Asymptotic behavior of fractional Laplacians in the half space

In this article, we first give the asymptotic behaviors of fractional Laplacians in the half space of RnRn. Next, we discuss the weighted boundary limits of modified fractional Laplacians. The asymptotic behaviors hold outside an explicitly defined exceptional set, whose size can be controlled and set to be as small as required. The estimation of the asymptotic behavior is the generalization of the argument reported in Qiao and Deng (2010).