Abstract
This paper is devoted to the study of fractional (q, p)-Sobolev-Poincaré in- equalities in irregular domains. In particular, the author establishes (essentially) sharp fractional (q, p)-Sobolev-Poincaré inequalities in s-John domains and in domains satisfying the quasihyperbolic boundary conditions. When the order of the fractional derivative tends to 1, our results tend to the results for the usual derivatives. Furthermore, the author verifies that those domains which support the fractional (q, p)-Sobolev-Poincaré inequalities together with a separation property are s-diam John domains for certain s, depending only on the associated data. An inaccurate statement in [Buckley, S. and Koskela, P., Sobolev-Poincaré implies John, Math. Res. Lett., 2(5), 1995, 577–593] is also pointed out.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adams, D. R. and Hedberg, L. I., Function spaces and potential theory, Grundlehren der Mathematischen Wissenschaften, Fundamental Principles of Mathematical Sciences, 314, Springer-Verlag, Berlin, 1996.
Buckley, S. and Koskela, P., Sobolev-Poincaré implies John, Math. Res. Lett., 2(5), 1995, 577–593.
Dyda, B., Ihnatsyeva, L. and Vähäkangas, A. V., On improved Sobolev-Poincaré inequalities, Ark. Mat., 2015, http://dx.doi.org/10.1007/s11512-015-0227-x
Gehring, F. W. and Osgood, B. G., Uniform domains and the quasihyperbolic metric, J. Analyse Math., 36, 1979, 50–74.
Gehring, F. W. and Palka, B. P., Quasiconformally homogeneous domains, J. Analyse Math., 30, 1976, 172–199.
Guo, C. Y. and Koskela, P., Generalized John disks, Cent. Eur. J. Math., 12(2), 2014, 349–361.
Hajlasz, P. and Koskela, P., Isoperimetric inequalities and imbedding theorems in irregular domains, J. London Math. Soc., 58(2), 1998, 425–450.
Hajlasz, P. and Koskela, P., Sobolev met Poincaré, Mem. Amer. Math. Soc., 145(688), 2000, 101 pages.
Hurri-Syrjänen, R. and Vähäkangas, A. V., On fractional Poincaré inequalities, J. Anal. Math., 120, 2013, 85–104.
Jiang, R. and Kauranen, A., A note on “quasihyperbolic boundary conditions and Poincaré domains”, Math. Ann., 357(3), 2013, 1199–1204.
John, F., Rotation and strain, Comm. Pure Appl. Math., 14, 1961, 391–413.
Kilpeläinen, T. and Malý, J., Sobolev inequalities on sets with irregular boundaries, Z. Anal. Anwendungen, 19(2), 2000, 369–380.
Koskela, P., Onninen, J. and Tyson, J. T., Quasihyperbolic boundary conditions and Poincaré domains, Math. Ann., 323(4), 2002, 811–830.
Martio, O. and Sarvas, J., Injectivity theorems in plane and space, Ann. Acad. Sci. Fenn. Ser. A I Math., 4(2), 1979, 383–401.
Maz’ya, V. G., Classes of domains and imbedding theorems for function spaces, Soviet Math. Dokl., 1, 1960, 882–885.
Maz’ya, V. G., Sobolev spaces with applications to elliptic partial differential equations, Second, revised and augmented edition, Grundlehren der Mathematischen Wissenschaften, Fundamental Principles of Mathematical Sciences, 342, Springer-Verlag, Heidelberg, 2011.
Smith, W. and Stegenga, D. A., Hölder and Poincaré domains, Trans. Amer. Math. Soc., 319, 1990, 67–100.
Acknowledgements
The author wants to express his gratitude to Antti V Vähäkangas. for posing the question in Jyväskylä analysis seminar, which is the main motivation of the current paper, for sharing the manuscript (see [3]), and for numerous useful comments on the manuscript. The author also wants to thank academy Professor Pekka Koskela for the helpful discussions.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by the Magnus Ehrnrooth Foundation.
Rights and permissions
About this article
Cite this article
Guo, CY. Fractional Sobolev-Poincaré inequalities in irregular domains. Chin. Ann. Math. Ser. B 38, 839–856 (2017). https://doi.org/10.1007/s11401-017-1099-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11401-017-1099-0