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Twisted Orlicz algebras

Ebrahim SameiUniversity of Saskatchewan, Canada.

A joint work with Serap Oztop

Banach algebras and applicationsOulu, Finland

July 5, 2017

Ebrahim SameiUniversity of Saskatchewan, Canada. (U of S)Twisted Orlicz algebras CAHAS 2017 1 / 13

Orlicz spaces

A nonzero function Φ : [0,∞)→ [0,∞] is called a Young function if Φ isconvex, Φ(0) = 0, and limx→∞ Φ(x) =∞. The complementary Young functionΨ of Φ is given by

Ψ(y) = supxy − Φ(x) : x ≥ 0 (y ≥ 0).

For a locally compact group G and a Young function Φ, the Orlicz space LΦ(G )is defined by

LΦ(G ) =

f : G → C :

∫G

Φ(α|f (s)|) ds <∞ for some α > 0.

There are two equivalent norms on LΦ(G ):

‖f ‖Φ = sup∫

G

|f (s)v(s)| ds :

∫G

Ψ(|v(s)|) ds ≤ 1,

where Ψ is a complementary function of Φ, and

NΦ(f ) = infk > 0 :

∫G

Φ

(|f (s)|k

)ds ≤ 1

.


Orlicz spacesA nonzero function Φ : [0,∞)→ [0,∞] is called a Young function if Φ isconvex, Φ(0) = 0, and limx→∞ Φ(x) =∞.

The complementary Young functionΨ of Φ is given by

Ψ(y) = supxy − Φ(x) : x ≥ 0 (y ≥ 0).


LΦ(G ) =

f : G → C :

∫G



‖f ‖Φ = sup∫

G

|f (s)v(s)| ds :

∫G

Ψ(|v(s)|) ds ≤ 1,


NΦ(f ) = infk > 0 :

∫G

Φ

(|f (s)|k

)ds ≤ 1

.


Orlicz spacesA nonzero function Φ : [0,∞)→ [0,∞] is called a Young function if Φ isconvex, Φ(0) = 0, and limx→∞ Φ(x) =∞. The complementary Young functionΨ of Φ is given by

Ψ(y) = supxy − Φ(x) : x ≥ 0 (y ≥ 0).


LΦ(G ) =

f : G → C :

∫G



‖f ‖Φ = sup∫

G

|f (s)v(s)| ds :

∫G

Ψ(|v(s)|) ds ≤ 1,


NΦ(f ) = infk > 0 :

∫G

Φ

(|f (s)|k

)ds ≤ 1

.



Ψ(y) = supxy − Φ(x) : x ≥ 0 (y ≥ 0).


LΦ(G ) =

f : G → C :

∫G



‖f ‖Φ = sup∫

G

|f (s)v(s)| ds :

∫G

Ψ(|v(s)|) ds ≤ 1,

where Ψ is a complementary function of Φ,

and

NΦ(f ) = infk > 0 :

∫G

Φ

(|f (s)|k

)ds ≤ 1

.



Ψ(y) = supxy − Φ(x) : x ≥ 0 (y ≥ 0).


LΦ(G ) =

f : G → C :

∫G



‖f ‖Φ = sup∫

G

|f (s)v(s)| ds :

∫G

Ψ(|v(s)|) ds ≤ 1,


NΦ(f ) = infk > 0 :

∫G

Φ

(|f (s)|k

)ds ≤ 1

.


Example

(1) The complementary Young function of Φ(x) = x is

Ψ(y) =

0 if 0 ≤ y ≤ 1,∞ otherwise.

In this case, LΦ(G ) = L1(G ) and LΨ(G ) = L∞(G ).(2) For 1 < p, q <∞ with 1

p + 1q = 1, if Φ(x) = xp

p , then Ψ(y) = yq

q . In thiscase, LΦ(G ) = Lp(G ) and LΨ(G ) = Lq(G ).(3) If Φ(x) = x ln(1 + x), then Ψ(x) ≈ cosh x − 1.(4) If Φ(x) = ex − x − 1, then Ψ(x) = (1 + x) ln(1 + x)− x .(5) Suppose that ϕ : [0,∞)→ [0,∞) is a continuous strictly increasing functionwith ϕ(0) = 0 and limx→∞ ϕ(x) =∞. Then

Φ(x) =

∫ x

0ϕ(y)dy and Ψ(y) =

∫ y

0ϕ−1(x)dx

are a complementary pair of continuous strictly increasing Young functions.


