Isoperimetric Inequalities for Non-Local Dirichlet Forms

Abstract

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.

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Acknowledgments

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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Correspondence to Jian Wang.

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Wang, FY., Wang, J. Isoperimetric Inequalities for Non-Local Dirichlet Forms. Potential Anal 53, 1225–1253 (2020). https://doi.org/10.1007/s11118-019-09804-9

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Keywords

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

Mathematics Subject Classification (2010)

  • 47G20
  • 47D62