Hostname: page-component-7479d7b7d-m9pkr Total loading time: 0 Render date: 2024-07-15T22:16:54.688Z Has data issue: false hasContentIssue false
  • English
  • Français

SOBOLEV INEQUALITIES FOR ORLICZ SPACES OF TWO VARIABLE EXPONENTS

Published online by Cambridge University Press:  25 November 2009

PETER HÄSTÖ ,
YOSHIHIRO MIZUTA ,
TAKAO OHNO  and
TETSU SHIMOMURA
PETER HÄSTÖ
Affiliation:
Department of Mathematical Sciences, PO Box 3000, FI-90014, University of Oulu, Oulu, Finland e-mail: peter.hasto@helsinki.fi
YOSHIHIRO MIZUTA
Affiliation:
Department of Mathematics, Graduate School of Science, Hiroshima University, Higashi-Hiroshima 739-8521, Japan e-mail: yomizuta@hiroshima-u.ac.jp
TAKAO OHNO
Affiliation:
General Arts, Hiroshima National College of Maritime Technology, Higashino Oosakikamijima Toyotagun 725-0231, Japan e-mail: ohno@hiroshima-cmt.ac.jp
TETSU SHIMOMURA
Affiliation:
Department of Mathematics, Graduate School of Education, Hiroshima University, Higashi-Hiroshima 739-8524, Japan e-mail: tshimo@hiroshima-u.ac.jp
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Our aim in this paper is to deal with Sobolev's embeddings for Sobolev–Orlicz functions with ∇u ∈ Lp(·) logLq(·)(Ω) for Ω ⊂ n. Here p and q are variable exponents satisfying natural continuity conditions. Also the case when p attains the value 1 in some parts of the domain is included in the results.

Keywords

46E30
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 52 , Issue 2 , May 2010 , pp. 227 - 240
Copyright
Copyright © Glasgow Mathematical Journal Trust 2009

References

REFERENCES

1.Almeida, A. and Samko, S., Pointwise inequalities in variable Sobolev spaces and applications, Z. Anal. Anwend. 26 (2) (2007), 179–193.Google Scholar
2.Capone, C., Cruz-Uribe, D. and Fiorenza, A., The fractional maximal operators on variable Lp spaces, Rev. Mat. Iberoam. 23 (3) (2007), 743–770.CrossRefGoogle Scholar
3.Çekiç, B., Mashiyev, R. and Alisoy, G. T., On the Sobolev-type inequality for Lebesgue spaces with a variable exponent, Int. Math. Forum 1 (25–28) (2006), 1313–1323.CrossRefGoogle Scholar
4.Cruz-Uribe, D. and Fiorenza, A., L log L results for the maximal operator in variable Lp spaces, Trans. Amer. Math. Soc. 361 (2009), 2631–2647.CrossRefGoogle Scholar
5.Diening, L., Riesz potential and Sobolev embeddings of generalized Lebesgue and Sobolev spaces L p(·) and W k,p(·), Math. Nachr. 263 (1) (2004), 31–43.Google Scholar
6.Diening, L., Hästö, P. and Nekvinda, A., Open problems in variable exponent Lebesgue and Sobolev spaces, in FSDONA04 Proceedings, Milovy, Czech Republic, 2004 (Drabek, P. and Rákosník, J., Editors) (Academy of Sciences of the Czech Republic, Prague, 2005), 38–58.Google Scholar
7.Edmunds, D. and Rákosník, J., Sobolev embeddings with variable exponent, Stud. Math. 143 (3) (2000), 267–293.CrossRefGoogle Scholar
8.Edmunds, D. and Rákosník, J., Sobolev embeddings with variable exponent. II, Math. Nachr. 246/247 (2002), 53–67.3.0.CO;2-T>CrossRefGoogle Scholar
9.Futamura, T., Mizuta, Y. and Shimomura, T., Sobolev embeddings for variable exponent Riesz potentials on metric spaces, Ann. Acad. Sci. Fenn. Math. 31 (2006), 495–522.Google Scholar
10.Harjulehto, P., Variable exponent Sobolev spaces with zero boundary values, Math. Bohem. 132 (2) (2007), 125–136.CrossRefGoogle Scholar
11.Harjulehto, P. and Hästö, P., A capacity approach to the Poincaré inequality and Sobolev imbeddings in variable exponent Sobolev spaces, Rev. Mat. Complut. 17 (2004), 129–146.Google Scholar
12.Harjulehto, P. and Hästö, P., Sobolev inequalities for variable exponents attaining the values 1 and n, Publ. Mat. 52 (2008), 347–363.CrossRefGoogle Scholar
13.Hästö, P., Local-to-global results in variable exponent spaces, Math. Res. Lett. 16 (2) (2009), 263–278.CrossRefGoogle Scholar
14.Kokilashvili, V. and Samko, S., On Sobolev theorem for Riesz type potentials in the Lebesgue spaces with variable exponent, Z. Anal. Anwend. 22 (4) (2003), 899–910.Google Scholar
15.Kurata, K. and Shioji, N., Compact embedding from W 1,20(Ω) to Lq(x)(Ω) and its application to nonlinear elliptic boundary value problem with variable critical exponent, J. Math. Anal. Appl. 339 (2) (2008), 1386–1394.Google Scholar
16.Mizuta, Y., Potential theory in Euclidean spaces (Gakkotosho, Tokyo, 1996).Google Scholar
17.Mizuta, Y., Sobolev's inequality for Riesz potentials with variable exponents approaching 1, in Proceedings of the 16th International Conference on Finite or Infinite Dimensional Complex Analysis and Applications (Dongguk University, Gyeongju, Korea, 2008), 189–195.Google Scholar
18.Mizuta, Y., Ohno, T. and Shimomura, T., Sobolev's inequalities and vanishing integrability for Riesz potentials of functions in the generalized Lebesgue space L p(·)(log L)q(·), J. Math. Anal. Appl. 345 (2008), 70–85.Google Scholar
19.Musielak, J.Orlicz spaces and modular spaces, Lecture Notes in Mathematics, vol. 1034 (Springer, Berlin, 1983).Google Scholar
20.O'Neil, R.Fractional integration in Orlicz spaces. I, Trans. Amer. Math. Soc. 115 (1965), 300–328.Google Scholar
21.Samko, S.On a progress in the theory of Lebesgue spaces with variable exponent: Maximal and singular operators, Integral Transform. Spec. Funct. 16 (5–6) (2005), 461–482.CrossRefGoogle Scholar
22.Samko, S., Shargorodsky, E. and Vakulov, B.Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators. II, J. Math. Anal. Appl. 325 (1) (2007), 745–751.CrossRefGoogle Scholar