Sharp boundary-inclusive eigenfunction bounds

Determine sharp global L^p and L^\infty estimates for eigenfunctions of the fractional Laplacian on bounded domains up to the boundary, including their optimal dependence on the fractional order \alpha.

Background

The paper proves a uniform L^\infty bound for normalized eigenfunctions of the fractional Laplacian on an interval, with a constant uniform in both the eigenvalue index and the fractional order. It contrasts this one-dimensional result with higher-dimensional work establishing Sogge-type interior L^p estimates.

The unresolved issue is to obtain sharp global estimates that remain valid up to the boundary, for both finite p and p=\infty, and to identify the optimal way those estimates depend on the fractional order \alpha.