Search

The equivalence of the Haar system in a rearrangement invariant space on and a sequence of pairwise disjoint functions in some Lorentz space is known to imply that up to the equivalence of norms. We show that the same holds for the class of uniform disjointly homogeneous rearrangement invariant spaces and obtain a few consequences for the properties of isomorphic embeddings of such spaces. In particular, the space with is the only uniform -disjointly homogeneous rearrangement invariant space on with nontrivial Boyd indices which has two rearrangement invariant representations on the half-axis . [ABSTRACT FROM AUTHOR]

Basing on geometric properties, we give a complete characterization of the Lorentz spaces that can be sharp extrapolation spaces for different types of the behavior of the growth of the norms of an operator in tending to a critical exponent. The results of this article are connected with the calculation of extrapolation constructions in about the same way as theorems of the Calderón-Mityagin type. [ABSTRACT FROM AUTHOR]

We calculate the norms of multipliers for the Haar system in some rearrangement invariant spaces for which the Haar system is not an absolute basis. [ABSTRACT FROM AUTHOR]

We consider the embeddings of the Nikol’skii classes of Lorentz spaces into Lorentz spaces. We obtain necessary and sufficient conditions for these embeddings under some restrictions on the fundamental functions of a Lorentz space. [ABSTRACT FROM AUTHOR]

We introduce a new class of rearrangement invariant spaces on the segment [0, 1] which contains the most common extrapolation spaces with respect to the L p-scale as p → ∞. We characterize the class and demonstrate under certain conditions that the Peetre [Figure not available: see fulltext.]-functional has extrapolatory description in the couple ( E, L∞) if and only if E belongs to the class. By way of application, we establish a new extrapolation theorem for the bounded operators in L p. [ABSTRACT FROM AUTHOR]

Under study is the absolute continuity of the functions satisfying the Poincaré inequality on s-regular metric spaces. [ABSTRACT FROM AUTHOR]

We consider the questions of convergence in Lorentz spaces for the Fourier-Walsh series of the functions with Denjoy integrable derivative. We prove that a condition on a function f sufficient for its Fourier-Walsh series to converge in the Lorentz spaces “near” L ∞ cannot be expressed in terms of the growth of the derivative f′. [ABSTRACT FROM AUTHOR]

We introduce the notion of a rearrangement invariant extrapolation space on [0, 1]. We obtain some sufficient conditions (also necessary in some cases) for the Marcinkiewicz and Lorentz spaces to be extrapolation spaces. We generalize and improve the previous results, which enables us to determine the possible limits of such description of spaces and thereby to establish assertions of the Yano-type theorem. In particular, we present some examples of the spaces "close" to L∞ in a sense but lacking this description. [ABSTRACT FROM AUTHOR]

We introduce the notion of a rearrangement invariant extrapolation space on [0, 1]. We obtain some sufficient conditions (also necessary in some cases) for the Marcinkiewicz and Lorentz spaces to be extrapolation spaces. We generalize and improve the previous results, which enables us to determine the possible limits of such description of spaces and thereby to establish assertions of the Yano-type theorem. In particular, we present some examples of the spaces "close" to L∞ in a sense but lacking this description. [ABSTRACT FROM AUTHOR]

The Haar system constitutes an unconditional basis for a separable rearrangement invariant (symmetric) spaceEif and only if the multiplier determined by the sequenceforn= 0 andk= 0, 1, . . . , 2n forn>1, is bounded inE. If the Lorentz space ?(?) differs fromL1 andL8 then there is a multiplier with respect to the Haar system which is bounded in ? (?) and unbounded inL8 andL1. [ABSTRACT FROM AUTHOR]