Spherical means and the weighted ball integral of space-dependent data

Assume the Axiom of Countable Choice. Let $n\ge 1$, let $f:{\mathbb{R}}^n\to \mathbb{R}$ be continuous, let $\sigma$ be the polar surface measure on $S^{n-1}$ (The polar surface set function on the unit sphere) and let ${\omega }_{n-1}=\sigma (S^{n-1})=n{V}_n>0$, where $V_n=\mathrm{\mid }{B}_1^n\mathrm{\mid }$ (Sphere and ball measures scale in Rn). The spherical mean of $f$ is $M_f(x,r):=\frac{1}{{\omega }_{n-1}}{\int }_{S^{n-1}}f(x+r\omega )d\sigma (\omega )(x\in {\mathbb{R}}^n,r>0),$ the average of $f$ over the sphere of centre $x$ and radius $r$ with respect to the polar measure; this is the mean of Spherical averages and local ball means in Rn with $u=f$, restricted to continuous data. One sets $M_f(x,0):=f(x)$; that value is a convention whose consistency as a limit is proved later on this page, not assumed here. The unnormalised sphere integral is $S_f(x,r):={\int }_{S^{n-1}}f(x+r\omega )d\sigma (\omega )={\omega }_{n-1}{M}_f(x,r)(x\in {\mathbb{R}}^n,r>0)\mathrm{.}$ For even $n$ and $r>0$ put $W_f(x,r):=\frac{1}{n!!V_n}{\int }_{B_r(x)}\frac{f(y)}{\sqrt{r^2-\mathrm{\mid }y-x{\mathrm{\mid }}^2}}dy,$ where $n!!=n(n-2)\cdots 2$ and $V_n=\mathrm{\mid }{B}_1^n\mathrm{\mid }$ is the volume of the unit ball (Sphere and ball measures scale in Rn); for $n=2$ this is the weighted disk integral appearing in the two-dimensional Poisson formula below. The integral defining $W_f(x,r)$ is absolutely convergent and hence well defined: the weight $y\mapsto (r^2-\mathrm{\mid }y-x{\mathrm{\mid }}^2{)}^{-1\mathrm{/}2}$ is integrable over $B_r(x)$ — by translation invariance (Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation) and the polar formula its integral is ${\omega }_{n-1}{\int }_0^r{s}^{n-1}(r^2-s^2{)}^{-1\mathrm{/}2}ds<\mathrm{\infty }$ (Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma) — while $f$ is bounded on the closed ball $\overline{B_r(x)}$ because that ball is compact (For $n\ge 1$, every Euclidean closed ball and every Euclidean sphere of positive radius is compact) and a continuous function is bounded on a compact set (A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value); the product of an integrable function and a bounded function is Lebesgue integrable. The ball average used on this page is the normalised mean $A_g(x,r)$ of The average of a locally integrable function over a Euclidean ball, and every sphere or ball integral below is read under the Axiom of Countable Choice of The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), which supplies the polar measure and its integrals.