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On S. Mazur's problems 8 and 88 from the Scottish Book

V. V. Peller (2007)

Studia Mathematica

The paper discusses Problems 8 and 88 posed by Stanisław Mazur in the Scottish Book. It turns out that negative solutions to both problems are immediate consequences of the results of Peller [J. Operator Theory 7 (1982)]. We discuss here some quantitative aspects of Problems 8 and 88 and give answers to open problems discussed in a recent paper of Pełczyński and Sukochev in connection with Problem 88.

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.

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