TY - JOUR
T1 - On Lp-estimates for a class of non-local elliptic equations
AU - Dong, Hongjie
AU - Kim, Doyoon
N1 - Funding Information:
* Corresponding author. Fax: +82 312048122. E-mail addresses: Hongjie_Dong@brown.edu (H. Dong), doyoonkim@khu.ac.kr (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).
PY - 2012/2/1
Y1 - 2012/2/1
N2 - 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.
AB - 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.
KW - Bessel potential spaces
KW - Lévy processes
KW - Non-local elliptic equations
KW - The martingale problem
UR - http://www.scopus.com/inward/record.url?scp=84155167485&partnerID=8YFLogxK
U2 - 10.1016/j.jfa.2011.11.002
DO - 10.1016/j.jfa.2011.11.002
M3 - Article
AN - SCOPUS:84155167485
SN - 0022-1236
VL - 262
SP - 1166
EP - 1199
JO - Journal of Functional Analysis
JF - Journal of Functional Analysis
IS - 3
ER -