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

Hongjie Dong; Doyoon Kim

doi:10.1016/j.jfa.2011.11.002

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

Hongjie Dong, Doyoon Kim

Research output: Contribution to journal › Article › peer-review

44 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 L_p, and the unique strong solvability of the corresponding non-local elliptic equations in L_p spaces. As a byproduct, we also obtain interior L_p-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
Pages (from-to)	1166-1199
Number of pages	34
Journal	Journal of Functional Analysis
Volume	262
Issue number	3
DOIs	https://doi.org/10.1016/j.jfa.2011.11.002
Publication status	Published - 2012 Feb 1
Externally published	Yes

Keywords

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

ASJC Scopus subject areas

Analysis

Access to Document

10.1016/j.jfa.2011.11.002

Cite this

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title = "On Lp-estimates for a class of non-local elliptic equations",

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.",

keywords = "Bessel potential spaces, L{\'e}vy processes, Non-local elliptic equations, The martingale problem",

author = "Hongjie Dong and Doyoon Kim",

note = "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).",

year = "2012",

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doi = "10.1016/j.jfa.2011.11.002",

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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

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