Spherical mean transform

Remarks

$M f(x, r)$ is the average of $f$ over the sphere of center $x$ and radius $r$, and $M f(x, 0) = f(x)$. If $f$ is of class $C^2$, then $M f$ satisfies the Darboux equation $$\Delta_x M f(x, r) = \left( \frac{\partial^2}{\partial r^2} + \frac{n - 1}{r} \frac{\partial}{\partial r} \right) M f(x, r), \qquad r > 0.$$

Example: $f(x) = \abs{x}^2$. Since $\abs{x + r\theta}^2 = \abs{x}^2 + 2r\, x \cdot \theta + r^2$ and $\theta \mapsto x \cdot \theta$ has mean $0$ over the sphere, $M f(x, r) = \abs{x}^2 + r^2$. Both sides of the Darboux equation equal $2n$.

Non-example: $f(y) = \abs{y}^{1-n}$ for $y \ne 0$ and $f(0) = 0$ is not continuous. Its mean over the sphere of center $x \ne 0$ and radius $r = \abs{x}$, which passes through $0$, is $+\infty$.

The function $f$ may also be only locally integrable, and then $M f(x, r)$ is defined for almost every $(x, r)$.