Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
weighted Lorentz space; weighted inequality; non-increasing rearrangement; Banach function space; associate space
weighted Lorentz space; weighted inequality; non-increasing rearrangement; Banach function space; associate space
Summary:
We study normability properties of classical Lorentz spaces. Given a certain general lattice-like structure, we first prove a general sufficient condition for its associate space to be a Banach function space. We use this result to develop an alternative approach to Sawyer's characterization of normability of a classical Lorentz space of type $\Lambda $. Furthermore, we also use this method in the weak case and characterize normability of $\Lambda _{v}^{\infty }$. Finally, we characterize the linearity of the space $\Lambda _{v}^{\infty }$ by a simple condition on the weight $v$.
References:
[1] Bennett, C., Sharpley, R.: Interpolation of Operators. Pure and Applied Mathematics 129 Academic Press, Boston (1988). MR 0928802 | Zbl 0647.46057
[2] Carro, M., Pick, L., Soria, J., Stepanov, V. D.: On embeddings between classical Lorentz spaces. Math. Inequal. Appl. 4 (2001), 397-428. MR 1841071 | Zbl 0996.46013
[3] Cwikel, M., Kamińska, A., Maligranda, L., Pick, L.: Are generalized Lorentz ``spaces'' really spaces?. Proc. Am. Math. Soc. 132 (2004), 3615-3625. DOI 10.1090/S0002-9939-04-07477-5 | MR 2084084 | Zbl 1061.46026
[4] Gogatishvili, A., Pick, L.: Embeddings and duality theorem for weak classical Lorentz spaces. Can. Math. Bull. 49 (2006), 82-95. DOI 10.4153/CMB-2006-008-3 | MR 2198721 | Zbl 1106.26018
[5] Gogatishvili, A., Pick, L.: Discretization and anti-discretization of rearrangement-invariant norms. Publ. Mat., Barc. 47 (2003), 311-358. DOI 10.5565/PUBLMAT_47203_02 | MR 2006487 | Zbl 1066.46023
[6] Lorentz, G. G.: On the theory of spaces $\Lambda$. Pac. J. Math. 1 (1951), 411-429. DOI 10.2140/pjm.1951.1.411 | MR 0044740 | Zbl 0043.11302
[7] Sawyer, E.: Boundedness of classical operators on classical Lorentz spaces. Stud. Math. 96 (1990), 145-158. DOI 10.4064/sm-96-2-145-158 | MR 1052631 | Zbl 0705.42014
Search
Partner of
