Displaying similar documents to “The dual of weak L p ”

Purely non-atomic weak L p spaces

Denny Leung (1997)

Studia Mathematica

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Let (Ω,∑,μ) be a purely non-atomic measure space, and let 1 < p < ∞. If L p , ∞ ( Ω , ∑ , μ ) is isomorphic, as a Banach space, to L p , ∞ ( Ω ' , ∑ ' , μ ' ) for some purely atomic measure space (Ω’,∑’,μ’), then there is a measurable partition Ω = Ω 1 ∪ Ω 2 such that ( Ω 1 , Σ ∩ Ω 1 , μ | Σ ∩ Ω 1 ) is countably generated and σ-finite, and that μ(σ) = 0 or ∞ for every measurable σ ⊆ Ω 2 . In particular, L p , ∞ ( Ω , ∑ , μ ) is isomorphic to ℓ p , ∞ .

The space Weak H¹

Robert Fefferman, Fernando Soria (1987)

Studia Mathematica

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The converse of the Hölder inequality and its generalizations

Janusz Matkowski (1994)

Studia Mathematica

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Let (Ω,Σ,μ) be a measure space with two sets A,B ∈ Σ such that 0 < μ (A) < 1 < μ (B) < ∞ and suppose that ϕ and ψ are arbitrary bijections of [0,∞) such that ϕ(0) = ψ(0) = 0. The main result says that if ʃ Ω x y d μ ≤ ϕ - 1 ( ʃ Ω ϕ ∘ x d μ ) ψ - 1 ( ʃ Ω ψ ∘ x d μ ) for all μ-integrable nonnegative step functions x,y then ϕ and ψ must be conjugate power functions. If the measure space (Ω,Σ,μ) has one of the following properties: (a) μ (A) ≤ 1 for every A ∈ Σ of finite measure; (b) μ (A) ≥ 1 for every A ∈ Σ of positive measure, then...

The level function in rearrangement invariant spaces.

Gord Sinnamon (2001)

Publicacions Matemàtiques

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An exact expression for the down norm is given in terms of the level function on all rearrangement invariant spaces and a useful approximate expression is given for the down norm on all rearrangement invariant spaces whose upper Boyd index is not one.

Isomorphism of certain weak L p spaces

Denny Leung (1993)

Studia Mathematica

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It is shown that the weak L p spaces ℓ p , ∞ , L p , ∞ [ 0 , 1 ] , and L p , ∞ [ 0 , ∞ ) are isomorphic as Banach spaces.