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On some properties of generalized Marcinkiewicz spaces

Evgeniy Pustylnik — 2001

Studia Mathematica

We give a full solution of the following problems concerning the spaces M φ ( X ) : (i) to what extent two functions φ and ψ should be different in order to ensure that M φ ( X ) M ψ ( X ) for any nontrivial Banach couple X⃗; (ii) when an embedding M φ ( X ) M ψ ( X ) can (or cannot) be dense; (iii) which Banach space can be regarded as an M φ ( X ) -space for some (unknown beforehand) Banach couple X⃗.

Real interpolation for non-distant Marcinkiewicz spaces.

Evgeniy Pustylnik — 2001

Revista Matemática Complutense

We describe the real interpolation spaces between given Marcinkiewicz spaces that have fundamental functions of the form t (ln (e/t) with the same exponent q. The spaces thus obtained are used for the proof of optimal interpolation theorem from [7], concerning spaces L.

Extreme cases of weak type interpolation.

Evgeniy Pustylnik — 2005

Revista Matemática Iberoamericana

We consider quasilinear operators T of joint weak type (a, b; p, q) (in the sense of [2]) and study their properties on spaces Lφ,E with the norm ||φ(t) f*(t)||Ê, where Ê is arbitrary rearrangement-invariant space with respect to the measure dt/t. A space Lφ,E is said to be "close" to one of the endpoints of interpolation if the corresponding Boyd index of this space is equal to 1/a or to 1/p. For...

Ultrasymmetric sequence spaces in approximation theory.

Evgeniy Pustylnik — 2006

Collectanea Mathematica

Let φ(t) be a positive increasing function and let Ê be an arbitrary sequence space, rearrangement-invariant with respect to the atomic measure µ(n) = 1/n. Let {a*} mean the decreasing rearrangement of a sequence {|a|}. A sequence space l with symmetric (quasi)norm || {φ(n)a*} || is called , because it is not only intermediate but also interpolation between the corresponding Lorentz and Marcinkiewicz spaces Λ and M. We study properties of the spaces l for all admissible parameters φ, E and use them...

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