On the Principle of Duality in Lorentz Spaces

Abstract

Abstractcharacterization of the spaces dual to weighted Lorentz spaces are given by means of reverse Hölder inequalities (Theorems 2.1, 2.2). This principle of duality is then applied to characterize weight functions for which the identity operator, the Hardy-Littlewood maximal operator and the Hilbert transform are bounded on weighted Lorentz spaces.

Discretization and anti-discretization of rearrangement-invariant norms

“…e.g. [1], [26], [8], [9], [10], [16], [30], [31], [32], [28]. A summary of the results on embeddings of classical Lorentz spaces known by the end of 1990's, as well as some more references, can be found in [7].…”

“…In some other cases necessary and sufficient conditions have been established, but formulated in a way which is not entirely satisfactory, as they might be quite difficult to verify. Typical examples of such conditions are those expressed in terms of the Halperin level function [7,Section 7] or in terms of discretizing sequences [16].…”

“…In [16], a new approach based on discretization techniques of [25] and [15] was applied to classical Lorentz spaces in order to obtain necessary and sufficient conditions on parameters p, q ∈ (0, ∞) and weights v, w such that the embedding Γ p (v) → Γ q (w) or the embedding Γ p (v) → Λ 1 (w) hold. The former embedding is useful in interpolation theory while a standard argument applied to the latter provides a characterization of the associate space to Γ q (w).…”

Embeddings of concave functions and duals of Lorentz spaces

“…The study of the collection of concave functions has also had its successes. See [4], [5], [10], [11] and references there. Concave functions arise naturally in interpolation theory and much of the recent work shows that they are of equal importance in weighted norm inequalities and function spaces.…”

“…A complete answer to the embedding question was given in [5] but the conditions given are complicated and difficult to apply. Our object here is to give simple necessary and sufficient weight conditions that characterize the embedding of the cone of quasi-concave functions from L p v to L q u .…”