# On the Generalized Weighted Lebesgue Spaces of Locally Compact Groups

Authors: I. Akbarbaglu; S. Maghsoudi
Journal: Abstract and Applied Analysis
 (2011)

---

## Abstract

LetG be a locally compact group with a fixed left Haar measure λ and Ω be a system of weights on G. In this paper, we deal with locally convex space Lp(G,Ω) equipped with the locally convex topology generated by the family of norms (∥.∥p,ω)ω∈Ω. We study various algebraic and topological properties of the locally convex space Lp(G,Ω). In particular, we characterize its dual space and show that it is a semireflexive space. Finally, we give some conditions under which Lp(G,Ω) with the convolution multiplication is a topological algebra and then characterize its closed ideals and its spectrum.


---

## Body

## 1. Introduction

Throughout this paper, letG denote a locally compact Hausdorff group with a fixed left Haar measure λ. By a weight function on G, we mean an arbitrary strictly positive measurable function on G, and, by a system of weights on G, a set of weight functions Ω such that given ω1, ω2 in Ω and c>0, there is an ν∈Ω such that cωi(x)≤ν(x) (i=1,2) for locally almost all x∈G.For a weight functionω and 1≤p<∞, let Lp(G,ω) denote the space of all complex-valued measurable functions f on G such that fω∈Lp(G), the usual Lebesgue space on G with respect to λ; see [1] for more details. Then, Lp(G,ω) with the norm ∥·∥p,ω defined by ∥f∥p,ω:=∥fω∥p for all f∈Lp(G,ω) is a Banach space. We also denote by L∞(G,1/ω) the space of all measurable complex-valued functions f on G such that f/ω∈L∞(G), the space defined in [1]. Then, L∞(G,1/ω) with the norm ∥·∥∞,ω defined by ∥f∥∞,ω:=∥f/ω∥∞ for all f∈L∞(G,1/ω) is a Banach space. Furthermore, for 1≤p<∞, the topological dual of Lp(G,ω) coincides with Lq(G,1/ω), where q is the exponential conjugate to p defined by 1/p+1/q=1. In fact, the mapping T from Lq(G,1/ω) to Lp(G,ω) defined by
(1.1)〈T(f),g〉=∫Gf(x)g(x)dλ(x)
is an isometric isomorphism; see for example [2]. For measurable functions f and g on G, the convolution multiplication
(1.2)(f*g)(x)=∫Gf(y)g(y-1x)dλ(y)
is defined at each point x∈G for which this makes sense. The algebraic and topological properties of weighted Lp-spaces have been studied extensively; see for example [2–5].Let1≤p<∞ and Ω be a system of weights on G, we set
(1.3)Lp(G,Ω)=⋂ω∈ΩLp(G,ω).In this paper, we equip the spaceLp(G,Ω) with the natural locally convex topology generated by the family of norms ∥·∥p,ω, where ω runs through Ω. For a similar study in other contexts, see [6–8]. We investigate certain algebraic and topological properties of the locally convex space Lp(G,Ω). Our results generalize and improve some interesting results of [5] and partially answer a question raised in [3].


