Positive derivatives and canonical representatives

Reconstructed by GPT-6 Astra (OpenAI), Ultra reasoning effort, October 2026. Earlier edition by GPT-6.1 Sol (OpenAI). Original exposition: CC0. Linked programme prerequisites retain their stated licences. Self-checked by the writing AI.

A distribution does not choose a value at a single point. Positivity can nevertheless select a particular representative. A positive first derivative produces an increasing function, a positive Hessian produces a finite continuous convex function, and a positive Laplacian produces an upper semicontinuous function recovered by its shrinking sphere averages. In the last case the selected value can be minus infinity.

All positivity statements concern real distributions, meaning real values on real tests. We use the positive-measure proof in Order, positivity and distributional limits, Theorem 4.1; the flux and surface measure in Boundary flux and weak identities, Theorem 2.1 and Corollary 2.2; the all-dimensional harmonic distribution lemma in Cauchy kernels and boundary limits; the Laplacian source in Point sources and complex Gaussian kernels, Theorem 1.1; and the compact-factor convolution, parameter and local approximation proofs in Convolution as addition of supports, B0–B3, Theorem 1.1, Proposition 5.1 and Theorem 5.2. Polar integration, sphere scaling and constants are proved in the supplied angular foundations, A4–A5. Precise free human-source comparisons are listed at the end.

Two elementary identification facts will be useful. If on an open interval , every compact smooth test of integral zero has a compact primitive in : integrate from the left of its support; the zero integral makes the primitive vanish on the right as well. Thus . Choose of integral one and apply this to . It gives , so is constant.

Also, a locally integrable function with zero distribution is zero almost everywhere. On a compact , multiply it by a compact cutoff equal to one near , and extend by zero to obtain an function . Its small smooth averages vanish on , since there they pair the original zero distribution with compact tests. The approximation theorem in the integration foundations, §15.4, gives . An exhaustion by compact sets proves the assertion. Consequently two locally integrable representatives of a distribution agree almost everywhere; if they are continuous, they agree everywhere, because a nonzero continuous difference has a nonzero sign on a ball of positive volume.

Positive slopes are cumulative measures

Increasing always means nondecreasing. Jump values are not determined by a distribution; we select the right-continuous convention.

Theorem 1.1 (positive first derivatives). On a nonempty open interval , a real distribution has if and only if it has an increasing locally bounded representative. There is exactly one right-continuous increasing representative. If and , it has the form

for a real .

Proof. Theorem 4.1 of the positivity lesson makes a positive Radon measure, finite on compact subsets. Let be (1.1) with . For in , additivity, including the case when the interval crosses , gives

Thus is increasing and locally bounded. It is measurable, since an upper or lower level set of an increasing function is an interval with a possible endpoint. If , the finite measures decrease to zero: the sets decrease to the empty set, and countable additivity gives continuity from above by subtracting from one finite containing interval. Hence is right-continuous. Its jump at has size , as encoded by the left branch.

For a test , choose so that its support lies strictly between them. The product-measure Fubini theorem of the integration foundations, §16.1, applies to the finite measure and bounded . Equation (1.2) yields

The constant part integrates to zero. Therefore , and the zero-derivative fact above gives . Reality gives .

Conversely, average an increasing locally bounded function with a nonnegative compact smooth unit-mass kernel. On every smaller interval the averages are increasing: compare the translated arguments under their integrals. They are smooth by the local convolution theorem, so their first derivatives are nonnegative. Their distributional limit is the original function. Testing the derivative limit on any nonnegative compact test proves positivity.

Finally two right-continuous representatives agree almost everywhere by the identification fact. For each , choose points of equality decreasing to , possible because every nonempty interval has positive measure. Their right limits prove equality at .

Theorem 1.2 (positive second derivatives on an interval). A real distribution satisfies if and only if it has a finite continuous convex representative. That representative is unique and locally Lipschitz.

Proof. Apply Theorem 1.1 to to obtain an increasing locally bounded . Set

On a compact interval, the bound for proves a Lipschitz estimate for . The same Fubini argument as (1.3), now for the signed locally finite density , proves . Thus is a real constant. If , compare the averages of on the two ordered intervals:

One direct justification is to integrate the nonnegative function for . Multiplying out (1.5) is the chord inequality, so plus a constant is convex.

