| Title:
|
Some notes on embedding for anisotropic Sobolev spaces (English) |
| Author:
|
Li, Hongliang |
| Author:
|
Sun, Quinxiu |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
61 |
| Issue:
|
1 |
| Year:
|
2011 |
| Pages:
|
97-111 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper, we prove new embedding theorems for generalized anisotropic Sobolev spaces, $W_{\Lambda ^{p,q}(w)}^{r_1,\dots ,r_n}$ and $W_{X}^{r_1,\dots ,r_n}$, where $\Lambda ^{p,q}(w)$ is the weighted Lorentz space and $X$ is a rearrangement invariant space in $\mathbb R^n$. The main methods used in the paper are based on some estimates of nonincreasing rearrangements and the applications of $B_p$ weights. (English) |
| Keyword:
|
Lorentz spaces |
| Keyword:
|
Sobolev spaces |
| Keyword:
|
Besov spaces |
| Keyword:
|
Sobolev embedding |
| Keyword:
|
rearrangement invariant spaces |
| MSC:
|
42B35 |
| MSC:
|
46E35 |
| idZBL:
|
Zbl 1224.46065 |
| idMR:
|
MR2782762 |
| DOI:
|
10.1007/s10587-011-0020-3 |
| . |
| Date available:
|
2011-05-23T12:34:27Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141521 |
| . |
| Reference:
|
[1] Bastero, J., Milman, M., Blasco, F. Ruiz: A note on $L(\infty,q)$ spaces and Sobolev embeddings.Indiana Univ. Math. J. 52 (2003), 1215-1230. MR 2010324, 10.1512/iumj.2003.52.2364 |
| Reference:
|
[2] Bennett, C., Sharpley, R.: Interpolation of Operators. Pure and Applied Mathematics, Vol. 129.Academic Press Boston (1988). MR 0928802 |
| Reference:
|
[3] Besov, O. V., Il'in, V. P., Nikol'skij, S. M.: Integral Representation of Functions and Imbedding Theorems, Vol. 1-2.V. H. Winston/John Wiley & Sons Washington, D. C./New York-Toronto-London (1978). |
| Reference:
|
[4] Boyd, D. W.: The Hilbert transform on rearrangement-invariant spaces.Can. J. Math. 19 (1967), 599-616. Zbl 0147.11302, MR 0212512, 10.4153/CJM-1967-053-7 |
| Reference:
|
[5] Carro, M. J., Raposo, J. A., Soria, J.: Recent Developments in the Theory of Lorentz Spaces and Weighted Inequalities. Mem. Amer. Math. Soc. Vol. 877.(2007). MR 2308059 |
| Reference:
|
[6] Kolyada, V. I.: On an embedding of Sobolev spaces.Mat. Zametki 54 (1993), 48-71; English transl.: Math. Notes {\it 54}, (1993), 908-922. Zbl 0821.46043, MR 1248284 |
| Reference:
|
[7] Kolyada, V. I.: Rearrangement of functions and embedding of anisotropic spaces of Sobolev type.East J. Approx. 4 (1998), 111-199. Zbl 0917.46019, MR 1638343 |
| Reference:
|
[8] Kolyada, V. I., Pérez, F. J.: Estimates of difference norms for functions in anisotropic Sobolev spaces.Math. Nachr. 267 (2004), 46-64. MR 2047384, 10.1002/mana.200310152 |
| Reference:
|
[9] Kudryavtsev, L. D., Nikol'skij, S. M.: Spaces of Differentiable Functions of Several Variables and Embedding Theorems. Current problems in mathematics. Fundamental directions.Russian Itogi nauki i Techniki, Akad. Nauk SSSR Moscow 26 (1988), 5-157. MR 1178111 |
| Reference:
|
[10] Martín, J.: Symmetrization inequalities in the fractional case and Besov embeddings.J. Math. Anal. Appl. 344 (2008), 99-123. MR 2416295, 10.1016/j.jmaa.2008.02.028 |
| Reference:
|
[11] Milman, M., Pustylnik, E.: On sharp higher order Sobolev embeddings.Commun. Contemp. Math. 6 (2004), 495-511. Zbl 1108.46029, MR 2068850, 10.1142/S0219199704001380 |
| Reference:
|
[12] Nikol'skij, S. M.: Approximation of Functions of Several Variables and Imbedding Theorems.Springer Berlin-Heidelberg-New York (1975). Zbl 0307.46024 |
| Reference:
|
[13] Lázaro, F. J. Pérez: A note on extreme cases of Sobolev embeddings.J. Math. Anal. Appl. 320 (2006), 973-982. MR 2226008, 10.1016/j.jmaa.2005.07.019 |
| Reference:
|
[14] Lázaro, F. J. Pérez: Embeddings for anisotropic Besov spaces.Acta Math. Hung. 119 (2008), 25-40. MR 2400793, 10.1007/s10474-007-6235-y |
| Reference:
|
[15] Sobolev, S. L.: On the theorem of functional analysis.Mat. Sb. 4(46) (1938), 471-497. |
| Reference:
|
[16] Soria, J.: Lorentz spaces of weak-type.Quart. J. Math. Oxf., II. Ser. 49 (1998), 93-103. Zbl 0943.42010, MR 1617343, 10.1093/qmathj/49.1.93 |
| Reference:
|
[17] Triebel, H.: Theory of Function Spaces.Birkhäuser Basel (1983). Zbl 0546.46028, MR 0781540 |
| Reference:
|
[18] Triebel, H.: Theory of Function Spaces II.Birkhäuser Basel (1992). Zbl 0763.46025, MR 1163193 |
| . |
