Musielak-Orlicz-Sobolev spaces with zero boundary values on metric measure spaces

Ohno, Takao ; Shimomura, Tetsu

Czechoslovak Mathematical Journal, Tome 66 (2016) no. 2, pp. 371-394

Abstract (VO)

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.

Détail

DOI : 10.1007/s10587-016-0262-1

Classification : 31B15, 46E35
Keywords: Sobolev space; metric measure space; Hajłasz-Sobolev space; Musielak-Orlicz space; capacity; variable exponent; zero boundary values

Bibliographie

[1] Adams, R. A.: Sobolev Spaces. Pure and Applied Mathematics 65 Academic Press, New York (1975). | MR | Zbl

[2] Adams, D. R., Hedberg, L. I.: Function Spaces and Potential Theory. Grundlehren der Mathematischen Wissenschaften 314 Springer, Berlin (1996). | DOI | MR

[3] ssaoui, N. Aï: Another extension of Orlicz-Sobolev spaces to metric spaces. Abstr. Appl. Anal. 2004 (2004), 1-26. | MR

[4] ssaoui, N. Aï: Strongly nonlinear potential theory of metric spaces. Abstr. Appl. Anal. 7 (2002), 357-374. | DOI | MR

[5] Bennett, C., Sharpley, R.: Interpolation of Operators. Pure and Applied Mathematics 129 Academic Press, Boston (1988). | MR | Zbl

[6] Björn, A., Björn, J.: Nonlinear Potential Theory on Metric Spaces. EMS Tracts in Mathematics, 17. European Mathematical Society Zürich (2011). | MR | Zbl

[7] Cruz-Uribe, D., Fiorenza, A.: Variable Lebesgue Spaces. Foundations and Harmonic Analysis. Applied and Numerical Harmonic Analysis Birkhäuser, Heidelberg (2013). | MR | Zbl

[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 | Zbl

[9] Evans, L. C., Gariepy, R. F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics CRC Press, Boca Raton (1992). | MR | Zbl

[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

[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

[12] Hajłasz, P.: Sobolev spaces on an arbitrary metric space. Potential Anal. 5 (1996), 403-415. | Zbl

[13] Hajłasz, P., Koskela, P.: Sobolev met Poincaré. Mem. Am. Math. Soc. 145 (2000), 101 pages. | MR | Zbl

[14] Hajłasz, P., Koskela, P., Tuominen, H.: Sobolev embeddings, extensions and measure density condition. J. Funct. Anal. 254 (2008), 1217-1234. | DOI | MR | Zbl

[15] Harjulehto, P.: Variable exponent Sobolev spaces with zero boundary values. Math. Bohem. 132 (2007), 125-136. | MR | Zbl

[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 | MR | Zbl

[17] Harjulehto, P., Hästö, P., Koskenoja, M.: Properties of capacities in variable exponent Sobolev spaces. J. Anal. Appl. 5 (2007), 71-92. | MR | Zbl

[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 | MR | Zbl

[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

[20] Harjulehto, P., Hästö, P., Pere, M.: Variable exponent Sobolev spaces on metric measure spaces. Funct. Approx. Comment. Math. 36 (2006), 79-94. | DOI | MR | Zbl

[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 | Zbl

[22] Heinonen, J.: Lectures on Analysis on Metric Spaces. Springer, New York (2001). | MR | Zbl

[23] Kilpeläinen, T.: A remark on the uniqueness of quasi continuous functions. Ann. Acad. Sci. Fenn. Math. 23 (1998), 261-262. | MR | Zbl

[24] Kilpeläinen, T., Kinnunen, J., Martio, O.: Sobolev spaces with zero boundary values on metric spaces. Potential Anal. 12 (2000), 233-247. | DOI | MR

[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 | MR | Zbl

[26] Kinnunen, J., Latvala, V.: Lebesgue points for Sobolev functions on metric spaces. Rev. Mat. Iberoam. 18 (2002), 685-700. | DOI | MR | Zbl

[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 | Zbl

[28] Kinnunen, J., Martio, O.: The Sobolev capacity on metric spaces. Ann. Acad. Sci. Fenn. Math. 21 (1996), 367-382. | MR | Zbl

[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 | MR | Zbl

[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 | Zbl

[31] Musielak, J.: Orlicz Spaces and Modular Spaces. Lecture Notes Math. 1034 Springer, Berlin (1983). | MR | Zbl

[32] Ohno, T., Shimomura, T.: Musielak-Orlicz-Sobolev spaces on metric measure spaces. Czech. Math. J. 65 (2015), 435-474. | DOI | MR | Zbl

[33] Shanmugalingam, N.: Newtonian space: An extension of Sobolev spaces to metric measure space. Rev. Mat. Iberoam. 16 (2000), 243-279. | DOI | MR

[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 | MR | Zbl