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Narrow operators (a survey)

Mikhail Popov (2011)

Banach Center Publications

Narrow operators are those operators defined on function spaces which are "small" at signs, i.e., at {-1,0,1}-valued functions. We summarize here some results and problems on them. One of the most interesting things is that if E has an unconditional basis then each operator on E is a sum of two narrow operators, while the sum of two narrow operators on L₁ is narrow. Recently this notion was generalized to vector lattices. This generalization explained the phenomena of sums: the set of all regular...

Nearly smooth points and near smoothness in Orlicz spaces

Ji Donghai, Yanming Lü, Ting Fu Wang (1998)

Commentationes Mathematicae Universitatis Carolinae

Nearly smooth points and near smoothness in Orlicz spaces are characterized. It is worth to notice that in the nonatomic case smooth points and nearly smooth points are the same, but in the sequence case they differ.

Necessary and sufficient conditions for boundedness of the maximal operator in local Morrey-type spaces

Viktor I. Burenkov, Huseyn V. Guliyev (2004)

Studia Mathematica

The problem of boundedness of the Hardy-Littewood maximal operator in local and global Morrey-type spaces is reduced to the problem of boundedness of the Hardy operator in weighted L p -spaces on the cone of non-negative non-increasing functions. This allows obtaining sufficient conditions for boundedness for all admissible values of the parameters. Moreover, in case of local Morrey-type spaces, for some values of the parameters, these sufficient conditions are also necessary.

New examples of K-monotone weighted Banach couples

Sergey V. Astashkin, Lech Maligranda, Konstantin E. Tikhomirov (2013)

Studia Mathematica

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...

Nonconvolution transforms with oscillating kernels that map Ḃ 1 0 , 1 into itself

G. Sampson (1993)

Studia Mathematica

We consider operators of the form ( Ω f ) ( y ) = ʃ - ∞ ∞ Ω ( y , u ) f ( u ) d u with Ω(y,u) = K(y,u)h(y-u), where K is a Calderón-Zygmund kernel and h ∈ L ∞ (see (0.1) and (0.2)). We give necessary and sufficient conditions for such operators to map the Besov space Ḃ 1 0 , 1 (= B) into itself. In particular, all operators with h ( y ) = e i | y | a , a > 0, a ≠ 1, map B into itself.

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