Conversely, nonnegative local averaging preserves the chord inequalities of a continuous convex function. The symmetric second difference of a smooth convex function is nonnegative; dividing by its squared increment and taking the limit proves its second derivative is nonnegative. Pass to the distributional limit of the averages. The construction just proved then supplies a locally Lipschitz representative, and the two continuous representatives agree everywhere.

Proposition 1.3 (one-sided and even averages). Let be smooth, compactly supported and of integral one. All arguments below must stay in . If and is increasing, then

is increasing in , decreasing in , and increases to as . The forward average tends to . If is even and is finite convex, then

is convex in , increasing in , and tends to as .

Proof. Ordered arguments give all inequalities for . A locally bounded increasing function has finite one-sided limits: the left value is a supremum and the right value an infimum on a sufficiently small interval, by real completeness. Dominated convergence then gives both limits.

The finite convex function is continuous locally. For an interior point , choose . Convexity bounds above there by the larger endpoint value ; the midpoint inequality bounds it below on by . For in , ordered secant slopes bound the slope on below by the slope on and above by the slope on . Both comparison intervals have length at least , and their value differences have modulus at most . Hence , proving local Lipschitz continuity without assuming it.

Pair and . For fixed ,

is even and convex. If , express as a convex combination of ; convexity and evenness give . Integrate against . Integrating shifted chord inequalities gives convexity in ; continuity and a bound on the compact range of arguments give the limit.

For complex distributions, a positive first derivative leaves an imaginary constant, and a positive second derivative leaves an imaginary affine function. This follows by applying the zero-derivative fact to the imaginary part once or twice; Solution 2 gives the precise statements.

Sphere averages reveal point sources

Let be the area of the unit sphere ; in dimension one use counting measure on and . The point-source theorem cited above proves

These kernels are locally integrable. Write for the radial value at . Differentiation gives

Set for , and . The kernel is smooth and harmonic away from zero. For smooth , define

Differentiating on the fixed compact sphere and using its scaling from A4 and the flux theorem gives

For , this is , the fundamental theorem. A harmonic function therefore has sphere mean , from the limit at zero.

We supply the surface facts needed when a sphere passes through a pole. Surface measure is invariant under every orthogonal map . To prove this from the supplied foundations, apply polar integration to and to its composition with , first for bounded Borel . The linear change-of-variables formula gives equal Euclidean integrals since . Cancelling the positive radial integral proves invariance on the sphere.

For , after such a rotation, a neighborhood of a chosen point is , . The graph density proved in U011 is . Moreover Indeed the squared chord length is . The rotation can be obtained by completing the unit vector to an orthonormal basis, as proved by Gram–Schmidt in the finite-algebra prerequisite.

It follows that a cap of chord radius has measure at most , by containing its projected coordinates in a box. Every nonempty open cap has positive measure, since its projection contains an open box and the density is positive. A point has measure zero for . On the dyadic rings of chord radii between and , the integral of the power , , is at most . For , the corresponding logarithmic bound is . Summing proves integrability of both singularities on the sphere, using only the geometric series and its differentiated sum.

Lemma 2.1 (the shell formula). For every and ,

The integral is finite even when the sphere meets the pole.

Proof. In dimension one direct evaluation gives

Suppose and . For , the translated kernel is harmonic on the closed ball, so its mean is . For , apply the flux theorem to with a small closed ball about removed. The kernel is harmonic there. On the small sphere, its outward radial flux is ; the hole's normal reverses its sign. Thus the flux on the outer sphere is one. Differentiation of its mean gives .

To identify the constants on the two sides, put . For ,

For , the absolute kernel is bounded by a constant depending on times . For , it is bounded by : the displayed lower bound controls the negative logarithm, and a uniform upper bound on the distance controls the positive logarithm. The integrability just proved and dominated convergence show that the mean is finite and continuous at . Thus the above- mean with derivative joins the value , and equals . If , the integrand is constant on the sphere.

Proposition 2.2 (positive potentials and their averages). For a compactly supported positive Radon measure , put

For , is finite and continuous. For , it takes values in and is upper semicontinuous. In every dimension it is locally integrable, represents , and satisfies . Its finite sphere means are

They increase with and decrease to as , including the value .

