A generalization of affine fractional $L^p$ Sobolev inequalities

General affine fractional $L^p$ Sobolev inequalities were established by Monika Ludwig and Julián Haddad in their foundational work for fractional parameters $s\in (0,1)$ and $1<p<n/s$. In this paper, we extend the admissible range of the fractional parameter to $s\in (0,n)\setminus \mathbb{N}$ and $1<p<n/s$. The affine fractional $L^p$ Sobolev inequalities are obtained for both the classical fractional case $s\in (0,1)$ and the higher-order fractional case. In the course of the proof, we define the higher-order fractional $L^p$ Sobolev polar projection body and use the method of symmetric decreasing rearrangement.