Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
Sobolev space; metric measure space; Hajłasz-Sobolev space; Musielak-Orlicz space; capacity; variable exponent; zero boundary values
Sobolev space; metric measure space; Hajłasz-Sobolev space; Musielak-Orlicz space; capacity; variable exponent; zero boundary values
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.
References:
[1] Adams, R. A.: Sobolev Spaces. Pure and Applied Mathematics 65 Academic Press, New York (1975). MR 0450957 | Zbl 0314.46030
[2] Adams, D. R., Hedberg, L. I.: Function Spaces and Potential Theory. Grundlehren der Mathematischen Wissenschaften 314 Springer, Berlin (1996). DOI 10.1007/978-3-662-03282-4 | MR 1411441
[3] ssaoui, N. Aï: Another extension of Orlicz-Sobolev spaces to metric spaces. Abstr. Appl. Anal. 2004 (2004), 1-26. MR 2058790
[4] ssaoui, N. Aï: Strongly nonlinear potential theory of metric spaces. Abstr. Appl. Anal. 7 (2002), 357-374. DOI 10.1155/S1085337502203024 | MR 1939129
[5] Bennett, C., Sharpley, R.: Interpolation of Operators. Pure and Applied Mathematics 129 Academic Press, Boston (1988). MR 0928802 | Zbl 0647.46057
[6] Björn, A., Björn, J.: Nonlinear Potential Theory on Metric Spaces. EMS Tracts in Mathematics, 17. European Mathematical Society Zürich (2011). MR 2867756 | Zbl 1231.31001
[7] Cruz-Uribe, D., Fiorenza, A.: Variable Lebesgue Spaces. Foundations and Harmonic Analysis. Applied and Numerical Harmonic Analysis Birkhäuser, Heidelberg (2013). MR 3026953 | Zbl 1268.46002
[8] Diening, L., Harjulehto, P., Hästö, P., Růžička, M.: Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Mathematics 2017 Springer, Berlin (2011). MR 2790542 | Zbl 1222.46002
[9] Evans, L. C., Gariepy, R. F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics CRC Press, Boca Raton (1992). MR 1158660 | Zbl 0804.28001
[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
[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
[12] Hajłasz, P.: Sobolev spaces on an arbitrary metric space. Potential Anal. 5 (1996), 403-415. Zbl 0859.46022
[13] Hajłasz, P., Koskela, P.: Sobolev met Poincaré. Mem. Am. Math. Soc. 145 (2000), 101 pages. MR 1683160 | Zbl 0954.46022
[14] Hajłasz, P., Koskela, P., Tuominen, H.: Sobolev embeddings, extensions and measure density condition. J. Funct. Anal. 254 (2008), 1217-1234. DOI 10.1016/j.jfa.2007.11.020 | MR 2386936 | Zbl 1136.46029
[15] Harjulehto, P.: Variable exponent Sobolev spaces with zero boundary values. Math. Bohem. 132 (2007), 125-136. MR 2338802 | Zbl 1174.46322
[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. DOI 10.5209/rev_REMA.2004.v17.n1.16790 | MR 2063945 | Zbl 1072.46021
[17] Harjulehto, P., Hästö, P., Koskenoja, M.: Properties of capacities in variable exponent Sobolev spaces. J. Anal. Appl. 5 (2007), 71-92. MR 2314780 | Zbl 1143.31003
[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. DOI 10.1007/s11118-006-9023-3 | MR 2255345 | Zbl 1120.46016
[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
[20] Harjulehto, P., Hästö, P., Pere, M.: Variable exponent Sobolev spaces on metric measure spaces. Funct. Approx. Comment. Math. 36 (2006), 79-94. DOI 10.7169/facm/1229616443 | MR 2296640 | Zbl 1140.46013
[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. MR 2126796 | Zbl 1072.42016
[22] Heinonen, J.: Lectures on Analysis on Metric Spaces. Springer, New York (2001). MR 1800917 | Zbl 0985.46008
[23] Kilpeläinen, T.: A remark on the uniqueness of quasi continuous functions. Ann. Acad. Sci. Fenn. Math. 23 (1998), 261-262. MR 1601887 | Zbl 0919.31006
[24] Kilpeläinen, T., Kinnunen, J., Martio, O.: Sobolev spaces with zero boundary values on metric spaces. Potential Anal. 12 (2000), 233-247. DOI 10.1023/A:1008601220456 | MR 1752853
[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. DOI 10.1007/s00526-011-0420-0 | MR 2875650 | Zbl 1238.31008
[26] Kinnunen, J., Latvala, V.: Lebesgue points for Sobolev functions on metric spaces. Rev. Mat. Iberoam. 18 (2002), 685-700. DOI 10.4171/RMI/332 | MR 1954868 | Zbl 1037.46031
[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. MR 1847522 | Zbl 0992.46023
[28] Kinnunen, J., Martio, O.: The Sobolev capacity on metric spaces. Ann. Acad. Sci. Fenn. Math. 21 (1996), 367-382. MR 1404091 | Zbl 0859.46023
[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. DOI 10.1007/s10587-013-0063-8 | MR 3165506 | Zbl 1313.46041
[30] Mizuta, Y., Shimomura, T.: Continuity of Sobolev functions of variable exponent on metric spaces. Proc. Japan Acad. Ser. A 80 (2004), 96-99. MR 2075449 | Zbl 1072.46506
[31] Musielak, J.: Orlicz Spaces and Modular Spaces. Lecture Notes Math. 1034 Springer, Berlin (1983). MR 0724434 | Zbl 0557.46020
[32] Ohno, T., Shimomura, T.: Musielak-Orlicz-Sobolev spaces on metric measure spaces. Czech. Math. J. 65 (2015), 435-474. DOI 10.1007/s10587-015-0187-0 | MR 3360438 | Zbl 1363.46027
[33] Shanmugalingam, N.: Newtonian space: An extension of Sobolev spaces to metric measure space. Rev. Mat. Iberoam. 16 (2000), 243-279. DOI 10.4171/RMI/275 | MR 1809341
[34] Tuominen, H.: Orlicz-Sobolev Spaces on Metric Spaces. Annales Academiæ Scientiarum Fennicæ. Mathematica. Dissertationes 135 (2004), Suomalainen Tiedeakatemia, Helsinki.
[35] Ziemer, W. P.: Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation. Graduate Texts in Mathematics 120 Springer, Berlin (1989). DOI 10.1007/978-1-4612-1015-3 | MR 1014685 | Zbl 0692.46022
Search
Partner of
