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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: 2016-04-07
Stable URL:
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


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