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

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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.

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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.

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Example

1) Let 1 < p < +∞ and 1

p + 1 q = 1. Then Φ(x) = |x|p 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 Ψ(x) = 0, 0 ≤ |x| ≤ 1, +∞, |x| > 1.

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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 LΦ

w(G) is defined by

LΦ

w(G) :=

• f : G → K | ∃α > 0,
• G

Φ(α|fw|)dµ < +∞

• .

Then LΦ

w(G) becomes a Banach space under the norm || · ||Φ,w (called the

weighted Orlicz norm) defined for f ∈ LΦ

w(G) by

||f ||Φ,w := sup

• G

|fwv|dµ | v ∈ LΨ(G),

• G

Ψ(|v|)dµ ≤ 1

• ,

where Ψ is the complementary function of Φ.

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For f ∈ LΦ

w(G), one can also define the norm

||f ||◦

Φ,w = inf

• k > 0 |
• G

Φ |fw| k

• dµ ≤ 1
• ,

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 LΦ w(G) becomes the

classical weighted Lebesgue space Lp(G). If Ψ(x) = 0, 0 ≤ |x| ≤ 1, +∞, |x| > 1, then LΨ

w(G) = L∞ w (G).

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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(2x) ≤ KΦ(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.

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Dual Space of LΦ

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 (LΦ

w(G), || · ||Φ,w) is LΨ w−1(G) formed by all

measurable functions g on G such that g

w ∈ LΨ(G) and endowed with the

norm || · ||◦

Ψ,w−1 defined for g ∈ LΨ w−1(G) by

||g||◦

Ψ,w−1 :=

• g

w

• Ψ = inf
• k > 0 :
• G

Ψ g kw

• dµ ≤ 1
• .

Corollary

Let (Φ, Ψ) be a complementary pair of Young functions such that Φ, Ψ ∈ ∆2. Then the weighted Orlicz space LΦ

w(G) is reflexive.

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Basic Properties of LΦ

w(G)

Proposition

Let Φ be a Young function such that Φ ∈ ∆2 and f ∈ LΦ

w(G). Then

i) Cc(G)

||·||Φ,w = LΦ w(G),

ii) for every x ∈ G, Lxf ∈ LΦ

w(G) and ||Lxf ||Φ,w ≤ w(x)||f ||Φ,w,

iii) the map G → LΦ

w(G)

x → Lxf is continuous.

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Weighted Orlicz Algebra with Respect to Convolution Multiplication

Theorem [H. Hudzik, 1985]

For G is a locally compact abelian group, LΦ(G) is a Banach algebra w.r.t. convolution ⇔ LΦ(G) ⊆ L1(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 LΦ

w(G) ⊆ L1 w(G), then the weighted Orlicz space

(LΦ

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,

Lp

w(G) is a Banach algebra (Kuznetsova, 2006), but it is not in L1 w(G).

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Observation

For Φ is a Young function with Φ′

+(0) > 0, then the inclusion

LΦ

w(G) ⊆ L1 w(G) is true. So LΦ w(G) becomes a weighted Orlicz algebra. In

particular, if G is non compact and abelian locally compact group, then Φ′

+(0) > 0 ⇔ LΦ w(G) ⊆ L1 w(G) (Hudzik, 1985).

Observation

Without any assumption on the Young function Φ, we can have the weighted Orlicz space LΦ

w(G) as a left Banach L1 w(G)-module

w.r.t. convolution.

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Henceforth, we assume that a Young function Φ satisfies the ∆2 condition.

Theorem

The weighted Orlicz algebra LΦ

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 LΦ

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 LΦ

w(G) admits no bounded

left approximate identity w.r.t. the || · ||Φ,w norm.

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Let LΦ

w(G) be a weighted Orlicz algebra. The closed left ideals of LΦ w(G)

turn out to be nothing but the closed left translation-invariant subspaces

• f LΦ

w(G).

Theorem

Let LΦ

w(G) be a weighted Orlicz algebra and let I be a closed linear

subspace of LΦ

w(G). Then

I is a left ideal ⇔ ∀x ∈ G, Lx(I) ⊆ I.

Observation

If w = 1, then the closed left ideals of the Orlicz algebra LΦ(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 LΦ

w(G) is a left ideal in L1 w(G).

