,w2),
H.G. FEICHTINGER AND A. T. GURKANLI
520
LEMMA
tinuous,
3.1.
i.e.
given weight functions
-
If A p*
tively one has:
p
W ,{D
(G) _c A p2
W2,2
p’-
llfll Wl, {2 .llfU W2,2
such that
A[,,(G)-spaces_
between
Inclusions
are
con-
automatically
6
w 1,w 2,to1,
and
respecG
on
2
(G) then there is some constant
C > 0
A pl CG).
Wl ,1
fo all f
LI(G).
Banach spaces, continuously embedded into
p
A
PROOF. The result follows from the fact, that the
W,
(G)-spaces are
Thus the closed graph theo-
rem can be applied in order to check that the inclusion mapping has closed
graph, hence is continuous.
p
(G} _c A p
THEOREM 3.3. h
Wl ,1
W2,{2
(G} if and only if
It is obvious that w2(w
p
CG)
hence A
Lp
2
Wl ,bl
PROOF.
p
and L
1
(6)
and
C6),
2(1.
L
implies
2(1
p
and
w2(w
W
L
(G)
(G)
W2
---> A W2,2 (G)
The converse implication follows by means of Lemma 3.1 with Lemmas 2.2
p
,,llLxfllw2,
and 2.3. (using the estimate
<2
COROLLARY 3.3. Two A
p
p
C.IILxfllwl,ol.... ).
are equal if and only if the correspon-
-spaces
ding weights are equivalent.
COROLLARY
norm for A
p
3.4.
There
.
an
is
(G} if and only if
POO.
f
equivalent,
w
Ilfll:
p
ten
character
strongly
invariant
1.
is
strongly
equivalent
an
(G)
character invariant norm on A
Conversely, if such an equivalent norm
p (G}
exists it is clear that the function
is bounded on A
IIIMxIII
weight
is
to
equivalent
the
W,{D
by Thm. 2.4. since any bounded Beurling
which implies the boundedness of
constant
(e.g.
the
trivial
w(x)
weight
weight, the assertion is verified.
COROLLARY 3.5. There is an equivalent,
p (G}
A
if and only if w=l
strongly
invariant
translation
norm for
Since the proof is similar to that of 3.4 it is left to the reader.
So far we have compared algebras A p
(G) with fixed p. In order to deal
with the situation of different values p and q we have to introduce a slight
extra condition: A weight
is said to satisfy the Beurlir-Domar condiw
tion (shortly: (BD}, Domar [7] or Reiter [2]}, if one has
Since
%zl n-2"lg(w(xn)) <
in
6.
PROOF.
Then A
p
W,b)
(G}
w satisfies {BD} and that
q (G)
A
if and only if p-<q
t--
for
Kn
6
n
support
such that
for
llfll pW,0)
-<
q
C-llfll W,O)
for f A
it is possible to find for any
such that
Ko’ o(0)
(ti+Ko)
e(X}
W,
By Lemma 3.1. we may assume
w(t)
compact set
G\K
G.
THEOREM 3.6. Assume that
m(t) z n
lies the (BD)-condition we cs_n find some
pact^
for all x
2
f
o
Since
n
such that
(t.+K^)
j
fn
@
for
some
w
satis-
j. We then define
k=nl -l(tk)’Mtkfo"
o
has com-
(tk)k=In
Choosing by induction a sequence
n
(G)
W,)
n z
for all t K
L wI{G)
p
f
n
by
in
521
FAMILY OF WEIGHTED CONVOLUTION ALGEBRAS
Therefore
n
r’knq O-lctk)’LtkO
fn,,llp,o
from below we observe first that by the arguIn order to estimate
one hms
ment used In the second pt of the ppoof o Lemmm 2.2
Ltko p,
e
Co.(tk).]lfo p
.
for 11 k
-
Since these fctlons have disjoint supports we have
n
P
(nllp,o)p rnlJt(tk)’(llLtkollp,o)P Cor=lfollp
fn
and thus
fn
n
p,o
1, w
q/p
Since t
k
t K
n
-
for all k ->
nllq_
C-n 1/p
Assume that
A p (G)
at infinity. Then
-I/q
"llfoll p,
-1
fo
,,,,llfnllPw,o
=
1, w"
-I/p
n
Since
implies
our first estimate
q z p.
