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Ball means and sphere means are related by a radial derivative

Statement

Assume the Axiom of Countable Choice, let $n \geq 1$ , let $g \in C^{0} (R^{n})$ , and for $x \in R^{n}$ , $r > 0$ let $A_{g} (x, r) : = ∣ B_{r} (x) ∣^{- 1} \int_{B_{r} (x)} g$ be the ball average of The average of a locally integrable function over a Euclidean ball and $M_{g}$ the spherical mean of Spherical means and the weighted ball integral of space-dependent data. Then for every $r > 0$ $A_{g} (x, r) = \frac{n}{r^{n}} \int_{0}^{r} s^{n - 1} M_{g} (x, s) d s, M_{g} (x, r) = A_{g} (x, r) + \frac{r}{n} \partial_{r} A_{g} (x, r) .$

Facts & Assumptions

Given: Countable Choice, $n \geq 1$ , $g \in C^{0} (R^{n})$ , and the means $A_{g}$ and $M_{g}$ of the statement.

[F1]

Under Countable Choice, $\int_{R^{n}} f d λ_{n} = \int_{0}^{\infty} \int_{S^{n - 1}} f (s θ) s^{n - 1} d σ (θ) d s$ for every Borel measurable $f : R^{n} \to [0, \infty]$ (Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma).

[F2]

$M_{g} (x, s) = ω_{n - 1}^{- 1} \int_{S^{n - 1}} g (x + s ω) d σ (ω)$ and $ω_{n - 1} = σ (S^{n - 1})$ for $x \in R^{n}$ , $s > 0$ (Spherical means and the weighted ball integral of space-dependent data).

[F3]

$∣ B_{r} (x) ∣ = ω_{n - 1} r^{n} / n$ for $r > 0$ (Sphere and ball measures scale in Rn).

[F4]

A continuous function on a compact metric space is uniformly continuous (Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous); Euclidean closed balls are compact (For $n \geq 1$ , every Euclidean closed ball and every Euclidean sphere of positive radius is compact, in the sense of Open cover, subcover, compact metric space, and compact subset of a metric space).

[F5]

If $I \subseteq R$ is order-convex with at least two elements and $f : I \to R$ is continuous on $I$ , then for $r_{0} \in I$ , $r \mapsto \int_{r_{0}}^{r} f$ is a primitive of $f$ on $I$ (Every continuous function on an interval has a primitive; two primitives differ by a constant; and $\int_{a}^{b} f = G (b) - G (a)$ for any primitive $G$ ).

Proof

1.1F1F2F3algebra

Polar coordinates applied to the positive and negative parts of the function $u \mapsto g (x + u) 1_{B_{r} (0)} (u)$ give, using translation invariance (Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation) to set $y = x + u$ , $∣ B_{r} (x) ∣ A_{g} (x, r) = \int_{B_{r} (x)} g = \int_{0}^{r} \int_{S^{n - 1}} g (x + s ω) s^{n - 1} d σ (ω) d s = ω_{n - 1} \int_{0}^{r} s^{n - 1} M_{g} (x, s) d s$ , where the inner integral is $ω_{n - 1} M_{g} (x, s)$ by the definition of the spherical mean; dividing by $∣ B_{r} (x) ∣ = ω_{n - 1} r^{n} / n$ gives the first identity.

1.2F4algebra

The function $s \mapsto M_{g} (x, s)$ is continuous on $(0, \infty)$ : for $s, s_{0} > 0$ with $∣ s - s_{0} ∣ < 1$ one has $∣ M_{g} (x, s) - M_{g} (x, s_{0}) ∣ \leq \sup_{ω \in S^{n - 1}} ∣ g (x + s ω) - g (x + s_{0} ω) ∣$ , and the two points $x + s ω$ and $x + s_{0} ω$ lie in the compact ball $\overline{B_{s_{0} + 1} (x)}$ at distance $∣ s - s_{0} ∣$ ; since $g$ is uniformly continuous on that ball by [F4], the supremum tends to $0$ as $s \to s_{0}$ .

1.3F5algebra

Hence $s \mapsto s^{n - 1} M_{g} (x, s)$ is continuous on $(0, \infty)$ , so locally at any $r > 0$ split the integral at a fixed $r_{0} \in (0, r)$ ; the first part is constant and [F5] differentiates the second part. Thus $r \mapsto \int_{0}^{r} s^{n - 1} M_{g} (x, s) d s$ is a primitive of $s^{n - 1} M_{g} (x, s)$ , and the first identity gives $\partial_{r} (r^{n} A_{g} (x, r)) = n r^{n - 1} M_{g} (x, r)$ for $r > 0$ .

2.1algebra∎

Differentiating the product gives $\partial_{r} (r^{n} A_{g} (x, r)) = n r^{n - 1} A_{g} (x, r) + r^{n} \partial_{r} A_{g} (x, r)$ , so equating with the previous display and dividing by $n r^{n - 1} > 0$ yields $M_{g} (x, r) = A_{g} (x, r) + \frac{r}{n} \partial_{r} A_{g} (x, r)$ , the second identity.

Depends on

Used by

The Euler–Poisson–Darboux equation for spherical means Lemma

Dependency tree · two levels

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Sources

Gerald Teschl, Partial Differential Equations: From Classical to Modern (2025 archived author manuscript, AMS Graduate Studies in Mathematics) (standard reference, not scraped)
John K. Hunter, Notes on Partial Differential Equations (revised 18 June 2014, UC Davis) (standard reference, not scraped)