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Newtonian potential of compactly supported data

Statement

Assume Countable Choice and $n \geq 2$ . For a measurable compactly supported $f$ , define $N f (x) = \int_{R^{n}} Φ (x - y) f (y) d y$ at each $x$ where the integral is absolutely finite. Whenever this integral is finite almost everywhere, $N f$ also denotes its almost-everywhere class. The following items establish everywhere convergence for bounded compact data and almost-everywhere convergence for compact $L^{1}$ data; the definition itself makes no unconditional convergence claim.

Definition

Let $A_{f} : = {x \in R^{n} : \int_{R^{n}} ∣ Φ (x - y) f (y) ∣ d y < \infty} .$ For $x \in A_{f}$ , the pointwise value $N f (x)$ is the displayed Lebesgue integral. If $A_{f}$ has full Lebesgue measure, the phrase “almost-everywhere class of $N f$ ” means the equivalence class under equality outside a Lebesgue null set; one may assign arbitrary values on $A_{f}^{c}$ to obtain a representative on all of $R^{n}$ . The potential is not asserted to be defined at points outside $A_{f}$ unless a representative extension is explicitly being used.

Facts & Assumptions

Given: Assume ${A C}_{ω}$ , let $n \geq 2$ , and let $f$ be a finite-valued, Lebesgue-measurable, compactly supported scalar function. Complex-valued data are handled by their real and imaginary parts.

[A1]

Countable Choice, written ${A C}_{ω}$ , says every sequence of nonempty sets has a choice function. (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

The kernel is given away from zero by the power formula when $n \geq 3$ and by the logarithmic formula when $n = 2$ . (Fundamental solution for the positive operator minus Laplacian).

[F2]

The kernel's value at zero may be assigned arbitrarily; its locally integrable class is unchanged by that point assignment. (Fundamental solution for the positive operator minus Laplacian).

[F3]

Lebesgue measurability on $R^{n}$ is the completion of the Borel Lebesgue measure. ( $L (R^{n})$ is exactly the completion of the restriction of $λ_{n}$ to the Borel sets).

[F4]

Under Countable Choice, a function measurable for a completed measure is almost everywhere equal to a function measurable for the original $σ$ -algebra. (A function measurable for a completion is almost everywhere equal to one measurable for the original sigma-algebra).

[F5]

For a nonnegative product-measurable function on a product of $σ$ -finite measure spaces, its section-integral functions are measurable. (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product).

[F6]

Lebesgue measure on $R^{n}$ is $σ$ -finite and finite on bounded sets. (Lebesgue measure is sigma-finite, and every metrically bounded subset of $R^{n}$ has finite outer measure).

[F7]

Under Countable Choice, every at most countable subset of $R^{n}$ , including a singleton, is Lebesgue null. (Every at most countable subset of $R^{n}$ is Lebesgue null; in particular $λ_{1} (Q) = 0$ ).

[F8]

If $\tilde{Φ}$ and $\tilde{f}$ are Borel representatives, then $(x, y) \mapsto \tilde{Φ} (x - y) \tilde{f} (y)$ is Borel on $R^{2 n}$ . (Borel representatives make the convolution integrand Borel measurable).

Proof

technique · direct

1.1F1cases

On $R^{n} ∖ {0}$ the formulas in [F1] are continuous, so assigning the finite value $0$ at the closed singleton ${0}$ gives a Borel representative $\tilde{Φ}$ of the kernel.

1.2A1F2F3F4F7

By [F3] and [F4], using [A1], choose a Borel representative $\tilde{f}$ equal to $f$ almost everywhere; in the complex-valued case apply [F4] to the real and imaginary parts. Since $f$ is finite-valued, the Borel set where the resulting extended-real representative is infinite is contained in its null exceptional set; reset it to zero there, which keeps it Borel and equal to $f$ almost everywhere and gives a finite-valued representative. For each fixed $x$ , changing $f$ to $\tilde{f}$ changes the integrand only on the fixed null exceptional set. Changing the assigned kernel value at zero, arbitrary by [F2], changes the integrand only at $y = x$ , a null singleton by [F7]. Thus both choices preserve the Lebesgue integral, including whether its absolute value is finite.

2.1F5F6F8step 1.1step 1.2

The function $H (x, y) = \tilde{Φ} (x - y) \tilde{f} (y)$ is Borel on $R^{n} \times R^{n}$ by [F8]. By [F6] both Lebesgue measure spaces are $σ$ -finite, so [F5] shows that $J (x) : = \int_{R^{n}} ∣ H (x, y) ∣ d y$ is measurable. Applying [F5] to the positive and negative parts of the real and imaginary parts of $H$ also makes their section integrals measurable.

3.1step 2.1F5

The set $A_{f} = {x : J (x) < \infty}$ is measurable. On $A_{f}$ , all four section integrals from step 2.1 are finite, and their signed combination is $N f (x)$ , so $N f$ is measurable on its pointwise domain. If $A_{f}$ has full measure, extending by zero on $A_{f}^{c}$ gives a measurable representative on $R^{n}$ ; any other extension is equal to it almost everywhere and hence represents the same class. No finiteness claim is made at points outside $A_{f}$ .

4.1A1F2F3F4F5F7step 1.2step 3.1cases∎

If $f = 0$ , then $A_{f} = R^{n}$ and $N f = 0$ . The pole assignment and the finite/infinite boundary of the defining integral are handled in steps 1.2–3.1. This definition assumes $n \geq 2$ ; it uses Countable Choice only for the Borel representative and the stated measure interfaces, and no full Axiom of Choice.

Source notes

Hunter §2.7, equation (2.24), printed p.36, names the integral $\int Γ (x - y) f (y) d y$ the Newtonian potential after proving its smooth compact-data convolution result; the displayed definition itself is not an almost-everywhere convergence theorem. Teschl §5.3, equation (5.21), printed p.117, gives the same integral formula while expressly treating the initial distributional computation as heuristic at that point. Schmidt §2.11, printed p.70, defines the integral for $f \in L_{c p t}^{\infty}$ and proves everywhere finiteness from $Φ \in L_{l o c}^{1}$ . The present statement separates that convergence question: it defines the integral only where absolutely finite and justifies its measurable almost-everywhere class when that domain has full measure. The next items prove the promised stronger convergence claims for bounded and compact $L^{1}$ data.

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Sources

John K. Hunter, Notes on Partial Differential Equations (2014) (standard reference, not scraped)
Thomas Schmidt, Partial Differential Equations I (2026) (standard reference, not scraped)
Gerald Teschl, Partial Differential Equations: From Classical to Modern (2025 archived author manuscript) (standard reference, not scraped)