Example


Ψ(y) =


In this case, LΦ(G ) = L1(G ) and LΨ(G ) = L∞(G ).

(2) For 1 < p, q <∞ with 1p + 1

q = 1, if Φ(x) = xp

p , then Ψ(y) = yq


Φ(x) =

∫ x


∫ y

0ϕ−1(x)dx



Example


Ψ(y) =



p + 1q = 1, if Φ(x) = xp

p , then Ψ(y) = yq

q .

In thiscase, LΦ(G ) = Lp(G ) and LΨ(G ) = Lq(G ).(3) If Φ(x) = x ln(1 + x), then Ψ(x) ≈ cosh x − 1.(4) If Φ(x) = ex − x − 1, then Ψ(x) = (1 + x) ln(1 + x)− x .(5) Suppose that ϕ : [0,∞)→ [0,∞) is a continuous strictly increasing functionwith ϕ(0) = 0 and limx→∞ ϕ(x) =∞. Then

Φ(x) =

∫ x


∫ y

0ϕ−1(x)dx



Example


Ψ(y) =



p + 1q = 1, if Φ(x) = xp

p , then Ψ(y) = yq

q . In thiscase, LΦ(G ) = Lp(G ) and LΨ(G ) = Lq(G ).

(3) If Φ(x) = x ln(1 + x), then Ψ(x) ≈ cosh x − 1.(4) If Φ(x) = ex − x − 1, then Ψ(x) = (1 + x) ln(1 + x)− x .(5) Suppose that ϕ : [0,∞)→ [0,∞) is a continuous strictly increasing functionwith ϕ(0) = 0 and limx→∞ ϕ(x) =∞. Then

Φ(x) =

∫ x


∫ y

0ϕ−1(x)dx



Example


Ψ(y) =



p + 1q = 1, if Φ(x) = xp

p , then Ψ(y) = yq

q . In thiscase, LΦ(G ) = Lp(G ) and LΨ(G ) = Lq(G ).(3) If Φ(x) = x ln(1 + x), then Ψ(x) ≈ cosh x − 1.

(4) If Φ(x) = ex − x − 1, then Ψ(x) = (1 + x) ln(1 + x)− x .(5) Suppose that ϕ : [0,∞)→ [0,∞) is a continuous strictly increasing functionwith ϕ(0) = 0 and limx→∞ ϕ(x) =∞. Then

Φ(x) =

∫ x


∫ y

0ϕ−1(x)dx



Example


Ψ(y) =



p + 1q = 1, if Φ(x) = xp

p , then Ψ(y) = yq

q . In thiscase, LΦ(G ) = Lp(G ) and LΨ(G ) = Lq(G ).(3) If Φ(x) = x ln(1 + x), then Ψ(x) ≈ cosh x − 1.(4) If Φ(x) = ex − x − 1, then Ψ(x) = (1 + x) ln(1 + x)− x .

(5) Suppose that ϕ : [0,∞)→ [0,∞) is a continuous strictly increasing functionwith ϕ(0) = 0 and limx→∞ ϕ(x) =∞. Then

Φ(x) =

∫ x


∫ y

0ϕ−1(x)dx



Example


Ψ(y) =



p + 1q = 1, if Φ(x) = xp

p , then Ψ(y) = yq

q . In thiscase, LΦ(G ) = Lp(G ) and LΨ(G ) = Lq(G ).(3) If Φ(x) = x ln(1 + x), then Ψ(x) ≈ cosh x − 1.(4) If Φ(x) = ex − x − 1, then Ψ(x) = (1 + x) ln(1 + x)− x .(5) Suppose that ϕ : [0,∞)→ [0,∞) is a continuous strictly increasing functionwith ϕ(0) = 0 and limx→∞ ϕ(x) =∞.

Then

Φ(x) =

∫ x


∫ y

0ϕ−1(x)dx



Example


Ψ(y) =



p + 1q = 1, if Φ(x) = xp

p , then Ψ(y) = yq


Φ(x) =

∫ x


∫ y

0ϕ−1(x)dx



Definition

Let SΦ(G ) be the closure of the linear space of all step functions in LΦ(G ).

Fact 1: SΦ(G ) = LΦ(G ) if Φ satisfies the ∆2-condition , i.e. there exist aconstant K > 0 such that

Φ(2x) ≤ KΦ(x) for all x ≥ 0.

