The $s$-polyharmonic extension problem and higher-order fractional Laplacians

Abstract

We provide a detailed description of the relationships between the fractional Laplacian of order $2s\in(0,n)$ on $\mathbb{R}^n$ and the $\textit{$s$-polyharmonic}$ extension operator to the upper half space $\mathbb{R}^{n+1}_+$.