## 2. Preliminaries and Some Basic Results

LetG be a locally compact Hausdorff group with a fixed left Haar measure λ and Ω be a system of weights on G. We equip Lp(G,Ω) with the locally convex topology generated by the family of norms (∥·∥p,ω)ω∈Ω and denote this topology by τΩ. So (Lp(G,Ω),τΩ) has a basis of closed absolutely convex neighbourhoods at the origin of the form
(2.1)Vp,ω={f∈Lp(G,Ω):‖f‖p,ω≤1},(ω∈Ω).Note that the topologyτΩ is Hausdorff, because if f∈Lp(G,Ω) and f≠0, we have λ({x∈G:f(x)≠0})>0. Put E={x∈G:f(x)≠0} and fix ω∈Ω. Then,
(2.2)‖f‖p,ω=(∫G(|f|ω)pdλ)1/p≥(∫E(|f|ω)pdλ)1/p>0,
and thus τΩ is Hausdorff.IfΩ and Γ are two systems of weights on G and for every ω∈Ω, there is a ν∈Γ such that ω≤ν (pointwise locally almost everywhere on G), then we write Ω≤Γ. In the case which Γ≤Ω and Ω≤Γ, we write Ω~Γ.Proposition 2.1.
LetΩ and Γ be two systems of weights on G and T:G→G be a measurable mapping such that Ω≤Γ∘T:={ν∘T:ν∈Γ}. If the Radon-Nikodym function h=d(λ∘T-1)/dλ belongs to L∞(G), then the mapping f↦f∘T is a continuous linear map from (Lp(G,Γ),τΓ) into (Lp(G,Ω),τΩ).Proof.
Givenf∈Lp(G,Γ) and ω∈Ω, choose ν∈Γ such that ω≤ν∘T. Then we have
(2.3)‖f∘T‖p,ω=(∫G(|f∘T(x)|ω(x))pdλ(x))1/p≤(∫G(|f∘T(x)|(ν∘T)(x))pdλ(x))1/p=(∫G(|f(x)|ν(x))pd(λ∘T-1)(x))1/p≤(∫G(|f(x)|ν(x))ph(x)dλ(x))1/p≤‖h‖∞‖f‖p,ν<∞.
Hence, ω(f∘T)∈Lp(G). Since ω∈Ω was arbitrary, f∘T∈Lp(G,Ω). Continuity also follows from the above relations.The space of all bounded Borel measurable functions onG with compact support will be denoted by Bc(G). Let us remark that if Bc(G)⊆Lp(G,Ω), then Bc(G) is norm dense in Lp(G,ω) for any weight ω on G; see for example [9].Corollary 2.2.
LetΩ and Γ be two systems of weights on G. Then,(i)
IfΩ≤Γ, then the induced topology τΩ on Lp(G,Γ) is weaker than τΓ.(ii)
IfBc(G)⊆Lp(G,Γ)⊆Lp(G,Ω) and τΩ⊆τΓ, then Ω≤Γ. In particular, Ω~Γ if and only if Lp(G,Γ)=Lp(G,Ω).Proof.
(i) is trivial. For (ii), we observe that for anyω∈Ω, there is a ν∈Γ such that Vp,ν⊆Vp,ω∩Lp(G,Γ). So the identity map I from (Lp(G,Ω),∥·∥p,ν) into (Lp(G,ω),∥·∥p,ω) is continuous. Since Lp(G,Γ) is dense in (Lp(G,ν),∥·∥p,ν), I can be extended continuously to a continuous linear mapping on Lp(G,ν). The extension map is again the identity map. So Lp(G,ν)⊆Lp(G,ω). Hence, there exists a constant c>0 such that ω(x)≤cν(x) locally almost everywhere; see Lemma 2.1 in [10]. This proves that Ω≤Γ.Let us recall the definition of the projective limit of a family of locally convex spaces. Let(Λ,≤) be a partially ordered set and {Xα:α∈Λ} be a family of locally convex spaces, and for α≤β, fα,β be a linear map from Xβ into Xα. Suppose that fαγ=fαβ∘fβγ for all α≤β≤γ and fαα be the identity map on Xα for all α∈Λ. Then, the projective limit of the family (Xα,fα,β) is defined as
(2.4)lim⁡α(Xα,fα,β)={(xα)∈∏α∈ΛXα:xα=fα,β(xβ),wheneverα≤β};
for more details see for example [11].Proposition 2.3.
LetΩ be a system of weights on G. Then (Lp(G,Ω),τΩ) is a complete space.Proof.
We note that for any two weightsω,ν∈Ω with ω≤ν, Lp(G,ν)⊆Lp(G,ω). Let the mapping Iω,ν:Lp(G,ν)→Lp(G,ω) be the canonical injection. Then, it is clear that (Lp(G,Ω),τΩ) is isomorphic to the projective limit system limω(Lp(G,ω),Iω,ν) of the Banach spaces (Lp(G,ω),∥·∥p,ω), ω∈Ω, and, hence, is complete; see Lemma 3.2.1 in [12].Proposition 2.4.
The locally convex space(Lp(G,Ω),τΩ) is normable if and only if the topology τΩ is generated by ∥·∥p,ω for some ω∈Ω.Proof.
IfLp(G,Ω) is normable, then it has a neighbourhood V of zero that is norm bounded with respect to ∥·∥p,ω for every ω∈Ω. Hence, there is ω′∈Ω so that Vp,ω′={f∈Lp(G,Ω):∥f∥p,ω′≤1} is norm bounded in the space (Lp(G,ν),∥·∥p,ν) for every ν∈Ω. This implies that there is a positive constant cν so that Vp,ω′⊆cνVp,ν, and our claim is proved. The converse is clear.


