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On sequential properties of Banach spaces, spaces of measures and densities

Piotr Borodulin-Nadzieja, Grzegorz Plebanek (2010)

Czechoslovak Mathematical Journal

We show that a conjunction of Mazur and Gelfand-Phillips properties of a Banach space E can be naturally expressed in terms of weak* continuity of seminorms on the unit ball of E * . We attempt to carry out a construction of a Banach space of the form C ( K ) which has the Mazur property but does not have the Gelfand-Phillips property. For this purpose we analyze the compact spaces on which all regular measures lie in the weak* sequential closure of atomic measures, and the set-theoretic properties of generalized...

On sharp reiteration theorems and weighted norm inequalities

Jesús Bastero, Mario Milman, Francisco Ruiz (2000)

Studia Mathematica

We prove sharp end forms of Holmstedt's reiteration theorem which are closely connected with a general form of Gehring's Lemma. Reverse type conditions for the Hardy-Littlewood-Pólya order are considered and the maximal elements are shown to satisfy generalized Gehring conditions. The methods we use are elementary and based on variants of reverse Hardy inequalities for monotone functions.

On Sobolev spaces of fractional order and ε-families of operators on spaces of homogeneous type

A. Gatto, Stephen Vági (1999)

Studia Mathematica

We introduce Sobolev spaces L α p for 1 < p < ∞ and small positive α on spaces of homogeneous type as the classes of functions f in L p with fractional derivative of order α, D α f , as introduced in [2], in L p . We show that for small α, L α p coincides with the continuous version of the Triebel-Lizorkin space F p α , 2 as defined by Y. S. Han and E. T. Sawyer in [4]. To prove this result we give a more general definition of ε-families of operators on spaces of homogeneous type, in which the identity operator is...

On some convexity properties in the Besicovitch-Musielak-Orlicz space of almost periodic functions with Luxemburg norm

Fazia Bedouhene, Amina Daoui, Hocine Kourat (2012)

Commentationes Mathematicae Universitatis Carolinae

In this article, it is shown that geometrical properties such as local uniform convexity, mid point local uniform convexity, H-property and uniform convexity in every direction are equivalent in the Besicovitch-Musielak-Orlicz space of almost periodic functions ( B ˜ ϕ a . p . ) endowed with the Luxemburg norm.

On some density theorems in regular vector lattices of continuous functions.

Francesco Altomare, Mirella Cappelletti Montano (2007)

Collectanea Mathematica

In this paper, we establish some density theorems in the setting of particular locally convex vector lattices of continuous functions de ned on a locally compact Hausdorff space, which we introduced and studied in [3,4] and which we named regular vector lattices. In this framework, by using properties of the subspace of the so-called generalized af ne functions, we give a simple description of the closed vector sublattice, the closed Stone vector sublattice and the closed subalgebra generated by...

On some equivalent geometric properties in the Besicovitch-Orlicz space of almost periodic functions with Luxemburg norm

Fazia Bedouhene, Mohamed Morsli, Mannal Smaali (2010)

Commentationes Mathematicae Universitatis Carolinae

The paper is concerned with the characterization and comparison of some local geometric properties of the Besicovitch-Orlicz space of almost periodic functions. Namely, it is shown that local uniform convexity, H -property and strict convexity are all equivalent. In our approach, we first prove some metric type properties for the modular function associated to our space. These are then used to prove our main equivalence result.

On some ergodic properties for continuous and affine functions

Charles J. K. Batty (1978)

Annales de l'institut Fourier

Two problems posed by Choquet and Foias are solved:(i) Let T be a positive linear operator on the space C ( X ) of continuous real-valued functions on a compact Hausdorff space X . It is shown that if n - 1 ∑ r = 0 n - 1 T r 1 converges pointwise to a continuous limit, then the convergence is uniform on X .(ii) An example is given of a Choquet simplex K and a positive linear operator T on the space A ( K ) of continuous affine real-valued functions on K , such that inf { ( T n 1 ) ( x ) : n ≥ } &lt; 1 for each x in ∂ ℓ K , but ∥ T n 1 ∥ does not converge to 0.

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