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Fractional Sobolev extension and imbedding


Author: Yuan Zhou
Journal: Trans. Amer. Math. Soc. 367 (2015), 959-979
MSC (2010): Primary 46E35; Secondary 42B35
DOI: https://doi.org/10.1090/S0002-9947-2014-06088-1
Published electronically: June 30, 2014
MathSciNet review: 3280034
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Abstract: Let $ \Omega $ be a domain of $ \mathbb{R}^n$ with $ n\ge 2$ and denote by $ W^{s,\,p}(\Omega )$ the fractional Sobolev space for $ s\in (0,\,1)$ and $ p\in (0, \,\infty )$. We prove that the following are equivalent:

(i) there exists a constant $ C_1>0$ such that for all $ x\in \Omega $ and $ r\in (0,\,1]$,

$\displaystyle \vert B(x,\,r)\cap \Omega \vert\ge C_1 r^n;$      

(ii) $ \Omega $ is a $ W^{s,\,p}$-extension domain for all $ s\in (0,\,1)$ and all $ p\in (0,\,\infty )$;

(iii) $ \Omega $ is a $ W^{s,\,p}$-extension domain for some $ s\in (0,\,1)$ and some $ p\in (0,\,\infty )$;

(iv) $ \Omega $ is a $ W^{s,\,p}$-imbedding domain for all $ s\in (0,\,1)$ and all $ p\in (0,\,\infty )$;

(v) $ \Omega $ is a $ W^{s,\,p}$-imbedding domain for some $ s\in (0,\,1)$ and some $ p\in (0,\,\infty )$.


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Additional Information

Yuan Zhou
Affiliation: Department of Mathematics, Beihang University, Haidian District Xueyuan Road 37#, Beijing 100191, People’s Republic of China
Email: yuanzhou@buaa.edu.cn

DOI: https://doi.org/10.1090/S0002-9947-2014-06088-1
Received by editor(s): September 7, 2011
Received by editor(s) in revised form: December 2, 2012
Published electronically: June 30, 2014
Additional Notes: The author was supported by Program for New Century Excellent Talents in University of Ministry of Education of China, New Teachers’ Fund for Doctor Stations of Ministry of Education of China (#20121102120031), and National Natural Science Foundation of China (#11201015).
Article copyright: © Copyright 2014 American Mathematical Society