## 3. The Dual and Bidual ofLP(G,Ω),1≤p<∞

In this section we deal with the dual space of(Lp(G,Ω),τΩ) and, among other things, characterize its equicontinuous subsets.Theorem 3.1.
If1≤p<∞ and Bc(G)⊆Lp(G,Ω), then the dual space of (Lp(G,Ω),τΩ) is Ω·Lq(G):={ωf:ω∈Ω,f∈Lq(G)} with 1/p+1/q=1.Proof.
Leth∈Lq(G,1/ω). We define the linear functional F:Lp(G,Ω)→ℂ by F(f)=∫Gfhdλ, then F∈(Lp(G,Ω),τΩ)*.
Conversely, letF∈(Lp(G,Ω),τΩ)*. First, we know that Bc(G)⊆Lp(G,Ω)⊆Lp(G,ω) for every ω∈Ω. So there is a ν∈Ω such that |F(f)|≤1 whenever f∈{h∈Lp(G,Ω):∥h∥p,ν≤1}. As F is bounded in the intersection of the unit ball of (Lp(G,ν),∥·∥p,ν) with (Lp(G,Ω),∥·∥p,ν), F is continuous on Lp(G,Ω) with the topology induced by the norm ∥·∥p,ν. Since Lp(G,Ω) is dense in (Lp(G,ν),∥·∥p,ν), F can be extended continuously to a continuous linear form on Lp(G,ν) which we denote by F̃. Then, we have F̃∈(Lp(G,ν),∥·∥p,ν)*, and hence there is a unique h∈Lq(G,1/ν) so that
(3.1)F̃(f)=∫Gfhdλ(f∈Lp(G,ν));
therefore, we obtain the following isomorphism:
(3.2)Φ:⋃ω∈ΩLq(G,1ω)⟶(Lp(G,Ω),τΩ)*,
defined by Φ(h)=Fh, where Fh(f)=∫Gfhdλ for all f∈Lp(G,Ω).Lemma 3.2.
LetΩ be a system of weights on G. For every ω∈Ω, define the mapping Tω:Lp(G,Ω)→Lp(G) by Tω(f)=fω. Then, Vp,ω∘=Tω*(B∘) for ω∈Ω, where B is the closed unit ball of Lp(G) and B∘ denotes its polar.Proof.
It is clear thatTω is a well-defined continuous linear map. Also, Tω(Lp(G,Ω)) is dense in (Lp(G),∥·∥p). Therefore Tω* (the adjoint of Tω) is weak* continuous and one to one linear map from Lq(G) into Ω·Lq(G), where 1/p+1/q=1. Now, since B∘ is σ(Lq(G),Lp(G))-compact by the Alaoglu theorem and so Tω*(B∘) is σ(Ω·Lq(G),Lp(G,Ω))-compact, while Tω*(B∘) is obviously convex. So we find that
(3.3)Vp,ω={f∈Lp(G,Ω):Tω(f)∈B}=Tω-1(B)={f∈Lp(G,Ω):Tω(f)∈B∘∘}={f∈Lp(G,Ω):|Tω*(g)(f)|≤1,foreveryg∈B∘}=Tω*(B∘)∘.
Form which it follows that
(3.4)Vp,ω∘=Tω*(B∘)∘∘=Tω*(B∘).We have the following characterization of the equicontinuous subsets ofΩ·Lq(G).Theorem 3.3.
Let1≤p<∞ and M be a subset of (Lp(G,Ω),τΩ)*=Ω·Lq(G). The following are equivalent.(a)
M is τΩ-equicontinuous.(b)
There areω∈Ω and an equicontinuous subset M′ of (Lp(G),∥·∥p)*=Lq(G) so that M⊆ω·M′.(c)
There areω∈Ω and α>0 such that sup{∥f/ω∥q:f∈M}≤α<∞ whenever 1/p+1/q=1.Proof.
(a⇒b) By (a), there is ω∈Ω so that M⊆Vp,ω∘, where Vp,ω={f∈Lp(G,Ω):∥f∥p,ω≤1}. According to Lemma 3.2, we have Vp,ω∘=Tω*(B∘), where B is the closed unit ball of Lp(G). Hence M⊆ωB∘.