Fact 2: SΦ(G ) contains Cc(G ) as a dense subspace and SΦ(G )∗ ' LΨ(G ) ifeither Φ is continuous and strictly increasing or G is discrete.

We assume that (Φ,Ψ) is a pair of complementary Young function with Φ beingcontinuous and strictly increasing on [0,∞).


Definition


Fact 1:

SΦ(G ) = LΦ(G ) if Φ satisfies the ∆2-condition , i.e. there exist aconstant K > 0 such that





Definition


Fact 1: SΦ(G ) = LΦ(G ) if Φ satisfies the ∆2-condition

, i.e. there exist aconstant K > 0 such that





Definition







Definition




Fact 2:

SΦ(G ) contains Cc(G ) as a dense subspace and SΦ(G )∗ ' LΨ(G ) ifeither Φ is continuous and strictly increasing or G is discrete.



Definition




Fact 2: SΦ(G ) contains Cc(G ) as a dense subspace and SΦ(G )∗ ' LΨ(G ) ifeither

Φ is continuous and strictly increasing or G is discrete.



Definition




Fact 2: SΦ(G ) contains Cc(G ) as a dense subspace and SΦ(G )∗ ' LΨ(G ) ifeither Φ is continuous and strictly increasing

or G is discrete.



Definition







Weighted Lp algebras

It is known that Lp(G ) is a Banach algebra under convolution iff G is compact(S. Saeki-1990).

However, adding a weight can help.

DefinitionA weight on G is a locally integrable measurable function ω : G → R+ withω(e) = 1, 1/ω ∈ L∞(G ), and such that

Ω(s, t) =ω(st)

ω(s)ω(t)∈ L∞(G × G ).

For a weight ω on G , define the weighted Lp(G ) space

Lpω(G ) := f : f ω ∈ Lp(G ) with the norm ‖f ‖p,ω := ‖f ω‖p.

Theorem (Wermer 1954-Kuznetsova 2006)

Lpω(G ) is a Banach algebra with respect to the convolution if there is C > 0 with

ω−q ∗ ω−q ≤ Cω−q a.e.on G .


Weighted Lp algebras

It is known that Lp(G ) is a Banach algebra under convolution iff G is compact(S. Saeki-1990). However, adding a weight can help.

DefinitionA weight on G is a locally integrable measurable function ω : G → R+ withω(e) = 1, 1/ω ∈ L∞(G ), and such that

Ω(s, t) =ω(st)

ω(s)ω(t)∈ L∞(G × G ).

For a weight ω on G , define the weighted Lp(G ) space

Lpω(G ) := f : f ω ∈ Lp(G ) with the norm ‖f ‖p,ω := ‖f ω‖p.

Theorem (Wermer 1954-Kuznetsova 2006)

Lpω(G ) is a Banach algebra with respect to the convolution if there is C > 0 with

ω−q ∗ ω−q ≤ Cω−q a.e.on G .


Weighted LΦ algebras?

Main Obstacle:The relation (xy)p ≈ xpyp may not hold for a general Youngfunction!(A. Osanclıol and S. Oztop-2015) One can obtain LΦ

ω(G ) algebra under theassumption that LΦ

ω(G ) ⊆ L1ω(G ).

Alternative path:Consider the mapping

LΦω(G )→ LΦ(G ) , f 7→ f ω

and defined the product as

(f ~ g)(t) :=

∫G

f (s)g(s−1t)Ω(s, s−1t)ds,

whereΩ(s, t) =

ω(st)

ω(s)ω(t).



Main Obstacle:

The relation (xy)p ≈ xpyp may not hold for a general Youngfunction!(A. Osanclıol and S. Oztop-2015) One can obtain LΦ


ω(G ) ⊆ L1ω(G ).


LΦω(G )→ LΦ(G ) , f 7→ f ω


(f ~ g)(t) :=

∫G


whereΩ(s, t) =

ω(st)

ω(s)ω(t).



Main Obstacle:The relation (xy)p ≈ xpyp may not hold for a general Youngfunction!

(A. Osanclıol and S. Oztop-2015) One can obtain LΦω(G ) algebra under the

assumption that LΦω(G ) ⊆ L1

ω(G ).Alternative path:Consider the mapping

LΦω(G )→ LΦ(G ) , f 7→ f ω


(f ~ g)(t) :=

∫G


whereΩ(s, t) =

ω(st)

ω(s)ω(t).





