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 language | English |
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Pages (from-to) | 1166-1199 |
Number of pages | 34 |
Journal | Journal of Functional Analysis |
Volume | 262 |
Issue number | 3 |
DOIs | |
Publication status | Published - 2012 Feb 1 |
Externally published | Yes |
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