(b⇒c) There is α>0 so that M′⊆αB∘ by (b). So M⊆αωB∘, and supf∈M∥f/ω∥q≤α.
(c⇒a) If p=1, it is clear that
(3.5)M⊆{f∈Lp(G,Ω):∫G|f(x)|ω(x)dλ(x)≤1α}∘,
and if 1<p<∞, by Hölder’s inequality, for h∈M and
(3.6)f∈W={f∈Lp(G,Ω):‖f‖p,ω≤1α},
we have
(3.7)|∫Ghfdλ|≤∫G|hω||fω|dλ≤‖hω‖q‖fω‖p≤1.
Hence, M⊆W∘, and this guarantees that M is τΩ-equicontinuous in both cases.Proposition 3.4.
LetΩ be a system of weights on G. Then, the set of extreme points of Vp,ω∘ is the set {ωf:f∈Lq(G),∥f∥q=1} for 1<p<∞, and {f∈L∞(G):|f|=1l.a.e.} for p=∞.Proof.
Fixω∈Ω and let Tω:Lp(G,Ω)→Lp(G) be the map defined in Lemma 3.2. From Lemma 3.2, it follows that for any extreme point h of Vp,ω∘, there is an extreme point f of B∘ so that h=Tω*(f)=fω.
Conversely, letω∈Ω be arbitrary and let h=ωf, where f is an extreme point of B∘. Clearly, h∈Vp,ω∘, and if h=cg+(1-c)k for some g,k∈Vp,ω∘ and 0<c<1, then there are m,n∈B∘ such that Tω*(m)=g and Tω*(n)=k. Thus, Tω*(f)=h=Tω*(cm+(1-c)n) and since Tω* is one to one, f=cm+(1-c)n. However f is an extreme point of B∘, which implies that f=m=n, and hence h=g=k, that is, h is an extreme point of Vp,ω∘. Now the rest of the proof is easy to complete; see for example Section 2.14 in [13].Let us recall that a locally convex space(E,τ) is called semireflexive if (E,τ)**=E.Theorem 3.5.
LetΩ be a system of weights on G. Then (Lp(G,Ω),τΩ) is semireflexive.Proof.
IfF∈(Lp(G,Ω),τΩ)**, then the restriction of F to Lq(G,1/ω), for every ω∈Ω, belongs to Lq(G,1/ω)*, where Lq(G,1/ω) was considered with the induced strong topology on (Lp(G,Ω),τΩ)*. Now if {hα}α∈I⊆Lq(G,1/ω) and hα→h for some h∈Lq(G,1/ω) in the norm ∥·∥q,1/ω, then for every weakly bounded set A in Lp(G,Ω),
(3.8)∫Gf(hα-h)dλ⟶0uniformlyon⁡A.
This means thathα→h in the strong topology of (Lp(G,Ω),τΩ)*. Hence, for every ω∈Ω, there is a unique fω∈Lp(G,ω) so that
(3.9)F(h)=∫Gfωhdλon⁡Lq(G,1ω).
Now note that ifω,ν∈Ω with ω≤ν, then Lp(G,ν)⊆Lp(G,ω) and Lq(G,1/ω)⊆Lq(G,1/ν). Therefore for every h∈Lq(G,1/ω),
(3.10)∫Gfωhdλ=∫Gfνhdλ,
and hence fω=fν almost everywhere. This implies that
(3.11)F∈lim⁡ω(Lp(G,ω),Iω,ν)=Lp(G,Ω).
Conversely, iff∈Lp(G,Ω), then it is obvious that the linear form
(3.12)F(h)=∫Gfhdλ(h∈(Lp(G,Ω),τΩ)*)
is continuous with respect to the strong topology on (Lp(G,Ω),τΩ)*. So the canonical imbedding J:Lp(G,Ω)→(Lp(G,Ω),τΩ)** is onto. Hence Lp(G,Ω) is semireflexive.


