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Integrals and Banach spaces for finite order distributions

Erik Talvila (2012)

Czechoslovak Mathematical Journal

Let ℬ c denote the real-valued functions continuous on the extended real line and vanishing at - ∞ . Let ℬ r denote the functions that are left continuous, have a right limit at each point and vanish at - ∞ . Define 𝒜 c n to be the space of tempered distributions that are the n th distributional derivative of a unique function in ℬ c . Similarly with 𝒜 r n from ℬ r . A type of integral is defined on distributions in 𝒜 c n and 𝒜 r n . The multipliers are iterated integrals of functions of bounded variation. For each n ∈ ℕ , the spaces...

Interpolation on families of characteristic functions

Michael Cwikel, Archil Gulisashvili (2000)

Studia Mathematica

We study a problem of interpolating a linear operator which is bounded on some family of characteristic functions. A new example is given of a Banach couple of function spaces for which such interpolation is possible. This couple is of the form Φ ¯ = ( B , L ∞ ) where B is an arbitrary Banach lattice of measurable functions on a σ-finite nonatomic measure space (Ω,Σ,μ). We also give an equivalent expression for the norm of a function ⨍ in the real interpolation space ( B , L ∞ ) θ , p in terms of the characteristic functions of...

Les topologies sygma-Lebesgue sur C(X).

Belmesnaoui Aqzzouz, Redouane Nouira (2004)

Extracta Mathematicae

We prove that if X is a compact topological space which contains a nontrivial metrizable connected closed subset, then the vector lattice C(X) does not carry any sygma-Lebesgue topology.

Measures of noncompactness and normal structure in Banach spaces

J. García-Falset, A. Jiménez-Melado, E. Lloréns-Fuster (1994)

Studia Mathematica

Sufficient conditions for normal structure of a Banach space are given. One of them implies reflexivity for Banach spaces with an unconditional basis, and also for Banach lattices.

Modulus of dentability in L ¹ + L ∞

Adam Bohonos, Ryszard Płuciennik (2008)

Banach Center Publications

We introduce the notion of the modulus of dentability defined for any point of the unit sphere S(X) of a Banach space X. We calculate effectively this modulus for denting points of the unit ball of the classical interpolation space L ¹ + L ∞ . Moreover, a criterion for denting points of the unit ball in this space is given. We also show that none of denting points of the unit ball of L ¹ + L ∞ is a LUR-point. Consequently, the set of LUR-points of the unit ball of L ¹ + L ∞ is empty.

Monotone substochastic operators and a new Calderón couple

Karol Leśnik (2015)

Studia Mathematica

An important result on submajorization, which goes back to Hardy, Littlewood and Pólya, states that b ⪯ a if and only if there is a doubly stochastic matrix A such that b = Aa. We prove that under monotonicity assumptions on the vectors a and b the matrix A may be chosen monotone. This result is then applied to show that ( L p ˜ , L ∞ ) is a Calderón couple for 1 ≤ p < ∞, where L p ˜ is the Köthe dual of the Cesàro space C e s p ' (or equivalently the down space L ↓ p ' ). In particular, ( L ¹ ˜ , L ∞ ) is a Calderón couple, which gives a...

M-weak and L-weak compactness of b-weakly compact operators

J. H'Michane, A. El Kaddouri, K. Bouras, M. Moussa (2013)

Commentationes Mathematicae Universitatis Carolinae

We characterize Banach lattices under which each b-weakly compact (resp. b-AM-compact, strong type (B)) operator is L-weakly compact (resp. M-weakly compact).

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