Riesz potential operator

Summary

The Riesz potential operator $I_\alpha$ of order $\alpha$, with $0 < \alpha < n$, is the integral operator on $\R^n$ with kernel a constant times $\abs{x - y}^{\alpha - n}$. It is the inverse of the fractional power $(-\Delta)^{\alpha/2}$ of the Laplacian, so it acts as integration of order $\alpha$ in $n$ variables. For $n = 3$ and $\alpha = 2$ it is the Newtonian potential of physics. Its mapping properties between Lp spaces, given by the Hardy–Littlewood–Sobolev inequality, underlie the Sobolev embedding theorems.