ω(G ) ⊆ L1ω(G ).


LΦω(G )→ LΦ(G ) , f 7→ f ω


(f ~ g)(t) :=

∫G


whereΩ(s, t) =

ω(st)

ω(s)ω(t).





ω(G ) ⊆ L1ω(G ).

Alternative path:

Consider the mapping

LΦω(G )→ LΦ(G ) , f 7→ f ω


(f ~ g)(t) :=

∫G


whereΩ(s, t) =

ω(st)

ω(s)ω(t).





ω(G ) ⊆ L1ω(G ).


LΦω(G )→ LΦ(G ) , f 7→ f ω


(f ~ g)(t) :=

∫G


where

Ω(s, t) =ω(st)

ω(s)ω(t).





ω(G ) ⊆ L1ω(G ).


LΦω(G )→ LΦ(G ) , f 7→ f ω


(f ~ g)(t) :=

∫G


whereΩ(s, t) =

ω(st)

ω(s)ω(t).


2-cocycles

Definition

Let G be locally compact groups. A (normalized) 2-cocycle on G with values inC∗ := C \ 0 is a Borel measurable map Ω : G × G → C∗ such that

Ω(r , s)Ω(rs, t) = Ω(s, t)Ω(r , st) (r , s, t ∈ G )

andΩ(r , eG ) = Ω(eG , r) = 1 (r ∈ G ).

The set of all normalized 2-cocycles will be denoted by Z2(G ,C∗).

Since C∗ = R+T as a (pointwise) direct product of groups, any 2-cocycle Ω on Gwith values in C∗ can be (uniquely) written as Ω(s, t) = |Ω(s, t)|e iθ for some0 ≤ θ < 2π. Therefore, if we put

|Ω|(s, t) := |Ω(s, t)| and ΩT(s, t) := e iθ,

then Ω = |Ω|ΩT (in a unique way) and the mappings |Ω| and ΩT are 2-cocycleson G with values in R+ and T, respectively.


2-cocycles

Definition



andΩ(r , eG ) = Ω(eG , r) = 1 (r ∈ G ).


Since C∗ = R+T as a (pointwise) direct product of groups, any 2-cocycle Ω on Gwith values in C∗ can be (uniquely) written as Ω(s, t) = |Ω(s, t)|e iθ for some0 ≤ θ < 2π.

Therefore, if we put




2-cocycles

Definition



andΩ(r , eG ) = Ω(eG , r) = 1 (r ∈ G ).


Since C∗ = R+T as a (pointwise) direct product of groups, any 2-cocycle Ω on Gwith values in C∗ can be (uniquely) written as Ω(s, t) = |Ω(s, t)|e iθ for some0 ≤ θ < 2π. Therefore, if we put




Definition

We denote Z2b (G ,C∗) to be the group of bounded 2-cocycles on G with

values in C∗ which consists of all elements Ω ∈ Z2(G ,C∗) satisfying:

(i) Ω ∈ L∞(G × G );(ii) ΩT is continuous.We also define Z2

bw (G ,C∗) to be consisting of elements Ω ∈ Z2b (G ,C∗) for which

|Ω|(s, t) =ω(st)

ω(s)ω(t)(s, t ∈ G ),

where ω : G → R+ is a weight on G . We say that |Ω| is the 2-coboundarydetermined by ω, or alternatively, ω is the weight associated to |Ω|.

Theorem

For G amenable, Z2b (G ,C∗) = Z2

bw (G ,C∗).


Definition


values in C∗ which consists of all elements Ω ∈ Z2(G ,C∗) satisfying:(i) Ω ∈ L∞(G × G );

(ii) ΩT is continuous.We also define Z2


|Ω|(s, t) =ω(st)

ω(s)ω(t)(s, t ∈ G ),


Theorem


bw (G ,C∗).


Definition


values in C∗ which consists of all elements Ω ∈ Z2(G ,C∗) satisfying:(i) Ω ∈ L∞(G × G );(ii) ΩT is continuous.

We also define Z2bw (G ,C∗) to be consisting of elements Ω ∈ Z2

b (G ,C∗) for which

|Ω|(s, t) =ω(st)

ω(s)ω(t)(s, t ∈ G ),


Theorem


bw (G ,C∗).


