| Title:
|
Metric Sobolev spaces (English) |
| Author:
|
Koskela, Pekka |
| Language:
|
English |
| Journal:
|
Nonlinear Analysis, Function Spaces and Applications |
| Volume:
|
Vol. 7 |
| Issue:
|
2002 |
| Year:
|
|
| Pages:
|
133-147 |
| . |
| Category:
|
math |
| . |
| Summary:
|
We describe an approach to establish a theory of metric Sobolev spaces based on Lipschitz functions and their pointwise Lipschitz constants and the Poincaré inequality. (English) |
| Keyword:
|
Lipschitz function |
| Keyword:
|
Poicaré inequality |
| Keyword:
|
upper gradient |
| Keyword:
|
Sobolev space |
| MSC:
|
46E35 |
| . |
| Date available:
|
2009-10-08T09:50:04Z |
| Last updated:
|
2012-08-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/702483 |
| . |
| Reference:
|
[1] Cheeger J.: Differentiability of Lipschitz functions on metric measure spaces.Geom. Funct. Anal. 9 (1999), 428–517. Zbl 0942.58018, MR 2000g:53043. Zbl 0942.58018, MR 1708448 |
| Reference:
|
[2] Franchi B., Hajłasz P., Koskela P.: Definitions of Sobolev classes on metric spaces.Ann. Inst. Fourier (Grenoble) 49 (1999), 1903–1924. Zbl 0938.46037, MR 2001a:46033. Zbl 0938.46037, MR 1738070 |
| Reference:
|
[3] Hajłasz P., Koskela P.: Sobolev meets Poincaré.C. R. Acad. Sci. Paris, Sér. I Math. 320 (1995), 1211–1215. Zbl 0837.46024, MR 96f:46062. MR 1336257 |
| Reference:
|
[4] Hajłasz P., Koskela P.: Sobolev met Poincaré.Mem. Amer. Math. Soc. 145 (2000). Zbl 0954.46022, MR 2000j:46063. MR 1683160 |
| Reference:
|
[5] Heinonen J., Koskela P.: Quasiconformal maps in metric spaces with controlled geometry.Acta Math. 181 (1998), 1–61. Zbl 0915.30018, MR 99j:30025. Zbl 0915.30018, MR 1654771 |
| Reference:
|
[6] Heinonen J., Koskela P., Shanmuganlingam N., Tyson J.: Sobolev spaces on metric measure spaces: an approach based on upper gradients.(in preparation). |
| Reference:
|
[7] Keith S.: A differentiable structure for metric measure spaces.Advances Math. (to appear). Zbl 1077.46027, MR 2703118 |
| Reference:
|
[8] Keith S.: Modulus and the Poincaré inequality on metric measure spaces.Math. Z. (to appear). Zbl 1037.31009, MR 2013501 |
| Reference:
|
[9] Keith S., Rajala K.: A remark on Poincaré inequalities on metric measure spaces.Preprint. Zbl 1070.31003, MR 2098359 |
| Reference:
|
[10] Koskela P., Onninen J.: Sharp inequalities via truncation.J. Math. Anal. Appl. 278 (2003), 324–334. Zbl 1019.26003, MR 1974010 |
| Reference:
|
[11] Liu Y., Lu G., Wheeden R. L.: Some equivalent definitions of high order Sobolev spaces on stratified groups and generalizations to metric spaces.Math. Ann. 323 (2002), 157–174. Zbl 1007.46034, MR 2003f:46048. Zbl 1007.46034, MR 1906913 |
| Reference:
|
[12] Lu G., Wheeden R. L.: High order representation formulas and embedding theorems on stratified groups and generalizations.Studia Math. 142 (2000), 101–133. Zbl 0974.46039, MR 2001k:46055. Zbl 0974.46039, MR 1792599 |
| Reference:
|
[13] Malý J., Pick L.: The sharp Riesz potential estimates in metric spaces.Indiana Univ. Math. J. 51 (2002), 251–268. Zbl pre01780940, MR 2003d:46045. Zbl 1038.46027, MR 1909289 |
| Reference:
|
[14] Rissanen J.: Wavelets on self-similar sets and the structure of the spaces $M^{1,p}(E,\mu )$.Ann. Acad. Sci. Fenn. Math. Diss. 125 (2002). Zbl 0993.42016, MR 2002k:42081. MR 1880640 |
| Reference:
|
[15] Semmes S.: Some novel types of fractal geometry.Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2001.Zbl 0970.28001, MR 2002h:53073. Zbl 0970.28001, MR 1815356 |
| Reference:
|
[16] Shanmugalingam N.: Newtonian spaces: an extension of Sobolev spaces to metric measure spaces.Rev. Mat. Iberoamericana 16 (2000), 243–279. Zbl 0974.46038, MR 2002b:46059. Zbl 0974.46038, MR 1809341 |
| . |
