| Title:
|
Musielak-Orlicz-Sobolev spaces with zero boundary values on metric measure spaces (English) |
| Author:
|
Ohno, Takao |
| Author:
|
Shimomura, Tetsu |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
66 |
| Issue:
|
2 |
| Year:
|
2016 |
| Pages:
|
371-394 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We define and study Musielak-Orlicz-Sobolev spaces with zero boundary values on any metric space endowed with a Borel regular measure. We extend many classical results, including completeness, lattice properties and removable sets, to Musielak-Orlicz-Sobolev spaces on metric measure spaces. We give sufficient conditions which guarantee that a Sobolev function can be approximated by Lipschitz continuous functions vanishing outside an open set. These conditions are based on Hardy type inequalities. (English) |
| Keyword:
|
Sobolev space |
| Keyword:
|
metric measure space |
| Keyword:
|
Hajłasz-Sobolev space |
| Keyword:
|
Musielak-Orlicz space |
| Keyword:
|
capacity |
| Keyword:
|
variable exponent |
| Keyword:
|
zero boundary values |
| MSC:
|
31B15 |
| MSC:
|
46E35 |
| idZBL:
|
Zbl 06604473 |
| idMR:
|
MR3519608 |
| DOI:
|
10.1007/s10587-016-0262-1 |
| . |
| Date available:
|
2016-06-16T12:46:03Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/145730 |
| . |
| Reference:
|
[1] Adams, R. A.: Sobolev Spaces.Pure and Applied Mathematics 65 Academic Press, New York (1975). Zbl 0314.46030, MR 0450957 |
| Reference:
|
[2] Adams, D. R., Hedberg, L. I.: Function Spaces and Potential Theory.Grundlehren der Mathematischen Wissenschaften 314 Springer, Berlin (1996). MR 1411441, 10.1007/978-3-662-03282-4 |
| Reference:
|
[3] ssaoui, N. Aï: Another extension of Orlicz-Sobolev spaces to metric spaces.Abstr. Appl. Anal. 2004 (2004), 1-26. MR 2058790 |
| Reference:
|
[4] ssaoui, N. Aï: Strongly nonlinear potential theory of metric spaces.Abstr. Appl. Anal. 7 (2002), 357-374. MR 1939129, 10.1155/S1085337502203024 |
| Reference:
|
[5] Bennett, C., Sharpley, R.: Interpolation of Operators.Pure and Applied Mathematics 129 Academic Press, Boston (1988). Zbl 0647.46057, MR 0928802 |
| Reference:
|
[6] Björn, A., Björn, J.: Nonlinear Potential Theory on Metric Spaces.EMS Tracts in Mathematics, 17. European Mathematical Society Zürich (2011). Zbl 1231.31001, MR 2867756 |
| Reference:
|
[7] Cruz-Uribe, D., Fiorenza, A.: Variable Lebesgue Spaces. Foundations and Harmonic Analysis.Applied and Numerical Harmonic Analysis Birkhäuser, Heidelberg (2013). Zbl 1268.46002, MR 3026953 |
| Reference:
|
[8] Diening, L., Harjulehto, P., Hästö, P., Růžička, M.: Lebesgue and Sobolev Spaces with Variable Exponents.Lecture Notes in Mathematics 2017 Springer, Berlin (2011). Zbl 1222.46002, MR 2790542 |
| Reference:
|
[9] Evans, L. C., Gariepy, R. F.: Measure Theory and Fine Properties of Functions.Studies in Advanced Mathematics CRC Press, Boca Raton (1992). Zbl 0804.28001, MR 1158660 |
| Reference:
|
[10] Futamura, T., Harjulehto, P., Hästö, P., Mizuta, Y., Shimomura, T.: Variable exponent spaces on metric measure spaces.More Progresses in Analysis. Proc. of the 5th international ISAAC congress, Catania 2005 H. G. W. Begehr et al. World Scientific (2009), 107-121. Zbl 1189.46027 |
| Reference:
|
[11] Futamura, T., Mizuta, Y., Shimomura, T.: Sobolev embedding for variable exponent Riesz potentials on metric spaces.Ann. Acad. Sci. Fenn. Math. 31 (2006), 495-522. MR 2248828 |
| Reference:
|
[12] Hajłasz, P.: Sobolev spaces on an arbitrary metric space.Potential Anal. 5 (1996), 403-415. Zbl 0859.46022 |
| Reference:
|
[13] Hajłasz, P., Koskela, P.: Sobolev met Poincaré.Mem. Am. Math. Soc. 145 (2000), 101 pages. Zbl 0954.46022, MR 1683160 |
| Reference:
|
[14] Hajłasz, P., Koskela, P., Tuominen, H.: Sobolev embeddings, extensions and measure density condition.J. Funct. Anal. 254 (2008), 1217-1234. Zbl 1136.46029, MR 2386936, 10.1016/j.jfa.2007.11.020 |
| Reference:
|
[15] Harjulehto, P.: Variable exponent Sobolev spaces with zero boundary values.Math. Bohem. 132 (2007), 125-136. Zbl 1174.46322, MR 2338802 |