Proof. The measure has finite total mass, since its support is compact. On compact sets of and , the kernel has a common finite upper bound . Its integral means , and is therefore well-defined. The kernel is continuous with extended value at zero for . Fatou applied to its nonnegative complement gives

This proves upper semicontinuity even at infinite negative values. For , the continuous kernel is uniformly bounded on every compact difference set; dominated convergence proves the assertion.

For compact and in the support of , the translated set is contained in one bounded ball. Translation invariance and local integrability of the kernel give

Tonelli gives an integrable absolute double integral over and the measure, so and is finite almost everywhere. Absolute Fubini identifies the distribution with the compact-factor convolution of U021. Derivative transfer and (2.1) give .

For a fixed sphere, subtract the same common upper bound before using nonnegative product-measure Tonelli. Lemma 2.1 then proves (2.9), including any initially extended integrals. Its right-hand integrand is bounded above and below because and ranges over a compact set. Thus the sphere integral is finite. Formula (2.2) proves monotonicity; as the integrands decrease to . Subtract them from a fixed common upper bound and apply monotone convergence, obtaining the asserted value at every .

A positive Laplacian selects one function

A function is upper semicontinuous if is open for every real . It is then Borel measurable and bounded above on compact sets: choose a local finite upper bound at each point and use a finite subcover. On a nonempty compact set its supremum is attained. Unless all values are , choose a sequence approaching the finite supremum, take a convergent subsequence, and apply upper semicontinuity.

We call subharmonic when it is upper semicontinuous, is not identically on any connected component, and satisfies

The sphere integral initially has an extended value: its positive part is bounded on that compact sphere, and its negative part may have infinite integral.

Theorem 3.1 (the canonical representative). If is a real distribution on an arbitrary open , , with , it has exactly one subharmonic representative . It belongs to ; every admissible sphere mean is finite and increases with radius. At each point,

Write . For every closed ball contained in ,

and

Both right sides are nonnegative, possibly infinite. When , the left side is the finite sphere mean minus , hence . In the plane the identity is

Proof. Positivity makes a positive Radon measure. For a closed ball strictly inside , choose , smooth and compactly supported in , equal to one on a neighborhood of the ball. The cutoff construction is supplied in the scalar/test foundations. Extend the compactly supported measure by zero to , and set

Then on . The harmonic distribution lemma in U013, “Local smoothing for both constant symbols,” proves that has a smooth harmonic representative on this arbitrary open set. Put , using the pointwise potential from Proposition 2.2. It is upper semicontinuous and locally integrable. Its sphere means on closed balls in are finite, increase with radius and tend to its center value, since the harmonic term's mean is constant.

The polar formula gives and

This also holds in dimension one by splitting an interval into its two half-intervals. For small, the sphere means are bounded above by . After subtracting from that finite bound, monotone convergence proves that (3.7) tends to , even when ; the weight has integral one.

Any two local constructions represent the same distribution. They agree almost everywhere on their overlap by the identification fact, so their small ball averages agree. Their limits in (3.7) show agreement at every point. They therefore define a function throughout . It is locally integrable, so it cannot equal throughout any nonempty open subset; components of an open Euclidean set are open because every small ball is connected. It has (3.1) and (3.2).

For a particular admissible closed ball, choose near that whole ball in (3.6). This represents the same at every point by the gluing argument. The harmonic contribution cancels from the mean difference. In (2.9) the kernel difference is zero for , and is the nonnegative quantity in (3.3) for , where . If the center potential is finite, ordinary subtraction applies. If it is , subtracting the center kernel from the bounded finite averaged kernel gives a nonnegative integral of value . Thus (3.3) holds in both cases. The same construction for any larger admissible ball proves monotonicity for every pair of admissible radii.

For , the fundamental theorem and (2.2) give

At , pass to the limit; it is finite when and infinite when , exactly as the chosen center values require. Nonnegative Tonelli now proves (3.4) from (3.3), using . The choice of an endpoint in each inner integral has no effect; an atom with contributes zero. Inserting the planar kernel gives (3.5). Uniqueness among all subharmonic representatives follows from the converse below.

Theorem 3.2 (submean functions have positive Laplacian). Every function satisfying the definition (3.1) is locally integrable, has positive distributional Laplacian, and agrees at every point with the representative in Theorem 3.1.

Proof. First suppose , and fix a closed ball about inside . Let be a finite upper bound there. On every smaller sphere, submean gives . Polar Tonelli, with its nonnegative integrand, proves integrability of on each smaller ball. Thus is integrable there.

