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Characterization of generalized Orlicz spaces

    https://doi.org/10.1142/S0219199718500797Cited by:2
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    Abstract

    The norm in classical Sobolev spaces can be expressed as a difference quotient. This expression can be used to generalize the space to the fractional smoothness case. Because the difference quotient is based on shifting the function, it cannot be used in generalized Orlicz spaces. In its place, we introduce a smoothed difference quotient and show that it can be used to characterize the generalized Orlicz–Sobolev space. Our results are new even in Orlicz spaces and variable exponent spaces.

    During the reviewing process of this manuscript, the authors became aware that Sibei Yang, Dachun Yang, and Wen Yuan had completed a related manuscript [35]. Their manuscript is dated of 28 January 2018, over one year after our work was posted on arXiv (see arXiv:1612.04566).

    Their work is based on the abstract assumption that the maximal operator is bounded, whereas we assume (2.3) (which implies the maximal inequality [25]). On the other hand, our results are more general in the sense that we cover arbitrary open sets Ω, instead of the entire Euclidean space ℝn, and we assume only local integrability, whereas they assume global integrability.

    AMSC: 46E35, 46E30

    References

    • 1. A. Almeida and P. Hästö , Besov spaces with variable smoothness and integrability, J. Funct. Anal. 258(5) (2010) 1628–1655. Crossref, ISI, Google Scholar
    • 2. G. Aubert and P. Kornprobst , Can the nonlocal characterization of Sobolev spaces by Bourgain et al. be useful for solving variational problems?, SIAM J. Numer. Anal. 47 (2009) 844–860. Crossref, ISI, Google Scholar
    • 3. P. Baroni, M. Colombo and G. Mingione , Harnack inequalities for double phase functionals, Nonlinear Anal. 121 (2015) 206–222. Crossref, ISI, Google Scholar
    • 4. P. Baroni, M. Colombo and G. Mingione , Non-autonomous functionals, borderline cases and related function classes, St. Petersburg Math. J. 27 (2016) 347–379. Crossref, ISI, Google Scholar
    • 5. P. Baroni, M. Colombo and G. Mingione , Regularity for general functionals with double phase, Calc. Var. Partial Differential Equations 57 (2018) Article 62, 48 pp. Crossref, ISI, Google Scholar
    • 6. J. Bourgain, H. Brézis and P. Mironescu , Another look at Sobolev spaces, in Proc. Conf. Optimal Control and Partial Differential Equations. In Honour of Professor Alain Bensoussan’s 60th Birthday (IOS Press, France, 2001), pp. 439–455. Google Scholar
    • 7. J. Bourgain, H. Brézis and P. Mironescu , Limiting embedding theorems for Ws,p when s ↑ 1 and applications, J. Anal. Math. 87 (2002) 77–101. Crossref, ISI, Google Scholar
    • 8. S.-S. Byun and J. Oh , Global gradient estimates for non-uniformly elliptic equations, Calc. Var. Partial Differential Equations 56 (2017) Article 46, 36 pp. Crossref, ISI, Google Scholar
    • 9. S.-S. Byun, S. Ryu and P. Shin , Calderón–Zygmund estimates for omega-minimizers of double phase variational problems, Appl. Math. Lett. 86 (2018) 256–263. Crossref, ISI, Google Scholar
    • 10. Y. Chen, S. Levine and R. Rao , Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math. 66(4) (2006) 1383–1406. Crossref, ISI, Google Scholar
    • 11. M. Colombo and G. Mingione , Regularity for double phase variational problems, Arch. Ration. Mech. Anal. 215(2) (2015) 443–496. Crossref, ISI, Google Scholar
    • 12. M. Colombo and G. Mingione , Bounded minimisers of double phase variational integrals, Arch. Ration. Mech. Anal. 218(1) (2015) 219–273. Crossref, ISI, Google Scholar
    • 13. M. Colombo and G. Mingione , Calderón–Zygmund estimates and non-uniformly elliptic operators, J. Funct. Anal. 270 (2016) 1416–1478. Crossref, ISI, Google Scholar
    • 14. D. Cruz-Uribe and A. Fiorenza , Variable Lebesgue Spaces: Foundations and Harmonic Analysis (Birkhäuser/Springer, New York, 2013). Crossref, Google Scholar