## 4.LP(G,Ω) As a Topological Algebra

In this section, we study conditions on a system of weightsΩ for that Lp(G,Ω) with the convolution multiplication to be a topological algebra. We commence with some definitions.Iff is a function on G, the left translate of f by x∈G is the function given by Lxf(y)=f(x-1y). A subset ℱ of functions on G is called left translation invariant if Lxf∈ℱ for all f∈ℱ and x∈G.A weight functionω on a locally compact group G is called left moderate if (4.1)l(s):=esssup⁡tω(st)ω(t)<∞,
for all s∈G. It is easy to see that ℓ(s)>0, ℓ(st)≤ℓ(s)ℓ(t); see [4] or [9]. Let us remark that any submultiplicative and any locally integrable left moderate measurable function is bounded and bounded away from zero on any compact subset of G; see Theorem 2.7 in [10]. In particular, ℓ is bounded on compact sets. The condition that ω is left moderate is equivalent to that the space Lp(G,ω) (for 1≤p≤∞) being translation invariant; see for more details [4]. Observe that for f∈Lp(G,ω) and x∈G,(4.2)‖Lxf‖p,ω=(∫G(|f(x-1t)|ω(t))pdλ(t))1/p=(∫G(|f(t)|ω(xt))pdλ(t))1/p≤(∫G(|f(t)|l(x)ω(t))pdλ(t))1/p=l(x)‖f‖p,ω.Lemma 4.1.
LetΩ be a system of weights on G. Then Lp(G,Ω) is left translation invariant if and only if every element of Ω is left moderate.Proof.
The “if” part is clear by the remarks above. For the converse, we need only to note thatLp(G,Ω) is dense in (Lp(G,ω),∥·∥p,ω) for ω∈Ω.Theorem 4.2.
LetΩ be a system of locally integrable left moderate weights on G and f∈Lp(G,Ω). Then, the map x↦Lxf from G into (Lp(G,Ω),τΩ) is continuous.Proof.
Assume first thatf∈Bc(G) with K=supp(f). Let x∈G, ω∈Ω, and (xα) be a net in G convergent to x. Choose a compact neighbourhood U of x, then supp(Lxf)⊆UK whenever x∈U. Let
(4.3)k=sup⁡{ω(s):s∈UK}<∞.
Choose α0 such that xα∈U for all α≤α0 and ∥Lxαf-Lxf∥p≤ϵ/k. Then
(4.4)‖Lxαf-Lxf‖p,ω=(∫UF(|f(xα-1t)-f(x-1t)|ω(t))pdλ(t))1/p≤k(∫UF(|f(xα-1t)-f(x-1t)|pdλ(t))1/p=k‖Lxαf-Lxf‖p≤ϵ,
for all α≥α0.
Finally, letf be an arbitrary element of Lp(G,Ω) and ϵ>0. Let M be an upper bound for the function ℓ on the compact neighbourhood U of x; recall that ℓ is submultiplicative. For every ω∈Ω, there exists gω∈Bc(G) such that ∥f-gω∥p,ω≤ϵ/3M. By the first part, we can choose α0 such that
(4.5)‖Lxαgω-Lxgω‖p,ω≤ϵ3,xα∈U,
for all α≥α0. One can conclude that
(4.6)‖Lxαf-Lxf‖p,ω≤‖Lxαf-Lxαgω‖p,ω+‖Lxαgω-Lxgω‖p,ω+‖Lxgω-Lxf‖p,ω≤l(xα)ϵ+ϵ/3+l(x)ϵ≤Mϵ3M+ϵ3+Mϵ3M=ϵ,
for all α≥α0. This finishes the proof.We now focus on some systems of weights for thatLp(G,Ω) to be an algebra under usual convolution (4.7)f*g(t)=∫Gf(s)g(s-1t)dλ(s)(f,g∈Lp(G,Ω)),
