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

TL;DR

This paper explores the connection between fractional Laplacians of order 2s and the s-polyharmonic extension operator in higher dimensions, providing insights into their mathematical relationship.

Contribution

It offers a detailed description of the relationship between fractional Laplacians and s-polyharmonic extension operators, advancing understanding of higher-order fractional PDEs.

Findings

Established the link between fractional Laplacians and s-polyharmonic extensions.

Clarified the mathematical structure of the s-polyharmonic extension problem.

Enhanced the theoretical framework for 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 $\mathbb{R}^{n+1}_+$.