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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) ∆(LΦ

w(G)) of the commutative weighted Orlicz algebra LΦ 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 LΦ

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, ii)

γ w ∈ LΨ(G).

Let GΨ(w) denote the set of all generalized characters of G equipped with the topology of uniform convergence on compact subsets of G.

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Theorem

Let G be a locally compact abelian group and let LΦ

w(G) be a weighted

Orlicz algebra. For γ ∈ GΨ(w), define ϕγ : LΦ

w(G) → C by

ϕγ(f ) =

• G

f (x)γ(x)dµ(x) , f ∈ LΦ

w(G).

Then ϕγ ∈ ∆(LΦ

w(G)), and the map γ → ϕγ is a bijection between

GΨ(w) and ∆(LΦ

w(G)).

Observation

If w = 1, then for the Orlicz algebra LΦ(G), ∆(LΦ(G)) ∼ = GΨ.

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 LΦ

w(G) and LΦ(G) become commutative

Banach algebras and LΦ

w(G) ⊆ LΦ(G) is true. Hence we have

• GΨ ⊆

GΨ(w).

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Proposition

The weighted Orlicz algebra LΦ

w(G) is not radical.

Theorem

The weighted Orlicz algebra LΦ

w(G) is semi-simple.

Sketch of Proof

Since LΦ

w(G) is not a radical algebra, there exists a ϕ ∈ ∆(LΦ w(G)) and

this ϕ is determined by γ ∈ GΨ(w) uniquely. For each α ∈ ˆ G, define ϕα ∈ (LΦ

w(G))∗ by

ϕα(f ) =

• G

f αγdµ. Then ϕα ∈ ∆(LΦ

w(G)) since αγ ∈

GΨ(w) for all α ∈ ˆ

• G. Let f be an

element of the radical of LΦ

w(G). Then

f γ ∈ L1(G) and

• f γ(α) = ϕα(f ) = 0

for all α ∈ ˆ

• G. So we get f = 0.

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Observation

If w = 1, then the Orlicz algebra LΦ(G) is not radical, but is semi-simple.

Note

If LΦ

w(G) ⊆ L1 w(G), then we have seen above that LΦ 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 LΦ

w(G) is a Banach algebra

w.r.t. convolution, then one can always assume that the inclusion LΦ

w(G) ⊆ L1(G) is true, similar to the case Lp w(G) (Kuznetsova, 2006).

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Lemma

Let w be a weight on G and let Φ be a Young function such that Φ ∈ ∆2 with the complementary function Ψ. Then the following are equivalent: i)

1 w ∈ LΨ(G),

ii) LΦ

w(G) ⊆ L1(G),

iii) L∞(G) ⊆ LΨ

w−1(G).

Theorem

Each weighted Orlicz algebra LΦ

w(G) is isometrically isomorphic to an

algebra LΦ

˜ w(G) with a weight ˜

w satisfying 1

˜ w ∈ LΨ(G).

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Weighted Orlicz Algebra with Respect to Pointwise Multiplication

• H. Hudzik (1985) gives necessary and sufficient conditions for an Orlicz

space to be a Banach algebra with respect to pointwise multiplication on the measure space (X, Σ, µ). We adapt the results of H. Hudzik to a locally compact group G.

Proposition

Let G be a locally compact group and w be a weight on G. If Φ is a strictly increasing Young function, then the following statements are equivalent for limx→∞

Φ(x) x

= +∞: i) LΦ

w(G) ⊆ L∞ w (G),

ii) G is discrete, iii) L1

w(G) ⊆ LΦ w(G).

We need the limit condition for iii) ⇒ ii).

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Corollary

If G = Z, then the weighted Orlicz sequence spaces denoted by LΦ

w(Z) = ℓΦ w satisfy

ℓ1

w ⊆ ℓΦ w ⊆ ℓ∞ w .

Theorem

Let G be a locally compact group and w a weight on G . If Φ is a strictly increasing Young function, then LΦ

w(G) is a Banach algebra

w.r.t. pointwise multiplication if and only if LΦ

w(G) ⊆ L∞ w (G).

Observation

Under the same conditions as in the previous theorem, the weighted Orlicz space LΦ

w(G) is a Banach algebra w.r.t. pointwise multiplication if and only

if G is discrete.

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