Let be the open set of points with an integrable neighborhood. It meets each connected component because the definition gives a finite point there. It is relatively closed as well. If , choose with . The nonempty open set contains a point with finite, since is finite almost everywhere locally on . The preceding argument, using a ball of radius greater than about still inside , proves integrability on , which contains . Therefore . Connectedness on each component implies .

Take a nonnegative smooth unit-mass kernel supported in radius . Local convolution gives a smooth . For each sphere with all its -translates compactly inside , integrate (3.1) at the shifted centers. The product integral involving is finite: for each , bound by its supremum and integrate on one common compact set, then multiply by the finite sphere area. Fubini is therefore legitimate and gives .

For smooth , Taylor's formula with a remainder uniform in gives

Here reflection of one coordinate makes the means of and , , zero. Coordinate permutations make all means of equal, and their sum is one, so each is . These measure symmetries were proved before Lemma 2.1 and hold directly on the two-point sphere when . The uniform follows by the twice-integrated fundamental theorem and continuity of the Hessian on a compact ball.

The smooth submean inequality and (3.9) imply . Local approximation from U021 gives distributionally, so testing on a nonnegative test proves .

Finally integrate (3.1) in the radius:

Local integrability makes the middle term finite. If is finite, upper semicontinuity bounds the last term by for small . If , it bounds it by any specified real number for small . Thus the ball average tends to the given value at every point. The representative supplied by Theorem 3.1 has the same distribution, hence the same ball averages, and the same limiting values. This proves agreement and the promised uniqueness.

Corollary 3.3 (radial smoothing recovers all point values). Let be the canonical representative and let be smooth, radial, supported in the unit ball and of integral one. At fixed , for , the smooth value increases with , is at least , and tends down to as . If , interpret the distance as infinite.

Proof. Write . Polar integration and sphere reflection invariance give

The weights are nonnegative and integrate to one. Theorem 3.1 proves the lower bound and monotonicity. For , a common upper bound is ; subtract from it and use monotone convergence for the limit, including .

Theorem 3.4 (maximum principles). If is nonempty and compact, then

If is connected and attains a finite global maximum in , it is constant.

Proof. A nonempty compact subset of , , has nonempty boundary: a ray from a point eventually leaves the bounded set and has a last initial point in it. If is throughout , the equality follows. Otherwise its finite maximum is attained. If a maximum point is interior to , let . Compactness of the boundary attains this distance; the closed ball lies in , and its sphere meets .

The sphere values are at most , while submean at makes their mean at least . If one sphere value were less than , upper semicontinuity would make it less by a fixed positive amount on an open cap of positive measure. This would make the mean strictly less than , a contradiction. In dimension one each endpoint has positive weight. Thus a boundary contact point also has value , proving (3.12).

For a finite global maximum , the same argument makes every sufficiently small sphere about a maximum point equal to . Those spheres fill a neighborhood, so is open. It is closed because is open and all values are at most . Connectedness proves the assertion.

Positive Hessians force convexity

On an arbitrary open set, convexity here means the chord inequality on every segment contained in that set; the domain itself need not be convex.

Theorem 4.1 (the distributional Hessian criterion). For a real distribution on open , the following are equivalent: every directional second derivative , , is positive; and has a finite continuous representative satisfying

whenever . The representative is unique and locally Lipschitz.

Proof. The directional condition in the coordinate directions implies . Let be the canonical representative and smooth with a nonnegative radial kernel. Each directional second derivative of is a positive distribution paired with a nonnegative translated kernel, hence is a nonnegative smooth function. Restricting to a segment, the fundamental theorem makes the first derivative increasing; its ordered secant slopes prove the chord inequality.

Any compact segment in has positive distance from . For small the preceding argument applies to it. Corollary 3.3 lets in the chord inequality, initially with possible endpoint values. If either endpoint tends to , its positive coefficient forces the right side to , since the other endpoint has a finite upper bound. Thus the extended inequality also holds.

There are in fact no points. If , choose a ball about in . Almost every in it has finite, by local integrability. The midpoint inequality forces . Affine change of variables shows that these midpoints fill almost all of a smaller ball of positive volume, contradicting integrability.

Choose and a finite upper bound for on it. For , apply the midpoint inequality to :