On Lp-estimates for a class of non-local elliptic equations

Hongjie Dong, Doyoon Kim

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61 Citations (Scopus)

Abstract

We consider non-local elliptic operators with kernel K(y)=a(y)/|y|d+σ, where 0<σ<2 is a constant and a is a bounded measurable function. By using a purely analytic method, we prove the continuity of the non-local operator L from the Bessel potential space Hpσ to Lp, and the unique strong solvability of the corresponding non-local elliptic equations in Lp spaces. As a byproduct, we also obtain interior Lp-estimates. The novelty of our results is that the function a is not necessarily to be homogeneous, regular, or symmetric. An application of our result is the uniqueness for the martingale problem associated to the operator L.

Original languageEnglish
Pages (from-to)1166-1199
Number of pages34
JournalJournal of Functional Analysis
Volume262
Issue number3
DOIs
Publication statusPublished - 2012 Feb 1
Externally publishedYes

Bibliographical note

Funding Information:
* Corresponding author. Fax: +82 312048122. E-mail addresses: [email protected] (H. Dong), [email protected] (D. Kim). 1 H. Dong was partially supported by the NSF under agreements DMS-0800129 and DMS-1056737. 2 D. Kim was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2011-0013960).

Keywords

  • Bessel potential spaces
  • Lévy processes
  • Non-local elliptic equations
  • The martingale problem

ASJC Scopus subject areas

  • Analysis

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