which in turn implies
COROLLARY 3.6.
that o -->
n
n
1, w
Combining both estimates it is clear that we have
,,,,llfnllqw,o
o
we also have
k--nqw-I (tk)" Mtkfo 1, w n-2knq fo
the same kind of estimate gives
n.C
w,o
-
p,q <
A q (G)
w,o
that
satisfies (BD) and
w
if and only if p
q
It is now important to reexamine the above proofs in order to verify
that the arguments used in the proofs of Theorems 3.2 and 3.6 are in fact
independent of each other. Thus we come to the following result:
THEOREM 3.7. Assume that
t -->
--"
for i=1,2.
in
Wl,W 2
Then
satisfies
A pl
(G)=
Wl, 01
(BD) and that
APw4G)if
for
oi(t)
and only if
w2w
01 and PI: P, i.e. if all parameters are equal.
PROOF
That these conditions are sufficient for the equality of the
spaces is evident. For the converse we only have to observe that the arguments in the proof of 3.2 also work if different exponents
are
and
Pl
involved.
Thus
P2
we may conclude that corresponding weights have equivalent.
Given this the equality of exponents follows from Theorem 3.6.
4. APPROXIMATE IDENTITIES AND CONSEQUENCES.
L/EMMA 4.1.
algebra
LI(G)
w
Let the weight
w
on
G
satisfy (BD). Then the Banach
has a BAI whose Fourier transforms have compact support.
K
Set
Aw
compact ). Condition (BD) implies
( f[ f Lw’ supp
K
that A
is a dense ideal in LI(G), Domar [7]. Since LI(G) has a BAI (Reiter
w
w
w
[2]), the proof is complete by Lemma 1.4. in Doran and Wichmam.n [8]
PROOF.
THEOREM 4.2. If w satisfies (BD), the following is true:
p (G)
A
is a dense Banach ideal in L
having an approximate identity,
w,o
w
bounded in the norm of
with compactly supported Fourier transforms.
w
p
Ap
Ap
In particular it is an essential Banach ideal, and L
A (G)
I(G),
LI(G),
does not have BAI, because G was assumed to be non-discrete.
PROOF
dense in L
w
That A
p (G) is a
Banach ideal,
W,O
is easy to verify.
containing
K
A W’ hence being
In view of Lemma 4.1 it will be sufficient
GRKANLI
H. G. FEICHTINGER AND A. T.
522
Ll-bundedw AI (ea)al
< .
L lw
in A
to show that any
-
First
supll
C
Given thus
-Cp,
(A,(G)
’(2+2C)
o
there Is some
<
Altogether
-
we choose
e
to
p,.
Lw
in
(G)
implies
compact set
locl
such that
W,(
that
Implies
= =o"
Altogether
o
p
w,
(G) is an essential Banach module over
p
w,
e=
llf-fo
Ap
or
-
all
coBpact sets.
ove
to
convergence
for
that
0 (for
uniformly
G=cE-c <.(2{{{{p,)
that
obvious
(ea)i
e IIl,w
g-g.
((t)I
el2 + e/2
follows
it
therefore
with
I1-11.
-(
o
lplies convergence
in A
AI
is also an
is translation invariant the estimate
Since
I(t)l II-(t)l
Kw
of
boundedness
that
observe
we
It is
fA
any
for
0
L1 (G),
W
which in turn implies the factorization results (cf. [S]).
The unboundedness of M AI in the case of m non-discrete group (i.e.
fsmifollows from the observation that
in case of non-compactness of
6
IM,
LP()
for p<,
over compsct sets cnot be bonded in
IfomlM to
which converges
t
(t) a>0 for all
because
to
left
(details
the
reader).
p (C) we also find
Interesw,
to potntwise multiplication. We de-
Besides the convolution properties of A
p (G)
ring stPucte of A
with
W,
note by
A
in
(G}nA
A
is also
p
is a dense subspace of A
w,
It
conseence
is
(u)j,_
in
constt
(G)A,
fctton
from
uf-flll,w
0
for
the
module over L
()
Y
same guments show
f
LI(G)’w
Since
e,-
usf-f
serve first that we c approximate
P
Aw,).