whenever this integral makes sense. For p=1, it is well known that L1(G,ω) is a convolution algebra if and only if ω is weakly submultiplicative; that is, for all x,y∈G, (4.8)ω(st)≤cω(s)ω(t),
for some c>0.For any two weight functionsω and ν on G, we set (4.9)Φ[ω,ν](x)=∫G(ω(x)ν(y)ν(y-1x))qdλ(y).
In the case where ω=ν, we simply write Φω=Φ[ω,ω].The following lemma is similar to Lemma 2.2 in [9].Lemma 4.3.
Let1<p<∞ and Ω be a system of weights on G. If Lp(G,Ω) is a convolution algebra, then ωp is locally integrable for each ω∈Ω.The next result gives a sufficient condition for thatLp(G,Ω) to be a convolution algebra.Theorem 4.4.
Let1<p<∞ and Ω be a system of weights on G. If for every ω∈Ω, there is a ν∈Ω such that Φ[ω,ν]∈L∞(G), where q is the conjugate exponent to p, then the space (Lp(G,Ω),τΩ) is a complete locally convex algebra with continuous multiplication.Proof.
We must show that(4.10)‖f*g‖p,ω≤‖f‖p,ν‖g‖p,ν
for all f,g∈Lp(G,Ω). By Lemma 4.3, Bc(G) is dense in (Lp(G,Ω),∥·∥p,ω), thus for any ω∈Ω, it suffices to show that
(4.11)‖f*g‖p,ω≤‖f‖p,ν‖g‖p,ν,
for all f,g∈Bc(G). For this, let f,g∈Bc(G). Writing
(4.12)f*g(x)=∫Gf(y)g(y-1x)ν(y)ν(y-1x)ν(y)ν(y-1x)dλ(y),
and using Hölder's inequality, we obtain
(4.13)|f*g(x)|≤(∫G(|f(y)|ν(y))p(|g(y-1x)|ν(y-1x))pdλ(y))1/p(∫G(1ν(y)ν(y-1x))qdλ(y))1/q.
This shows that
(4.14)|∫G(|f*g(x)|ω(x))pdλ(x)|≤∫G(∫G(|f(y)|ν(y))p(|g(y-1x)|ν(y-1x))pdλ(y))Φ[ω,ν](x)p/qdλ(x)≤‖f‖p,νp‖g‖p,νp‖Φ[ω,ν]‖∞p/q.
Whence
(4.15)‖f*g‖p,ω≤‖Φ[ω,ν]‖∞1/q‖f‖p,ν‖g‖p,ν.
This completes the proof.The following corollary is a direct consequence of Theorem4.4.Corollary 4.5.
Let1<p<∞ and Ω be a system of weights on G such that for every ω∈Ω, Φω∈L∞(G). Then Lp(G,Ω) is a complete locally multiplicative convex algebra.The next result provides us with a class of weightsω on the additive group ℝn for which the usual weighted Lebesgue space Lp(ℝn,ω) becomes a Banach algebra.Proposition 4.6.
Let1<p<∞ and n be a natural number. Let ϖ:ℝn→(0,+∞) be a function such that
(i)
If∥x∥≤∥y∥, then ϖ(x)≤ϖ(y).(ii)
ϖ-1∈L1(ℝn).(iii)
There exists a positive numberM such that ϖ(2x)≤Mϖ(x) for all x∈ℝn.
ThenLp(ℝn,ϖq) is a Banach algebra, where q is the conjugate exponent to p.Proof.
For anyx∈ℝn, let Ax={y∈ℝn:2∥y∥≥∥x∥} and observe that
(4.16)ϖ(y)≥ϖ(x2),ify∈Ax,ϖ(x-y)≥ϖ(x2),ify∈Rn∖Ax.
Hence, forx∈ℝn,
(4.17)Φϖq(x)=∫Rnϖ(x)ϖ(y)ϖ(x-y)dy=∫Rn∖Axϖ(x)ϖ(y)ϖ(x-y)dy+∫Axϖ(x)ϖ(y)ϖ(x-y)dy≤(ϖ(x)ϖ(x/2))(∫Rn∖Ax1ϖ(y)dy+∫Ax1ϖ(x-y)dy)≤2M‖ϖ‖1.
Thus,Φϖq∈L∞(ℝn), and now the result follows from Corollary 4.5.Example 4.7.
Let1≤p<∞, q be the conjugate exponent to p, and n∈ℕ. Set
(4.18)ω(x)=(a+‖x‖r)s/qb(1/q)ln⁡(c+‖x‖t)(x∈Rn),
