Abstract
This paper is devoted to interior estimates for eigenfunctions of the restricted fractional Laplacian on a bounded domain in Rd. We prove that the eigenfunctions satisfy the expected Lp bounds analogous to the classical results by Sogge [24]. As the fractional Laplacian is nonlocal, the standard method for Laplacian eigenfunction estimates can no longer work here. In the proof, we mainly reduce Lp bounds to a kind of L2 commutator estimates, which can be handled by the explicit integral expression of the restricted fractional Laplacian and its heat kernel estimates.
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