BM
be continuous. Now
A
sets.
in
is
On the other hd we have
LP(G)
is
a
which is a dense ideal as
compact
over
In order to check the density of
norm of
A
that
theorem
essential
is
has
a
BAI
tends to the
tn
implies
uB
(e)
Bach convolution
we have at the same time
(uf-flIp,
e
inversion
X(G}A
which by the guments given in m.4.2.
LI(G).
ideal
the
containing
As in Lemma 4.1 one derives that
iformly
for some BAI in
Is a dense Bach
(G)
(G).
evident
(BD).
of
w,
(uo}B
pointwise Bach algebra, containing
a
p
for
b} Any bounded approximate identity
Jp
p (G) d
X(G) n A
approximate identity for A
as a dense ideal,
PROOF.
A
respect to pointwise multiplication,
with
in X(G) n A
A
with it’s natal norm.
satisfies (BD) then
EOM 4.3. a} If
ideal
Pespect
-1 ’tLltu,,
the Bach algebra
lip
0
(G) nA
p
fA
W,
uh
(e)j
p
A
in
W,
in
(G)
L(6).
let us ob-
K
by functions in A W (In the
the inversion theorem we see thmt
appPoxmates h
APw,(G)"
for
for any BAI
h
mmM be sssumed to
d the pPoof of b) is complete.
523
FAMILY OF WEIGHTED CONVOLUTION ALGEBRAS
Our next step is to check that A
sense
([2])
that
A w (6)
((6)
n A
result
with
norm,
p
w,
A W (6)
A
Let
by
f-->llfll p
Since
we
know
(BD) (see [2]) und that
satisfies
p the assertion follows
from
A w,
ApW, (6)).
and B
is dense in
(by choosing A
w
is a Wiener algebra in Reiter’s
w,0
given
is a Wiener algebra if
LEMMA 4.4.
Ica.
natural
its
p
the following
be a Banach algebra of continuous functions on
group G (under pointwise multiplication) which is a Wiener algebra in
Reiter’s sense.
A n ((G)
liB)
(B,
any Banach ideal
Then
A
in
which
contains
as a dense subspace is again a Wiener algebra.
The conditions to be checked will be clear from the proof:
PROOF.
all it is clear that (B, B) is a normed function algebra
h h(x) is continuous for any
(upon renormalization, if necessary), and
A implies that lh(x)l s
xG, hB, since B
B
A
Furthermore we have to check a regularity condition (functions in B
separate points from open sets) and the local inversion property (given
G)
there exists he B
function
f B
non-vanishing over a compact set K
First
of
Cxhll
with
I/f (y)
h(y)
for all y
-
Cx.Chll
K
coinB
cides with A under the given circumstances. In fact, given any compact set
B with
K G for any function f A one can find some
for all y K. This is verified as follows: By the density of ((G)oA in A we
0 for all y K. Choosing now
can find some g ((G)oA
B such that g(y)
K (which exists, since A has local
over
A as a local inverse to
g
El
inversion) we obtain
the function
g’gl (Bo((G)).A B(G). Now it is
Both properties follow from
the observation
that
’locally’
fl
we
evident
that
f1=f-gE1
A.B c B
B (with the
liB
have
f(y)
f(y).gg1(y)
=:
as was required. Finally,
f1(y)
-nrm) is part of the assumption, since ((G)oB
S. IDEAL THEORY AND NONFACTORIZATION IN A
y
all
for
the density
of
G,
with
((G)oB
in
((G)nA.
p (G).
w,(
p
In this section the ideal theorem for the algebras A w, is discussed.
(BD) there is a one to one corresponw
satisfies
THEOREM S.I.
If
p
LI(G)
(G), given by
and those of A
dence between the closed ideals of
W
the following two mappings which are inverse to each other:
LI(G)
is a closed
(a) Given a closed ideal
J nA p
the set
J
w
ideal in A
p
(G).