where sr>n,b>1 and a,c,t>0. Then Lp(ℝn,ω) is a Banach algebra.We are going to prove the converse of Theorem4.4. For this, we fix some notation. If f,g be two complex-valued functions on G, then f⊗g denotes the function on G×G given by f⊗g(x,y)=f(x)g(y) for all x,y∈G. Also for any two sets ℱ and 𝒦 of functions on G we set ℱ⊗𝒦={f⊗g:f∈ℱ,g∈𝒦}. For a locally compact group G, note that the cartesian product G×G is a locally compact group by defining the product (x,y)(s,t)=(xs,yt) for all x,y,s,t∈G.We need the following easy lemma in the sequel.Lemma 4.8.
Let1<p<∞ and Ω be a system of weights on G such that Bc(G)⊆Lp(G,Ω). Then Bc(G)⊗Bc(G) is dense in (Lp(G×G,ω⊗ω),∥·∥p,ω⊗ω).Proof.
SinceBc(G) is norm dense in Lp(G,ω), then Bc(G)⊗Bc(G) is projective tensor norm dense in Lp(G,ω)⊗̂πLp(G,ω), where ⊗̂ is the projective tensor product. Hence Lp(G,ω)⊗Lp(G,ω) is π-dense in Lp(G,ω)⊗̂πLp(G,ω). On the other hand, it is known that Lp(G,ω)⊗̂πLp(G,ω) is isometric with (Lp(G×G,ω⊗ω),∥·∥p,ω⊗ω). In fact, the linear map
(4.19)ϱ:Lp(G,ω)⊗̂πLp(G,ω)⟶Lp(G×G,ω⊗ω),ϱ(f⊗g)(x,y)=f(x)g(y)
for all f,g∈Lp(G,ω) and x,y∈G, can be extended to a surjective isometry; for more details, see for example [14]. Now we conclude that Bc(G)⊗Bc(G) is ∥·∥p,ω⊗ω-dense in Lp(G×G,ω⊗ω).The next theorem is our main result in this section.Theorem 4.9.
Let1<p<∞, G be σ-compact, and Ω be a system of weights on G. If the space (Lp(G,Ω),τΩ) is an algebra with continuous multiplication, then for every ω∈Ω there exists a ν∈Ω such that Φ[ω,ν]∈L∞(G).Proof.
Choose an arbitraryω∈Ω. Then, by assumption, there exists some ν∈Ω such that for every f,g∈Lp(G,Ω), ∥f*g∥p,ω≤∥f∥p,ν∥g∥p,ν. Now for every h∈Lq(G,1/ω),
(4.20)F(f)=∫Gf(x)h(x)dλ(x)(f∈Bc(G))
defines a continuous linear functional on Bc(G) with the norm ∥F∥=∥h∥q,1/ω. Also for every f,g∈Bc(G), f*g∈Bc(G), and we have
(4.21)F(f*g)=∫Gf*g(x)h(x)dλ(x)=∫G(∫Gf(y)g(y-1x)dλ(y))h(x)dλ(x)=∫G∫Gf(y)g(x)h(yx)dλ(x)dλ(y)<∞.
SetF(f⊗g)=F(f*g)=∫G×Gf(y)g(x)h(yx)dλ×λ(x,y) for f,g∈Bc(G). By Lemma 4.8, F can be extended to a ∥·∥p.ν×ν-continuous functional on Lp(G×G,ν⊗ν). Since G is σ-compact, by Exercise 15.14 in [1], it follows that the function (x,y)↦h(yx) belongs to Lq(G×G,1/(ν⊗ν)). But
(4.22)∫G∫G(h(yx)ν(x)ν(y))qdλ(x)dλ(y)=∫G∫G(h(x)ν(y)ν(y-1x))qdλ(x)dλ(y)=∫G(h(x)ω(x))qΦ[ω,ν](x)dλ(x)<∞.
Since (h/ω)q∈L1(G) is arbitrary, we conclude that Φ[ω,ν]∈L∞(G); see Section 14 in [15] or Theorem  20.15 in [1].As an immediate consequence of Theorem4.9, we obtain the following corollary that partially answers a question raised in [3].Corollary 4.10.
Letω be a weight on σ-compact group G and 1<p<∞. Then Lp(G,ω) is a convolution algebra if and only if Φω∈L∞(G).


