A NOTE ON WEIGHTED ORLICZ ALGEBRAS Serap OZTOP Istanbul - PowerPoint PPT Presentation

A NOTE ON WEIGHTED ORLICZ ALGEBRAS Serap ¨ OZTOP ˙ Istanbul University Joint work with Alen OSANC ¸LIOL 1 / 20

Young Function Definition (Young Function)[H. Hudzik, 1985] A non-zero function Φ : R → [0 , + ∞ ] is called a Young function if (i) Φ is convex, (ii) Φ is even, (iii) Φ(0) = 0. Note Note that this definition of Young functions allows them to take the value ∞ , and hence they may be discontinuous at the point where they take the value infinity. However, unless otherwise specified we will consider only real-valued Young functions. Such a Φ is necesserily continuous, and tends to infinity as x tends to infinity. 2 / 20

Complementary Function Definition (Complementary Function) Given a Young function Φ, the complementary function Ψ of Φ is given by Ψ( y ) = sup { x | y | − Φ( x ) | x ≥ 0 } for y ∈ R . If Ψ is the complementary function of Φ, then Φ is the complementary function of Ψ and (Φ , Ψ) is called a complementary pair of Young functions. Note Even if Φ is finite valued it may happen that Ψ takes infinite values. 3 / 20

Example q = 1. Then Φ( x ) = | x | p 1) Let 1 < p < + ∞ and 1 p + 1 p , x ∈ R , and Ψ( x ) = | x | q q , x ∈ R , are a complementary pair of Young functions. Example 2) In particular, when p = 1, the complementary function of Φ( x ) = | x | is � 0 , 0 ≤ | x | ≤ 1 , Ψ( x ) = + ∞ , | x | > 1 . 4 / 20

Weighted Orlicz Spaces Definition (Weighted Orlicz Space) Let G be a locally compact group with left Haar measure µ and w be a weight on G (i.e., w is a positive, Borel measurable function such that w ( xy ) ≤ w ( x ) w ( y ) for all x , y ∈ G ). Given a Young function Φ, the weighted Orlicz space L Φ w ( G ) is defined by � � � L Φ w ( G ) := f : G → K | ∃ α > 0 , Φ( α | fw | ) d µ < + ∞ . G Then L Φ w ( G ) becomes a Banach space under the norm || · || Φ , w (called the weighted Orlicz norm) defined for f ∈ L Φ w ( G ) by �� � � | fwv | d µ | v ∈ L Ψ ( G ) , || f || Φ , w := sup Ψ( | v | ) d µ ≤ 1 , G G where Ψ is the complementary function of Φ. 5 / 20

For f ∈ L Φ w ( G ), one can also define the norm � � | fw | � � � || f || ◦ Φ , w = inf k > 0 | Φ d µ ≤ 1 , k G which is called the weighted Luxemburg norm and is equivalent to the weighted Orlicz norm. Recall... Notice that if Φ( x ) = | x | p p , 1 ≤ p < + ∞ , then L Φ w ( G ) becomes the classical weighted Lebesgue space L p ( G ). If � 0 , 0 ≤ | x | ≤ 1 , Ψ( x ) = + ∞ , | x | > 1 , then L Ψ w ( G ) = L ∞ w ( G ). 6 / 20

Definition (∆ 2 Condition) Let Φ be a Young function. We say that Φ satisfies the ∆ 2 condition whenever there exists a K > 0 such that Φ(2 x ) ≤ K Φ( x ) for all x ≥ 0, and we write Φ ∈ ∆ 2 in such a case. Mostly we consider the ∆ 2 condition for the Young function Φ. Examples • For 1 ≤ p < ∞ , if Φ( x ) = | x | p p , x ∈ R , then Φ ∈ ∆ 2 . • If Φ( x ) = e | x |− 1 , x ∈ R , then Φ �∈ ∆ 2 . 7 / 20

Dual Space of L Φ w ( G ) Theorem (Dual Space) Let G be a locally compact group and w be a weight on G . If Φ be a Young function such that Φ ∈ ∆ 2 and Ψ is the complementary function of Φ, then the dual space of ( L Φ w ( G ) , || · || Φ , w ) is L Ψ w − 1 ( G ) formed by all measurable functions g on G such that g w ∈ L Ψ ( G ) and endowed with the norm || · || ◦ Ψ , w − 1 defined for g ∈ L Ψ w − 1 ( G ) by � � � � � � g � � � g � ◦ || g || ◦ Ψ , w − 1 := � Ψ = inf k > 0 : Ψ d µ ≤ 1 . w kw G Corollary Let (Φ , Ψ) be a complementary pair of Young functions such that Φ , Ψ ∈ ∆ 2 . Then the weighted Orlicz space L Φ w ( G ) is reflexive. 8 / 20

Basic Properties of L Φ w ( G ) Proposition Let Φ be a Young function such that Φ ∈ ∆ 2 and f ∈ L Φ w ( G ). Then ||·|| Φ , w = L Φ i) C c ( G ) w ( G ), ii) for every x ∈ G , L x f ∈ L Φ w ( G ) and || L x f || Φ , w ≤ w ( x ) || f || Φ , w , iii) the map L Φ G → w ( G ) x �→ L x f is continuous. 9 / 20

Weighted Orlicz Algebra with Respect to Convolution Multiplication Theorem [H. Hudzik, 1985] For G is a locally compact abelian group, L Φ ( G ) is a Banach algebra w.r.t. convolution ⇔ L Φ ( G ) ⊆ L 1 ( G ). Theorem (Weighted Orlicz Algebra) Let G be a locally compact group, w be a weight on G and let Φ be a Young function. If L Φ w ( G ) ⊆ L 1 w ( G ), then the weighted Orlicz space ( L Φ w ( G ) , || · || Φ , w ) is a Banach algebra w.r.t. convolution, which we call the weighted Orlicz algebra. Note that the converse is not true in general. For Φ( x ) = | x | p p , p > 1, L p w ( G ) is a Banach algebra (Kuznetsova, 2006), but it is not in L 1 w ( G ). 10 / 20