(b) Given any closed ideal
a closed ideal in
_= A p (G)
w,
the closure of
in
LI(G)
w
is
w
p (G)
is an essential Banach ideal in LI(G) (by Thm.4.2,
PROOF. Since A
w
w,
and L I(G) has bounded approximate units, the result follows from the ideal
w
theorem
for
abstract
Segal algebras
detailed results of this type).
(Relter
[10],
and
Felchtlnger
[11]
for
H. G. FEIGHTINGER AND A. T.
524
GRKANLI
The abstract bijection between the set of closed ideals of two spaces
A
p
belonging to the same Beurling algebra
i=1,2,
w,
direct
Theorem
of
application
5. I,
be
cn
LI(G),
w
resulting from a
more
described
If w satisfies (BD) for any two weights
THEOREM 5.2.
explicitly:
,,
on
6
the
following mappings establish a bijection between the set of closed ideals of
p (G):
A p (G) and those of A
w,
w,
(the closure here in
a)
I1-->I1-nAP,w p
b) 12-->(12nAKw)(12A 1 )- (the
W,
L1)w
and
closure being taken in A
p
W’(I
(G)).
c) Moreover corresponding ideals have the same cospectrum.
PROOF. a) There is just one way to describe the composition of the two
bijections as described in Thm.5.1. As for b) it is sufficient to verify
K
p
in an A
-algebra can be recovered from IA w (by
that any closed ideal
w,(
is trivial it is suftaking the closure). Since the inclusion
ficient to check that any
= (IAK)w
f(IAK)
w
can be approximated by elements in I.
K
(according to Lemma 4.1) such
g
K
K
p
c
A w the result is proved.
< e. Since f.g I.Aw
that llf-f.gll
w,
Finally, c) follows easily from the observation that two spaces with the
f and e > 0
Thus, given
same L
l-closure
w
w
2
w
and
CAw
have the same cospectrum.
a) A pl
THEOREM 5.3.
if
we choose
(G)
A p2
as a Banach ideal if and only
(G)
W2,{
WI ,1
In the positive case the bijection between ideals
I 2"
is established by taking the closure in the larger space, o fo the inverse
msppi the intersection with the smslle one (ms in m..).
PROOF. It is easy to check thst the conditions e sficient to verify
the popetles of
thst
w2{w
we
have
Bsch ides1.
by m..2.
d
2{
f, EPW1 ,1
ConvePsely the inclusion slesdy
In ode to prove
fPW1 ,1 .EPW2,
fo
w{w 2
fAPWl ,1
implies
note thst
EAPW2,
erefore
w1 ,1
FON Lemms
eivslent
2.q
to
the left side is
w
2.
W2,
W1 ,1
WI ,1
the right
to
d
eivslent
wq
e bove estimate
eoem 5.q.
then
implies
side
wq(w2. e
is
finsl
ststement of obvious from
The question, whether
A
p (G)
has the weak factorization property can
w,w
be answered to the negative:
w(-x) for all x G.
w(x)
THEOREM 5.4. If w is symmetric, i.e. if
p (G)
(with
respect to convolution).
does not have weak factorization
Then A
w,
PROOF. We deduce this result from Corollay 1.4 in [9]. We first check
p
is a weakly self adjoint Banach algebra. This follows from the
A w,
that
L w is closed with respect to
fat that the symmetry of w
implies that
p
weak selfL
Since
is a Banach ideal in
complex cortjugation.
adointness
LP(G),
follows
A
therefrom.
w,(
Since
by definition
A p
w,(
w
is
contained
in
and the Haa_ measure on G is unbounded for non-discrete groups G we
FAMILY OF WEIGHTED CONVOLUTION ALGEBRAS
can apply Corollary 1.4
ACKNOWLEDGEMENTS.
525
of [9] and the proof is complete.
The major part of this work was prepared at the
Univer-
sity of Wien. The second named author would like to thank the Department
Mathematics of the University of Vienna for its reception,
tific and Technical Research Council of Turkey
for
also would like to thank Doz. Hans G. Feichtinger for
and
financial
his
the
He
support.
hospitality
and
cooperation. The first named author has to carry the responsibility for
late presentation of the revised version. He wants to thnk the
of
Scien-
the
Mathematics
Department of the University of Maryland at College Park for its hospitality, as a long term visit there gave him the chance to finish the paper.
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