Definition


values in C∗ which consists of all elements Ω ∈ Z2(G ,C∗) satisfying:(i) Ω ∈ L∞(G × G );(ii) ΩT is continuous.We also define Z2


|Ω|(s, t) =ω(st)

ω(s)ω(t)(s, t ∈ G ),

where ω : G → R+ is a weight on G .

We say that |Ω| is the 2-coboundarydetermined by ω, or alternatively, ω is the weight associated to |Ω|.

Theorem


bw (G ,C∗).


Definition


values in C∗ which consists of all elements Ω ∈ Z2(G ,C∗) satisfying:(i) Ω ∈ L∞(G × G );(ii) ΩT is continuous.We also define Z2


|Ω|(s, t) =ω(st)

ω(s)ω(t)(s, t ∈ G ),


Theorem


bw (G ,C∗).



Suppose that Ω ∈ Z2b (G ,C∗) and f and g are measurable functions on G .

Ifthere is a σ-finite measurable subset E of G such that f = g = 0 on G \ E , thenwe define the twisted convolution of f and g under Ω to be

(f ~ g)(t) =

∫G

f (s)g(s−1t)Ω(s, s−1t)ds (t ∈ G ).

Definition

Let G be a locally compact group, let Ω ∈ Z2b (G ,C∗), and let ~ be the twisted

convolution coming from Ω. We say that (LΦ(G ),~) is a twisted Orlicz algebraif (LΦ(G ),~, ‖ · ‖Φ) is a Banach algebra, i.e. there is C > 0 such that for everyf , g ∈ LΦ(G ), f ~ g ∈ LΦ(G ) with

‖f ~ g‖Φ ≤ C‖f ‖Φ‖g‖Φ.



Suppose that Ω ∈ Z2b (G ,C∗) and f and g are measurable functions on G . If

there is a σ-finite measurable subset E of G such that f = g = 0 on G \ E , thenwe define the twisted convolution of f and g under Ω to be

(f ~ g)(t) =

∫G

f (s)g(s−1t)Ω(s, s−1t)ds (t ∈ G ).

Definition



‖f ~ g‖Φ ≤ C‖f ‖Φ‖g‖Φ.





(f ~ g)(t) =

∫G

f (s)g(s−1t)Ω(s, s−1t)ds (t ∈ G ).

Definition


convolution coming from Ω. We say that (LΦ(G ),~) is a twisted Orlicz algebraif (LΦ(G ),~, ‖ · ‖Φ) is a Banach algebra,

i.e. there is C > 0 such that for everyf , g ∈ LΦ(G ), f ~ g ∈ LΦ(G ) with

‖f ~ g‖Φ ≤ C‖f ‖Φ‖g‖Φ.





(f ~ g)(t) =

∫G

f (s)g(s−1t)Ω(s, s−1t)ds (t ∈ G ).

Definition



‖f ~ g‖Φ ≤ C‖f ‖Φ‖g‖Φ.


Theorem (S. Oztop and E. Samei)


convolution coming from Ω.

(i) LΦ(G ) is a Banach L1(G )-bimodule with respect to the twisted convolutionhaving SΦ(G ) as an essential Banach L1(G )-submodule.(ii) If there exist non-negative measurable functions u, v ∈ LΨ(G ) such that

|Ω(s, t)| ≤ u(s) + v(t) (s, t ∈ G ),

then (LΦ(G ),~) is a twisted Orlicz algebra with SΦ(G ) as a closed subalgebra.(iii) If, in addition, Ψ is continuous and increasing and u, v ∈ SΨ(G ), then(LΦ(G ),~) is an Arens regular, dual Banach algebra.




convolution coming from Ω.(i) LΦ(G ) is a Banach L1(G )-bimodule with respect to the twisted convolutionhaving SΦ(G ) as an essential Banach L1(G )-submodule.

(ii) If there exist non-negative measurable functions u, v ∈ LΨ(G ) such that

|Ω(s, t)| ≤ u(s) + v(t) (s, t ∈ G ),





convolution coming from Ω.(i) LΦ(G ) is a Banach L1(G )-bimodule with respect to the twisted convolutionhaving SΦ(G ) as an essential Banach L1(G )-submodule.(ii) If there exist non-negative measurable functions u, v ∈ LΨ(G ) such that

|Ω(s, t)| ≤ u(s) + v(t) (s, t ∈ G ),

then (LΦ(G ),~) is a twisted Orlicz algebra with SΦ(G ) as a closed subalgebra.