Observation For Φ is a Young function with Φ ′ + (0) > 0, then the inclusion L Φ w ( G ) ⊆ L 1 w ( G ) is true. So L Φ w ( G ) becomes a weighted Orlicz algebra. In particular, if G is non compact and abelian locally compact group, then Φ ′ + (0) > 0 ⇔ L Φ w ( G ) ⊆ L 1 w ( G ) (Hudzik, 1985). Observation Without any assumption on the Young function Φ, we can have the weighted Orlicz space L Φ w ( G ) as a left Banach L 1 w ( G )-module w.r.t. convolution. 11 / 20

Henceforth, we assume that a Young function Φ satisfies the ∆ 2 condition. Theorem The weighted Orlicz algebra L Φ w ( G ) has a left approximate identity consisting of compactly supported functions that are bounded w.r.t. the || · || 1 , w norm. Theorem The weighted Orlicz algebra L Φ w ( G ) has an identity if and only if G is discrete. Theorem Let the complementary function of Φ satisfy the ∆ 2 condition. If G is non-discrete, then the weighted Orlicz algebra L Φ w ( G ) admits no bounded left approximate identity w.r.t. the || · || Φ , w norm. 12 / 20

Let L Φ w ( G ) be a weighted Orlicz algebra. The closed left ideals of L Φ w ( G ) turn out to be nothing but the closed left translation-invariant subspaces of L Φ w ( G ). Theorem Let L Φ w ( G ) be a weighted Orlicz algebra and let I be a closed linear subspace of L Φ w ( G ). Then I is a left ideal ⇔ ∀ x ∈ G , L x ( I ) ⊆ I . Observation If w = 1, then the closed left ideals of the Orlicz algebra L Φ ( G ) coincide with the closed left translation-invariant subspaces. Proposition Let Φ be a Young function such that Φ ′ + (0) > 0. Then the weighted Orlicz algebra L Φ w ( G ) is a left ideal in L 1 w ( G ). 13 / 20

Let G be a locally compact abelian group, w be a weight and let Φ be a Young function. We now describe the maximal ideal space (spectrum) ∆( L Φ w ( G )) of the commutative weighted Orlicz algebra L Φ w ( G ) in terms of the so-called generalized characters of G determined by the complementary function Ψ of Φ and a weight w . Note If G is abelian, then the weighted Orlicz algebra L Φ w ( G ) is a commutative. Definition Let G be a locally compact abelian group, w be a weight and Φ be a Young function with the complementary function Ψ. A generalized character determined by the function Ψ and a weight w on G is a continuous function γ : G → C \{ 0 } satisfying the conditions i) γ ( x + y ) = γ ( x ) γ ( y ) for all x , y ∈ G , w ∈ L Ψ ( G ). ii) γ Let � G Ψ ( w ) denote the set of all generalized characters of G equipped with the topology of uniform convergence on compact subsets of G . 14 / 20

Theorem Let G be a locally compact abelian group and let L Φ w ( G ) be a weighted Orlicz algebra. For γ ∈ � G Ψ ( w ), define ϕ γ : L Φ w ( G ) → C by � f ( x ) γ ( x ) d µ ( x ) , f ∈ L Φ ϕ γ ( f ) = w ( G ) . G w ( G )), and the map γ �→ ϕ γ is a bijection between � Then ϕ γ ∈ ∆( L Φ G Ψ ( w ) and ∆( L Φ w ( G )). Observation If w = 1, then for the Orlicz algebra L Φ ( G ), ∆( L Φ ( G )) ∼ = � G Ψ . Observation Let Φ be a Young function with Φ ′ + (0) > 0 and w be a weight such that w ( x ) ≥ 1 for all x ∈ G . Then L Φ w ( G ) and L Φ ( G ) become commutative Banach algebras and L Φ w ( G ) ⊆ L Φ ( G ) is true. Hence we have G Ψ ⊆ � � G Ψ ( w ). 15 / 20

Proposition The weighted Orlicz algebra L Φ w ( G ) is not radical. Theorem The weighted Orlicz algebra L Φ w ( G ) is semi-simple. Sketch of Proof Since L Φ w ( G ) is not a radical algebra, there exists a ϕ ∈ ∆( L Φ w ( G )) and this ϕ is determined by γ ∈ � G Ψ ( w ) uniquely. For each α ∈ ˆ G , define w ( G )) ∗ by ϕ α ∈ ( L Φ � ϕ α ( f ) = f αγ d µ. G w ( G )) since αγ ∈ � G Ψ ( w ) for all α ∈ ˆ Then ϕ α ∈ ∆( L Φ G . Let f be an w ( G ). Then � element of the radical of L Φ f γ ∈ L 1 ( G ) and � f γ ( α ) = ϕ α ( f ) = 0 for all α ∈ ˆ G . So we get f = 0. 16 / 20

Observation If w = 1, then the Orlicz algebra L Φ ( G ) is not radical, but is semi-simple. Note If L Φ w ( G ) ⊆ L 1 w ( G ), then we have seen above that L Φ w ( G ) is a Banach algebra w.r.t. convolution, but the converse is not true in general. However, the following theorem shows that if L Φ w ( G ) is a Banach algebra w.r.t. convolution, then one can always assume that the inclusion w ( G ) ⊆ L 1 ( G ) is true, similar to the case L p L Φ w ( G ) (Kuznetsova, 2006). 17 / 20