    • 15. D. Cruz-Uribe and P. Hästö , Extrapolation and interpolation in generalized Orlicz spaces, Trans. Amer. Math. Soc. 370(6) (2018) 4323–4349. Crossref, ISI, Google Scholar
    • 16. L. Diening and P. Hästö , Variable exponent trace spaces, Studia Math. 183 (2007) 127–141. Crossref, ISI, Google Scholar
    • 17. L. Diening, P. Harjulehto, P. Hästö and M. Růžička , Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, Vol. 2017 (Springer-Verlag, Berlin, 2011), ix+509 pp. Crossref, Google Scholar
    • 18. L. Diening, P. Hästö and S. Roudenko , Function spaces of variable smoothness and integrability, J. Funct. Anal. 256(6) (2009) 1731–1768. Crossref, ISI, Google Scholar
    • 19. R. Ferreira, C. Kreisbeck and A. M. Ribeiro , Characterization of polynomials and higher-order Sobolev spaces in terms of functionals involving difference quotients, Nonlinear Anal. 112 (2015) 199–214. Crossref, ISI, Google Scholar
    • 20. F. Giannetti and A. Passarelli di Napoli , Regularity results for a new class of functionals with non-standard growth conditions, J. Differential Equations 254 (2013) 1280–1305. Crossref, ISI, Google Scholar
    • 21. P. Harjulehto and P. Hästö , Riesz potential in generalized Orlicz Spaces, Forum Math. 29(1) (2017) 229–244. Crossref, ISI, Google Scholar
    • 22. P. Harjulehto, P. Hästö and R. Klén , Generalized Orlicz spaces and related PDE, Nonlinear Anal. 143 (2016) 155–173. Crossref, ISI, Google Scholar
    • 23. P. Harjulehto, P. Hästö, V. Latvala and O. Toivanen , Critical variable exponent functionals in image restoration, Appl. Math. Lett. 26 (2013) 56–60. Crossref, ISI, Google Scholar
    • 24. P. Harjulehto, P. Hästö and O. Toivanen , Hölder regularity of quasiminimizers under generalized growth conditions, Calc. Var. Partial Differential Equations 56 (2017) Article 22, 26 pp. Crossref, ISI, Google Scholar
    • 25. P. Hästö , The maximal operator on generalized Orlicz spaces, J. Funct. Anal. 269 (2015) 4038–4048; Corrigendum, J. Funct. Anal. 271(1) (2016) 240–243. Crossref, ISI, Google Scholar
    • 26. P. Hästö and A. Ribeiro , Characterization of the variable exponent Sobolev norm without derivatives, Commun. Contemp. Math. 19(3) (2017) 1650022, 13 pp. Link, ISI, Google Scholar
    • 27. G. Leoni and D. Spector , Characterization of Sobolev and BV spaces, J. Funct. Anal. 261(10) (2011) 2926–2958. Crossref, ISI, Google Scholar
    • 28. F.-Y. Maeda, Y. Mizuta, T. Ohno and T. Shimomura , Boundedness of maximal operators and Sobolev’s inequality on Musielak–Orlicz–Morrey spaces, Bull. Sci. Math. 137 (2013), 76–96. Crossref, ISI, Google Scholar
    • 29. Y. Mizuta, T. Ohno and T. Shimomura , Sobolev’s inequalities and vanishing integrability for Riesz potentials of functions in the generalized Lebesgue space Lp(⋅)logLq(⋅), J. Math. Anal. Appl. 345 (2008) 70–85. Crossref, ISI, Google Scholar
    • 30. J. Musielak , Orlicz Spaces and Modular Spaces, Lecture Notes in Mathematics, Vol. 1034 (Springer, Berlin, 1983), vi+226 pp. Crossref, Google Scholar
    • 31. J. Ok , Gradient estimates for elliptic equations with Lp(⋅)logL growth, Calc. Var. Partial Differential Equations 55(2) (2016) Article 26, 30 pp. Crossref, ISI, Google Scholar
    • 32. J. Ok , Regularity results for a class of obstacle problems with nonstandard growth, J. Math. Anal. Appl. 444(2) (2016) 957–979. Crossref, ISI, Google Scholar
    • 33. J. Ok , Regularity of ω-minimizers for a class of functionals with non-standard growth, Calc. Var. Partial Differential Equations 56 (2017) Article 48, 31 pp. Crossref, ISI, Google Scholar
    • 34. L. Rudin, S. Osher and E. Fatemi , Nonlinear total variation based noise removal algorithms, Physica D 60(1–4) (1992) 259–268. Crossref, ISI, Google Scholar
    • 35. S. Yang, D. Yang and W. Yuan, New characterizations of Musielak–Orlicz–Sobolev spaces via sharp ball averaging functions; arXiv:1809.00284. Google Scholar
    • 36. A. Świerczewska-Gwiazda , Nonlinear parabolic problems in Musielak–Orlicz spaces, Nonlinear Anal. 98 (2014) 48–65. Crossref, ISI, Google Scholar
    Published: 26 November 2018
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