(iii) If, in addition, Ψ is continuous and increasing and u, v ∈ SΨ(G ), then(LΦ(G ),~) is an Arens regular, dual Banach algebra.




convolution coming from Ω.(i) LΦ(G ) is a Banach L1(G )-bimodule with respect to the twisted convolutionhaving SΦ(G ) as an essential Banach L1(G )-submodule.(ii) If there exist non-negative measurable functions u, v ∈ LΨ(G ) such that

|Ω(s, t)| ≤ u(s) + v(t) (s, t ∈ G ),



Compactly generated groups with polynomial growth (PG)

DefinitionLet G be a compactly generated group with PG, and let U be a generatingcompact neighborhood of G .

A length function | · | on G is defined by

|s|U = infn ∈ N : s ∈ Un for s 6= e, |e| = 0.

If ν : N ∪ 0 → R+ is an increasing subadditive function with ν(0) = 0 andlimn→∞ ν(n) =∞, then ω(x) = eν(|x|) is a weight on G .

Example

Let 0 < α ≤ 1, β ≥ 0, γ > 0, and C > 0. Then one can consider the polynomialweight ωβ of order β and the subexponential weights σα,C and ργ,C on G:

ωβ(s) = (1 + |s|)β , σα,C (s) = eC |s|α

, ργ,C (s) = eC|s|

(ln(e+|s|))γ (s ∈ G ).


Compactly generated groups with polynomial growth (PG)

DefinitionLet G be a compactly generated group with PG, and let U be a generatingcompact neighborhood of G . A length function | · | on G is defined by

|s|U = infn ∈ N : s ∈ Un for s 6= e, |e| = 0.

If ν : N ∪ 0 → R+ is an increasing subadditive function with ν(0) = 0 andlimn→∞ ν(n) =∞, then ω(x) = eν(|x|) is a weight on G .

Example

Let 0 < α ≤ 1, β ≥ 0, γ > 0, and C > 0. Then one can consider the polynomialweight ωβ of order β and the subexponential weights σα,C and ργ,C on G:

ωβ(s) = (1 + |s|)β , σα,C (s) = eC |s|α

, ργ,C (s) = eC|s|

(ln(e+|s|))γ (s ∈ G ).



Let G be a compactly generated group of polynomial growth, let Ω ∈ Z2bw (G ,C∗),

and let ω be the weight associated to Ω. Then there are u, v ∈ SΨ(G ) satisfying

|Ω(s, t)| ≤ u(s) + v(t) (s, t ∈ G ),

if ω is either one of the following weights:

(i) ω = ωβ with 1/ω ∈ SΨ(G );(ii) ω = σα,C ;(iii) ω = ργ,C .In particular, (LΦ(G ),~) is an Arens regular dual Banach algebra.

Note: 1/ωβ ∈ SΨ(G ) if β > dl , where d := d(G ) is the degree of the growth of

G and l ≥ 1 is such that limx→0+Ψ(x)x l exists.





|Ω(s, t)| ≤ u(s) + v(t) (s, t ∈ G ),

if ω is either one of the following weights:(i) ω = ωβ with 1/ω ∈ SΨ(G );

(ii) ω = σα,C ;(iii) ω = ργ,C .In particular, (LΦ(G ),~) is an Arens regular dual Banach algebra.







|Ω(s, t)| ≤ u(s) + v(t) (s, t ∈ G ),

if ω is either one of the following weights:(i) ω = ωβ with 1/ω ∈ SΨ(G );(ii) ω = σα,C ;

(iii) ω = ργ,C .In particular, (LΦ(G ),~) is an Arens regular dual Banach algebra.







|Ω(s, t)| ≤ u(s) + v(t) (s, t ∈ G ),

if ω is either one of the following weights:(i) ω = ωβ with 1/ω ∈ SΨ(G );(ii) ω = σα,C ;(iii) ω = ργ,C .

In particular, (LΦ(G ),~) is an Arens regular dual Banach algebra.







|Ω(s, t)| ≤ u(s) + v(t) (s, t ∈ G ),

if ω is either one of the following weights:(i) ω = ωβ with 1/ω ∈ SΨ(G );(ii) ω = σα,C ;(iii) ω = ργ,C .In particular, (LΦ(G ),~) is an Arens regular dual Banach algebra.




Thank you!


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