## 5. Ideals and the Spectrum of the AlgebraLP(G,Ω)

We commence this section with the following proposition.Proposition 5.1.
Let1≤p<∞ and let Lp(G,Ω) be a translation invariant algebra. Then(i)
(Lp(G,Ω),τΩ) has an approximate identity.(ii)
(Lp(G,Ω),τΩ) has a bounded approximate identity or an identity if and only if G is discrete.Proof.
(i) LetU be a fixed relatively compact neighbourhood of the identity element e, and let 𝒰 be the family of all neighbourhoods of e contained in U directed by reverse inclusion. Set eV∶=χV/λ(V), and note that since elements of Ω are locally integrable, eV∈Lp(G,Ω). Given ϵ>0 and ω∈Ω, then by Theorem 4.2, there exists a neighbourhood W of the identity such that ∥f-Ltf∥p,ω<ϵ for t∈W. Now, for V∈𝒰 with V⊆W, and g∈Lq(G,1/ω), we have
(5.1)|〈eV*f-f,g〉|=|∫G(eV*f-f)(x)g(x)dλ(x)|≤∫G∫V|f(t-1x)-f(x)|λ(V)dλ(t)|g(x)|dλ(x)≤1λ(V)∫V〈|Ltf-f|,|g|〉dλ(t)≤sup⁡t∈V⁡‖Ltf-f‖p,ω‖g‖q,1/ω<ϵ‖g‖q,1/ω.
Hence, ∥eV*f-f∥p,ω≤ϵ for all neighborhoods V⊆W, from which it follows that eV*f→f in τΩ-topology.
(ii) Let(eα)α be a bounded left approximate identity for Lp(G,Ω). Fix an ω∈Ω, then ∥eα∥p,ω≤M for some positive number M. Let f∈Lp(G,ω). Since Lp(G,Ω) is dense in Lp(G,ω) with the norm ∥·∥p,ω, then given ϵ>0, there exists g∈Lp(G,Ω) such that ∥f-g∥p,ω≤ϵ/3(M+1). Choose α0 such that ∥eα*g-g∥p,ω≤ϵ/3 for all α≥α0. Then it follows that
(5.2)‖eα*f-f‖p,ω≤‖eα*f-eα*g‖p,ω+‖eα*g-g‖p,ω+‖f-g‖p,ω≤Mϵ3(M+1)+ϵ3+ϵ3(M+1)<ϵ,
for all α≥α0. This means that (Lp(G,ω),∥·∥p,ω has a bounded left approximate identity. But according to Theorem 4.2 in [9], this is equivalent to that G is discrete.The next theorem shows that closed ideals of the algebra(Lp(G,Ω),τΩ) are exactly translation invariant subspaces.Theorem 5.2.
Let1≤p<∞ and Lp(G,Ω) be a translation invariant algebra. Then a closed linear subspace of Lp(G,Ω) is an ideal in Lp(G,Ω) if and only if it is two-sided translation invariant.Proof.
Suppose thatI is a τΩ-closed two-sided translation invariant subspace of Lp(G,Ω). We have to show that g*f∈I and f*g∈I for all f∈I and g∈Lp(G,Ω). Let h∈Lq(G,1/ω), for some ω∈G, such that ∫Gf(x)h(x)dλ(x)=0 for all f∈I. Then, for f∈I and any g∈Lp(G,Ω),
(5.3)∫G(g*f)(x)h(x)dλ(x)=∫Gh(x)(∫Gg(y)f(y-1x)dλ(y))dλ(x)=∫Gg(y)(∫GLyf(x)h(x)dλ(x))dλ(y)=0.
Since (Lp(G,Ω),τΩ)*=Ω·Lq(G), the Hahn-Banach theorem implies that g*f∈I for all f∈I and g∈Lp(G,Ω). Thus I is a left ideal, and using the right translation invariance of I, it is readily seen, in the same way, that I is also a right ideal.
Conversely, letI be a closed ideal of (Lp(G,Ω),τΩ), and x∈G. Let (eα) be an approximate identity for Lp(G,Ω). Then for each f∈Lp(G,Ω), we have
(5.4)‖Lx(eα)*f-Lxf‖p,ω≤l(x)‖eα*f-f‖p,ω⟶0.
Hence, Lx(eα)*f→Lxf in τΩ-topology. As I is a τΩ-closed left ideal, it follows that Lxf∈I; that is, I is left translation invariant. Similarly, it is shown that I is also right translation invariant.We denote byΔ(Lp(G,Ω)) the spectrum of (Lp(G,Ω),τΩ) consisting of all τΩ-continuous nonzero linear functionals Φ on Lp(G,Ω) which are multiplicative; that is,(5.5)Φ(f*g)=Φ(f)Φ(g)(f,g∈Lp(G,Ω)).We conclude this work with the following result which is a characterization of the spectrum of(Lp(G,Ω),τΩ).Proposition 5.3.
LetΩ be a system of weights on 6-compact group G. Then
(5.6)Δ(Lp(G,Ω))={Φρ:ρ∈Lq(G,1ω),ω∈Ω,ρ(xy)=ρ(x)ρ(y)},
where
(5.7)Φρ(f)=∫Gf(x)ρ(x)dλ(x)(f∈Lp(G,Ω)).Proof.
Letρ∈Lq(G,1/ω) for some ω∈Ω such that ρ(xy)=ρ(x)ρ(y) for almost all x,y∈G. Then, Φρ is ∥·∥p,ω-continuous and so τΩ-continuous. Moreover, for f,g∈Lp(G,Ω),
(5.8)Φρ(f*g)=∫G∫Gf(x)g(y)ρ(xy)dλ(x)dλ(y)=∫G∫Gf(x)g(y)ρ(x)ρ(y)dλ(x)dλ(y)=Φρ(f)Φρ(g).
That is, Φρ∈Δ(Lp(G,Ω)).
Conversely, letΦ∈Δ(Lp(G,Ω)). Then Φ is bounded on a τΩ-neighbourhood of zero. Thus Φ is bounded on the set {f∈Lp(G,Ω):∥f∥p,ω<1}∩Lp(G,ω) for some ω∈Ω. Therefore Φ can be extended to an element Φ¯ in (Lp(G,ω),∥·∥p,ω)*. It follows that there exists a function ρ∈Lq(G,1/ω) such that
(5.9)Φ¯(f)=∫Gfρdλ,
for all f∈Lp(G,ω). Since for f,g∈Bc(G), Φ(f)Φ(g)=Φ(f*g), we infer that
(5.10)∫G×Gf(y)g(x)ρ(y)ρ(x)dλ×λ(x,y)=∫G∫Gf(y)g(x)ρ(y)ρ(x)dλ(y)dλ(x)=∫G∫Gf(y)g(x)ρ(yx)dλ(y)dλ(x)=∫G×Gf(y)g(x)ρ(yx)dλ×λ(x,y).
By an argument similar to the proof of Theorem 4.9, we deduce that ρ(xy)=ρ(x)ρ(y) for almost all x,y∈G.



---

*Source: 947908-2011-10-19.xml*