| Title:
|
Normability of Lorentz spaces—an alternative approach (English) |
| Author:
|
Gogatishvili, Amiran |
| Author:
|
Soudský, Filip |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
64 |
| Issue:
|
3 |
| Year:
|
2014 |
| Pages:
|
581-597 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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 $v$. (English) |
| Keyword:
|
weighted Lorentz space |
| Keyword:
|
weighted inequality |
| Keyword:
|
non-increasing rearrangement |
| Keyword:
|
Banach function space |
| Keyword:
|
associate space |
| MSC:
|
46E30 |
| idZBL:
|
Zbl 06391513 |
| idMR:
|
MR3298548 |
| DOI:
|
10.1007/s10587-014-0120-y |
| . |
| Date available:
|
2014-12-19T15:53:56Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144045 |
| . |
| Reference:
|
[1] Bennett, C., Sharpley, R.: Interpolation of Operators.Pure and Applied Mathematics 129 Academic Press, Boston (1988). Zbl 0647.46057, MR 0928802 |
| Reference:
|
[2] Carro, M., Pick, L., Soria, J., Stepanov, V. D.: On embeddings between classical Lorentz spaces.Math. Inequal. Appl. 4 (2001), 397-428. Zbl 0996.46013, MR 1841071 |
| Reference:
|
[3] Cwikel, M., Kamińska, A., Maligranda, L., Pick, L.: Are generalized Lorentz ``spaces'' really spaces?.Proc. Am. Math. Soc. 132 (2004), 3615-3625. Zbl 1061.46026, MR 2084084, 10.1090/S0002-9939-04-07477-5 |
| Reference:
|
[4] Gogatishvili, A., Pick, L.: Embeddings and duality theorem for weak classical Lorentz spaces.Can. Math. Bull. 49 (2006), 82-95. Zbl 1106.26018, MR 2198721, 10.4153/CMB-2006-008-3 |
| Reference:
|
[5] Gogatishvili, A., Pick, L.: Discretization and anti-discretization of rearrangement-invariant norms.Publ. Mat., Barc. 47 (2003), 311-358. Zbl 1066.46023, MR 2006487, 10.5565/PUBLMAT_47203_02 |
| Reference:
|
[6] Lorentz, G. G.: On the theory of spaces $\Lambda$.Pac. J. Math. 1 (1951), 411-429. Zbl 0043.11302, MR 0044740, 10.2140/pjm.1951.1.411 |
| Reference:
|
[7] Sawyer, E.: Boundedness of classical operators on classical Lorentz spaces.Stud. Math. 96 (1990), 145-158. Zbl 0705.42014, MR 1052631, 10.4064/sm-96-2-145-158 |
| . |
