Riesz potential

Summary

The Riesz potential of order $\alpha$, for $0 < \alpha < n$, is the operator $I_\alpha$ that convolves a function on $\R^n$ with a constant multiple of $\abs{x}^{\alpha - n}$. It is the inverse of the fractional Laplacian $(-\Delta)^{\alpha/2}$, so it integrates a function to the fractional order $\alpha$. For $\alpha = 2$ and $n \ge 3$ it is, up to a constant factor, the Newtonian potential, which solves Poisson's equation. Its main estimate is the Hardy–Littlewood–Sobolev inequality, which underlies the Sobolev embedding theorem.