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

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

2011-02-20

arxiv.org/abs/1102.4073v3

Journal of Functional Analysis 262, Issue 3, (2012), Pages 1166-1199

Mathematics

Analysis of PDEs

Minor revision, to appear in J. Funct. Anal

Scientific paper

10.1016/j.jfa.2011.11.002

We consider non-local elliptic operators with kernel $K(y)=a(y)/|y|^{d+\sigma}$, where $0 < \sigma < 2$ 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 $H^\sigma_p$ 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 $L$.

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