On embeddings between classical Lorentz spaces
On embeddings between classical Lorentz spaces
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DOI：
10.7153/mia-04-37
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发表时间：
2001
影响因子：
1
通讯作者：
M. Carro;L. Pick;Javier Soria;V. Stepanov
中科院分区：
文献类型：
--
作者：
M. Carro;L. Pick;Javier Soria;V. Stepanov
Let p ∈ (0,∞) , let v be a weight on (0,∞) and let Λp(v) be the classical Lorentz space, determined by the norm ‖f ‖Λp(v) := ( ∫∞ 0 (f ∗(t))pv(t) dt)1/p . When p ∈ (1,∞) , this space is known to be a Banach space if and only if v is non-increasing, while it is only equivalent to a Banach space if and only if Λp(v) = Γp(v) , where ‖f ‖Γp(v) := ( ∫∞ 0 (f ∗∗(t))pv(t) dt)1/p . We may thus conclude that, for p ∈ (1,∞) , the space Λp(v) is equivalent to a Banach space if and only if the norm of a function f in it can be expressed in terms of f ∗∗ . We study the question whether an analogous assertion holds when p = 1 . Motivated by this problem, we consider general embeddings between four types of classical and weak Lorentz spaces, namely, Λp(v) , Λp,∞(v) , Γp(v) , Γp,∞(v) , where Λp,∞(v) and Γp,∞(v) are certain weak analogues of the spaces Λp(v) and Γp(v) , respectively. We present a unified approach to these embeddings, based on rearrangement techniques. We survey all the known results and prove new ones. Our main results concern the embedding Γp,∞(v) ↪→ Λq(w) which had not been characterized so far. We apply our results to the characterization of associate spaces of classical and weak Lorentz spaces and we give a characterization of fundamental functions for which the endpoint Lorentz space and the endpoint Marcinkiewicz space coincide. Mathematics subject classification (2000): 46E30, 26D15.