| Reference:
|
[16] Harjulehto, P., Hästö, P.: A capacity approach to the Poincaré inequality and Sobolev imbeddings in variable exponent Sobolev spaces.Rev. Mat. Complut. 17 (2004), 129-146. Zbl 1072.46021, MR 2063945, 10.5209/rev_REMA.2004.v17.n1.16790 |
| Reference:
|
[17] Harjulehto, P., Hästö, P., Koskenoja, M.: Properties of capacities in variable exponent Sobolev spaces.J. Anal. Appl. 5 (2007), 71-92. Zbl 1143.31003, MR 2314780 |
| Reference:
|
[18] Harjulehto, P., Hästö, P., Koskenoja, M., Varonen, S.: The Dirichlet energy integral and variable exponent Sobolev spaces with zero boundary values.Potential Anal. 25 (2006), 205-222. Zbl 1120.46016, MR 2255345, 10.1007/s11118-006-9023-3 |
| Reference:
|
[19] Harjulehto, P., Hästö, P., Koskenoja, M., Varonen, S.: Sobolev capacity on the space $W^{1,p(\cdot)}(\mathbb R^n)$.J. Funct. Spaces Appl. 1 (2003), 17-33. MR 2011498 |
| Reference:
|
[20] Harjulehto, P., Hästö, P., Pere, M.: Variable exponent Sobolev spaces on metric measure spaces.Funct. Approx. Comment. Math. 36 (2006), 79-94. Zbl 1140.46013, MR 2296640, 10.7169/facm/1229616443 |
| Reference:
|
[21] Harjulehto, P., Hästö, P., Pere, M.: Variable exponent Lebesgue spaces on metric spaces: the Hardy-Littlewood maximal operator.Real Anal. Exch. 30 (2005), 87-103. Zbl 1072.42016, MR 2126796 |
| Reference:
|
[22] Heinonen, J.: Lectures on Analysis on Metric Spaces.Springer, New York (2001). Zbl 0985.46008, MR 1800917 |
| Reference:
|
[23] Kilpeläinen, T.: A remark on the uniqueness of quasi continuous functions.Ann. Acad. Sci. Fenn. Math. 23 (1998), 261-262. Zbl 0919.31006, MR 1601887 |
| Reference:
|
[24] Kilpeläinen, T., Kinnunen, J., Martio, O.: Sobolev spaces with zero boundary values on metric spaces.Potential Anal. 12 (2000), 233-247. MR 1752853, 10.1023/A:1008601220456 |
| Reference:
|
[25] Kinnunen, J., Korte, R., Shanmugalingam, N., Tuominen, H.: A characterization of Newtonian functions with zero boundary values.Calc. Var. Partial Differ. Equ. 43 (2012), 507-528. Zbl 1238.31008, MR 2875650, 10.1007/s00526-011-0420-0 |
| Reference:
|
[26] Kinnunen, J., Latvala, V.: Lebesgue points for Sobolev functions on metric spaces.Rev. Mat. Iberoam. 18 (2002), 685-700. Zbl 1037.46031, MR 1954868, 10.4171/RMI/332 |
| Reference:
|
[27] Kinnunen, J., Martio, O.: Choquet property for the Sobolev capacity in metric spaces.S. K. Vodopyanov Proc. on Analysis and Geometry Sobolev Institute Press, Novosibirsk (2000), 285-290. Zbl 0992.46023, MR 1847522 |
| Reference:
|
[28] Kinnunen, J., Martio, O.: The Sobolev capacity on metric spaces.Ann. Acad. Sci. Fenn. Math. 21 (1996), 367-382. Zbl 0859.46023, MR 1404091 |
| Reference:
|
[29] Maeda, F.-Y., Mizuta, Y., Ohno, T., Shimomura, T.: Approximate identities and Young type inequalities in Musielak-Orlicz spaces.Czech. Math. J. 63 (2013), 933-948. Zbl 1313.46041, MR 3165506, 10.1007/s10587-013-0063-8 |
| Reference:
|
[30] Mizuta, Y., Shimomura, T.: Continuity of Sobolev functions of variable exponent on metric spaces.Proc. Japan Acad. Ser. A 80 (2004), 96-99. Zbl 1072.46506, MR 2075449 |
| Reference:
|
[31] Musielak, J.: Orlicz Spaces and Modular Spaces.Lecture Notes Math. 1034 Springer, Berlin (1983). Zbl 0557.46020, MR 0724434 |
| Reference:
|
[32] Ohno, T., Shimomura, T.: Musielak-Orlicz-Sobolev spaces on metric measure spaces.Czech. Math. J. 65 (2015), 435-474. Zbl 1363.46027, MR 3360438, 10.1007/s10587-015-0187-0 |
| Reference:
|
[33] Shanmugalingam, N.: Newtonian space: An extension of Sobolev spaces to metric measure space.Rev. Mat. Iberoam. 16 (2000), 243-279. MR 1809341, 10.4171/RMI/275 |
| Reference:
|
[34] Tuominen, H.: Orlicz-Sobolev Spaces on Metric Spaces.Annales Academiæ Scientiarum Fennicæ. Mathematica. Dissertationes 135 (2004), Suomalainen Tiedeakatemia, Helsinki. |
| Reference:
|
[35] Ziemer, W. P.: Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation.Graduate Texts in Mathematics 120 Springer, Berlin (1989). Zbl 0692.46022, MR 1014685, 10.1007/978-1-4612-1015-3 |
| . |
