## Arkiv för Matematik

• Ark. Mat.
• Volume 54, Number 2 (2016), 437-454.

### On improved fractional Sobolev–Poincaré inequalities

#### Abstract

We study a certain improved fractional Sobolev–Poincaré inequality on domains, which can be considered as a fractional counterpart of the classical Sobolev–Poincaré inequality. We prove the equivalence of the corresponding weak and strong type inequalities; this leads to a simple proof of a strong type inequality on John domains. We also give necessary conditions for the validity of an improved fractional Sobolev–Poincaré inequality, in particular, we show that a domain of finite measure, satisfying this inequality and a ‘separation property’, is a John domain.

#### Note

L.I. and A.V.V. were supported by the Finnish Academy of Science and Letters, Vilho, Yrjö and Kalle Väisälä Foundation. B.D. was supported in part by NCN grant 2012/07/B/ST1/03356. The authors would like to thank the referee for a careful reading of the manuscript and for the comments.

#### Article information

Source
Ark. Mat. Volume 54, Number 2 (2016), 437-454.

Dates
Revised: 21 September 2015
First available in Project Euclid: 30 January 2017

https://projecteuclid.org/euclid.afm/1485802743

Digital Object Identifier
doi:10.1007/s11512-015-0227-x

Zentralblatt MATH identifier
1362.35012

Rights

#### Citation

Dyda, Bartłomiej; Ihnatsyeva, Lizaveta; Vähäkangas, Antti V. On improved fractional Sobolev–Poincaré inequalities. Ark. Mat. 54 (2016), no. 2, 437--454. doi:10.1007/s11512-015-0227-x. https://projecteuclid.org/euclid.afm/1485802743

#### References

• Adams, R. A., Sobolev Spaces, Pure and Applied Mathematics 65, Academic Press, New York, 1975.
• Adams, D. R. and Hedberg, L. I., Function Spaces and Potential Theory, Springer, Berlin, 1996.
• Bellido, J. C. and Mora-Corral, C., Existence for nonlocal variational problems in peridynamics, SIAM J. Math. Anal. 46 (2014), 890–916.
• Bojarski, B., Remarks on Sobolev imbedding inequalities, in Complex Analysis, Joensuu 1987, Lecture Notes in Math. 1351, pp. 52–68, Springer, Berlin, 1988.
• Buckley, S. and Koskela, P., Sobolev–Poincaré implies John, Math. Res. Lett. 2 (1995), 577–593.
• Dyda, B., On comparability of integral forms, J. Math. Anal. Appl. 318 (2006), 564–577.
• Dyda, B. and Vähäkangas, A. V., Characterizations for fractional Hardy inequality, Adv. Calc. Var. 8 (2015), 173–182.
• Hajłasz, P., Sobolev inequalities, truncation method, and John domains, in Papers on Analysis, Rep. Univ. Jyväskylä Dep. Math. Stat. 83, pp. 109–126, Univ. Jyväskyä, Jyväskylä, 2001.
• Jonsson, A. and Wallin, H., A Whitney extension theorem in $L^{p}$ and Besov spaces, Ann. Inst. Fourier 28 (1978), 139–192.
• Harjulehto, P. and Hurri-Syrjänen, R., On a $(q,p)$-Poincaré inequality, J. Math. Anal. Appl. 337 (2008), 61–68.
• Harjulehto, P., Hurri-Syrjänen, R. and Vähäkangas, A. V., On the $(1,p)$-Poincaré inequality, Illinois J. Math. 56 (2012), 905–930.
• Hurri-Syrjänen, R. and Vähäkangas, A. V., Fractional Sobolev–Poincaré and fractional Hardy inequalities in unbounded John domains, Mathematika 61 (2015), 385–401.
• Hurri-Syrjänen, R. and Vähäkangas, A. V., On fractional Poincaré inequalities, J. Anal. Math. 120 (2013), 85–104.
• Martio, O., John domains, bi-Lipschitz balls and Poincaré inequality, Rev. Roumaine Math. Pures Appl. 33 (1988), 107–112.
• Maz’ya, V. G., Sobolev Spaces, Springer, Berlin, 1985.
• Reshetnyak, Y. G., Integral representations of differentiable functions, in Partial Differential Equations, pp. 173–187, Nauka, Novosibirsk, 1986.
• Shvartsman, P., Local approximations and intrinsic characterization of spaces of smooth functions on regular subsets of ${\mathbf{R}}^{n}$, Math. Nachr. 279 (2006), 1212–1241.
• Väisälä, J., Exhaustions of John domains, Ann. Acad. Sci. Fenn., Ser. A 1 Math. 19 (1994), 47–57.
• Zhou, Y., Criteria for optimal global integrability of Hajłasz–Sobolev functions, Illinois J. Math. 55 (2011), 1083–1103.
• Zhou, Y., Fractional Sobolev extension and imbedding, Trans. Amer. Math. Soc. 367 (2015), 959–979.