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A strengthening of a theorem of Marcinkiewicz

100%

Tikhomirov K.

Banach Center Publications

2011

tom 95

nr 1

369-383

We consider a problem of intervals raised by I. Ya. Novikov in [Israel Math. Conf. Proc. 5 (1992), 290], which refines the well-known theorem of J. Marcinkiewicz concerning structure of closed sets [A. Zygmund, Trigonometric Series, Vol. I, Ch. IV, Theorem 2.1]. A positive solution to the problem for some specific cases is obtained. As a result, we strengthen the theorem of Marcinkiewicz for generalized Cantor sets.

New examples of K-monotone weighted Banach couples

51%

Astashkin S., Maligranda L., Tikhomirov K.

Studia Mathematica

2013

tom 218

nr 1

55-88

Some new examples of K-monotone couples of the type (X,X(w)), where X is a symmetric space on [0,1] and w is a weight on [0,1], are presented. Based on the property of w-decomposability of a symmetric space we show that, if a weight w changes sufficiently fast, all symmetric spaces X with non-trivial Boyd indices such that the Banach couple (X,X(w)) is K-monotone belong to the class of ultrasymmetric Orlicz spaces. If, in addition, the fundamental function of X is $t^{1/p}$ for some p ∈ [1,∞], then $X = L_{p}$. At the same time a Banach couple (X,X(w)) may be K-monotone for some non-trivial w in the case when X is not ultrasymmetric. In each of the cases where X is a Lorentz, Marcinkiewicz or Orlicz space, we find conditions which guarantee that (X,X(w)) is K-monotone.

Ograniczanie wyników

1 Banach Center Publications

1 Studia Mathematica

2 Tikhomirov K.

1 Astashkin S.

1 Maligranda L.

1 2013

1 2011

Wyniki wyszukiwania

A strengthening of a theorem of Marcinkiewicz

New examples of K-monotone weighted Banach couples