Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
Lorentz spaces; Sobolev spaces; Besov spaces; Sobolev embedding; rearrangement invariant spaces
Lorentz spaces; Sobolev spaces; Besov spaces; Sobolev embedding; rearrangement invariant spaces
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.
References:
[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. DOI 10.1512/iumj.2003.52.2364 | MR 2010324
[2] Bennett, C., Sharpley, R.: Interpolation of Operators. Pure and Applied Mathematics, Vol. 129. Academic Press Boston (1988). MR 0928802
[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).
[4] Boyd, D. W.: The Hilbert transform on rearrangement-invariant spaces. Can. J. Math. 19 (1967), 599-616. DOI 10.4153/CJM-1967-053-7 | MR 0212512 | Zbl 0147.11302
[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
[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. MR 1248284 | Zbl 0821.46043
[7] Kolyada, V. I.: Rearrangement of functions and embedding of anisotropic spaces of Sobolev type. East J. Approx. 4 (1998), 111-199. MR 1638343 | Zbl 0917.46019
[8] Kolyada, V. I., Pérez, F. J.: Estimates of difference norms for functions in anisotropic Sobolev spaces. Math. Nachr. 267 (2004), 46-64. DOI 10.1002/mana.200310152 | MR 2047384
[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
[10] Martín, J.: Symmetrization inequalities in the fractional case and Besov embeddings. J. Math. Anal. Appl. 344 (2008), 99-123. DOI 10.1016/j.jmaa.2008.02.028 | MR 2416295
[11] Milman, M., Pustylnik, E.: On sharp higher order Sobolev embeddings. Commun. Contemp. Math. 6 (2004), 495-511. DOI 10.1142/S0219199704001380 | MR 2068850 | Zbl 1108.46029
[12] Nikol'skij, S. M.: Approximation of Functions of Several Variables and Imbedding Theorems. Springer Berlin-Heidelberg-New York (1975). Zbl 0307.46024
[13] Lázaro, F. J. Pérez: A note on extreme cases of Sobolev embeddings. J. Math. Anal. Appl. 320 (2006), 973-982. DOI 10.1016/j.jmaa.2005.07.019 | MR 2226008
[14] Lázaro, F. J. Pérez: Embeddings for anisotropic Besov spaces. Acta Math. Hung. 119 (2008), 25-40. DOI 10.1007/s10474-007-6235-y | MR 2400793
[15] Sobolev, S. L.: On the theorem of functional analysis. Mat. Sb. 4(46) (1938), 471-497.
[16] Soria, J.: Lorentz spaces of weak-type. Quart. J. Math. Oxf., II. Ser. 49 (1998), 93-103. DOI 10.1093/qmathj/49.1.93 | MR 1617343 | Zbl 0943.42010
[17] Triebel, H.: Theory of Function Spaces. Birkhäuser Basel (1983). MR 0781540 | Zbl 0546.46028
[18] Triebel, H.: Theory of Function Spaces II. Birkhäuser Basel (1992). MR 1163193 | Zbl 0763.46025
Search
Partner of
