Isoperimetric Inequalities for Non-Local Dirichlet Forms

  • Feng-Yu Wang
  • Jian WangEmail author


Let \((E,{\mathscr{E}} F,\mu )\) be a σ-finite measure space. For a non-negative symmetric measurable function J(x,y) on E × E, consider the quadratic form

$$ \mathscr{E}(f,f):= \frac{1}{2}{\int}_{E\times E} (f(x)-f(y))^{2} J(x,y) \mu(\text{\text{d}} x) \mu(\text{\text{d}} y) $$

in L2(μ). We characterize the relationship between the isoperimetric inequality and the super Poincaré inequality associated with \({\mathscr{E}}\). In particular, sharp Orlicz-Sobolev type and Poincaré type isoperimetric inequalities are derived for stable-like Dirichlet forms on \(\mathbb {R}^{n}\), which include the existing fractional isoperimetric inequality as a special example.


Isoperimetric inequality Non-local Dirichlet form Super Poincaré inequality Orlicz norm 

Mathematics Subject Classification (2010)

47G20 47D62 


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Supported in part by NNSFC (11431014, 11522106, 11626245, 11626250, 11771326, 11831014), the Program for Probability and Statistics: Theory and Application (No. IRTL1704) and the Program for Innovative Research Team in Science and Technology in Fujian Province University (IRTSTFJ). The authors would like to thank Professor Takashi Kumagai and the referee for their helpful comments.


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Center of Applied MathematicsTianjin UniversityTianjinChina
  2. 2.Department of MathematicsSwansea UniversitySingleton ParkUK
  3. 3.College of Mathematics and Informatics & Fujian Key Laboratory of Mathematical Analysis and Applications (FJKLMAA)Fujian Normal UniversityFuzhouChina

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