Uniform convexity and associate spaces

Petteri Harjulehto; Peter Hästö

Czechoslovak Mathematical Journal (2018)

  • Volume: 68, Issue: 4, page 1011-1020
  • ISSN: 0011-4642

Abstract

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We prove that the associate space of a generalized Orlicz space L φ ( · ) is given by the conjugate modular φ * even without the assumption that simple functions belong to the space. Second, we show that every weakly doubling Φ -function is equivalent to a doubling Φ -function. As a consequence, we conclude that L φ ( · ) is uniformly convex if φ and φ * are weakly doubling.

How to cite

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Harjulehto, Petteri, and Hästö, Peter. "Uniform convexity and associate spaces." Czechoslovak Mathematical Journal 68.4 (2018): 1011-1020. <http://eudml.org/doc/294167>.

@article{Harjulehto2018,
abstract = {We prove that the associate space of a generalized Orlicz space $L^\{\phi (\cdot )\}$ is given by the conjugate modular $\phi ^*$ even without the assumption that simple functions belong to the space. Second, we show that every weakly doubling $\Phi $-function is equivalent to a doubling $\Phi $-function. As a consequence, we conclude that $L^\{\phi (\cdot )\}$ is uniformly convex if $\phi $ and $\phi ^*$ are weakly doubling.},
author = {Harjulehto, Petteri, Hästö, Peter},
journal = {Czechoslovak Mathematical Journal},
keywords = {generalized Orlicz space; Musielak-Orlicz space; nonstandard growth; variable exponent; double phase; uniform convexity; associate space},
language = {eng},
number = {4},
pages = {1011-1020},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Uniform convexity and associate spaces},
url = {http://eudml.org/doc/294167},
volume = {68},
year = {2018},
}

TY - JOUR
AU - Harjulehto, Petteri
AU - Hästö, Peter
TI - Uniform convexity and associate spaces
JO - Czechoslovak Mathematical Journal
PY - 2018
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 68
IS - 4
SP - 1011
EP - 1020
AB - We prove that the associate space of a generalized Orlicz space $L^{\phi (\cdot )}$ is given by the conjugate modular $\phi ^*$ even without the assumption that simple functions belong to the space. Second, we show that every weakly doubling $\Phi $-function is equivalent to a doubling $\Phi $-function. As a consequence, we conclude that $L^{\phi (\cdot )}$ is uniformly convex if $\phi $ and $\phi ^*$ are weakly doubling.
LA - eng
KW - generalized Orlicz space; Musielak-Orlicz space; nonstandard growth; variable exponent; double phase; uniform convexity; associate space
UR - http://eudml.org/doc/294167
ER -

References

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  1. Adams, R., Sobolev Spaces, Pure and Applied Mathematics 65, Academic Press, New York (1975). (1975) Zbl0314.46030MR0450957
  2. Avci, M., Pankov, A., 10.1515/anona-2016-0043, Adv. Nonlinear Anal. 7 (2018), 35-48. (2018) Zbl06837817MR3757454DOI10.1515/anona-2016-0043
  3. Baroni, P., Colombo, M., Mingione, G., 10.1090/spmj/1392, St. Petersbg. Math. J. 27 (2016), 347-379 translation from Algebra Anal. 27 2015 6-50. (2016) Zbl1335.49057MR3570955DOI10.1090/spmj/1392
  4. Colombo, M., Mingione, G., 10.1007/s00205-014-0785-2, Arch. Ration. Mech. Anal. 215 (2015), 443-496. (2015) Zbl1322.49065MR3294408DOI10.1007/s00205-014-0785-2
  5. Cruz-Uribe, D., Hästö, P., 10.1090/tran/7155, Trans. Am. Math. Soc. 370 (2018), 4323-4349. (2018) Zbl06853979MR3811530DOI10.1090/tran/7155
  6. Diening, L., Harjulehto, P., Hästö, P., Růžička, M., 10.1007/978-3-642-18363-8, Lecture Notes in Mathematics 2017, Springer, Berlin (2011). (2011) Zbl1222.46002MR2790542DOI10.1007/978-3-642-18363-8
  7. Fan, X.-L., Guan, C.-X., 10.1016/j.na.2010.03.010, Nonlinear Anal., Theory Methods Appl., Ser. A 73 (2010), 163-175. (2010) Zbl1198.46010MR2645841DOI10.1016/j.na.2010.03.010
  8. Gwiazda, P., Wittbold, P., Wróblewska-Kamińska, A., Zimmermann, A., 10.1016/j.na.2015.08.017, Nonlinear Anal., Theory Methods Appl., Ser. A 129 (2015), 1-36. (2015) Zbl1331.35173MR3414919DOI10.1016/j.na.2015.08.017
  9. Harjulehto, P., Hästö, P., 10.1515/forum-2015-0239, Forum Math. 29 (2017), 229-244. (2017) MR3592600DOI10.1515/forum-2015-0239
  10. Harjulehto, P., Hästö, P., Klén, R., 10.1016/j.na.2016.05.002, Nonlinear Anal., Theory Methods Appl., Ser. A 143 (2016), 155-173. (2016) Zbl1360.46029MR3516828DOI10.1016/j.na.2016.05.002
  11. Harjulehto, P., Hästö, P., Toivanen, O., 10.1007/s00526-017-1114-z, Calc. Var. Partial Differ. Equ. 56 (2017), Article No. 2, 26 pages. (2017) Zbl1366.35036MR3606780DOI10.1007/s00526-017-1114-z
  12. Hästö, P., 10.1016/j.jfa.2015.10.002, J. Funct. Anal. 269 (2015), 4038-4048. (2015) Zbl1338.47032MR3418078DOI10.1016/j.jfa.2015.10.002
  13. Hudzik, H., Uniform convexity of Musielak-Orlicz spaces with Luxemburg's norm, Ann. Soc. Math. Pol., Ser. I, Commentat. Math. 23 (1983), 21-32. (1983) Zbl0595.46027MR0709167
  14. Hudzik, H., A criterion of uniform convexity of Musielak-Orlicz spaces with Luxemburg norm, Bull. Pol. Acad. Sci., Math. 32 (1984), 303-313. (1984) Zbl0565.46020MR0785989
  15. Maeda, F.-Y., Mizuta, Y., Ohno, T., Shimomura, T., 10.1016/j.bulsci.2012.03.008, Bull. Sci. Math. 137 (2013), 76-96. (2013) Zbl1267.46045MR3007101DOI10.1016/j.bulsci.2012.03.008
  16. Musielak, J., 10.1007/BFb0072210, Lecture Notes in Mathematics 1034, Springer, Berlin (1983). (1983) Zbl0557.46020MR0724434DOI10.1007/BFb0072210
  17. Ok, J., 10.1007/s00526-016-0965-z, Calc. Var. Partial Differ. Equ. 55 (2016), Article No. 26, 30 pages. (2016) Zbl1342.35090MR3465442DOI10.1007/s00526-016-0965-z
  18. Rao, M. M., Ren, Z. D., Theory of Orlicz Spaces, Pure and Applied Mathematics 146, Marcel Dekker, New York (1991). (1991) Zbl0724.46032MR1113700

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