ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6.1-sol)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The heat smoothing time exponent is forced by scaling

Example

Assume Countable Choice. For $n \geq 1$ and $1 \leq p \leq q \leq \infty$ , an estimate $∥ H_{t} f ∥_{q} \leq C t^{- β} ∥ f ∥_{p}$ valid for every $t > 0$ and every $f \in L^{p} (R^{n})$ with a finite constant $C$ independent of $t, f$ requires $β = \frac{n}{2} (\frac{1}{p} - \frac{1}{q})$ . This asserts the necessary power, not the optimal Young constant.

Facts & Assumptions

Given: Countable Choice, $n \geq 1$ , $1 \leq p \leq q \leq \infty$ , a real $β$ , a finite constant $C$ with $∥ H_{t} f ∥_{q} \leq C t^{- β} ∥ f ∥_{p}$ for all $t > 0$ and all $f \in L^{p} (R^{n})$ , and $λ > 0$ .

[A1]

Countable Choice is the hypothesis carried by the evolution and integration suppliers below (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

$H_{t} f$ is the $L^{p}$ (respectively $L^{q}$ ) class of $Γ_{t} * f$ whenever $f$ lies in the corresponding space (The heat evolution $H_{t}$ of initial data).

[F2]

For every $s > 0$ the kernel satisfies $Γ (z, s) = (4 π s)^{- n / 2} e^{- ∣ z ∣^{2} / (4 s)}$ and the scaling identity $Γ (λ u, λ^{2} s) = λ^{- n} Γ (u, s)$ ; it is positive with unit mass (Normalisation, parabolic scaling, heat equation and derivative bounds for the heat kernel).

[F3]

For $1 \leq p \leq q \leq \infty$ the heat flow satisfies $∥ H_{1} g ∥_{q} \leq C_{n, p, q} ∥ g ∥_{p}$ with a finite constant, so $∥ H_{1} g ∥_{q} < \infty$ for every $g \in L^{p}$ ( $L^{p}$ to $L^{q}$ smoothing estimate for the heat flow).

[F4]

For a measurable $g \geq 0$ , $\int_{R^{n}} g = 0$ if and only if $g = 0$ almost everywhere (A nonnegative measurable function has integral $0$ exactly when it vanishes almost everywhere).

[F5]

The unit ball $B_{1} \subseteq R^{n}$ is measurable with $0 < ∣ B_{1} ∣ < \infty$ (Euclidean balls have positive finite Lebesgue measure), so its indicator is measurable (An indicator function is measurable exactly when its set is measurable).

[F6]

For a $C^{1}$ diffeomorphism $T$ of open sets and $g \in L^{1} (V)$ , $\int_{V} g (y) d y = \int_{U} g (T (x)) ∣ \det D T (x) ∣ d x$ (A C^1 diffeomorphism satisfies the change-of-variables formula for L^1 functions); the mutually inverse maps $x \mapsto λ x$ and $z \mapsto λ^{- 1} z$ used below qualify with $∣ \det D T ∣ = λ^{\pm n}$ .

Verification

technique · direct

1.1A1F5given

The base datum: put $f : = 1_{B_{1}}$ . By [F5] the function $f$ is measurable, nonnegative and nonzero, and $∥ f ∥_{p} = ∣ B_{1} ∣^{1 / p}$ for $1 \leq p < \infty$ while $∥ f ∥_{\infty} = 1$ , so $f \in L^{p} (R^{n})$ with $0 < ∥ f ∥_{p} < \infty$ for every $1 \leq p \leq \infty$ (with the usual $1 / \infty = 0$ reading of the exponent).

2.1step 1.1F6givenalgebra

Dilated data: for $λ > 0$ put $f_{λ} (x) : = f (λ x)$ . Substituting $z = λ x$ in the $L^{p}$ integral through [F6] gives $∥ f_{λ} ∥_{p} = λ^{- n / p} ∥ f ∥_{p}$ for $1 \leq p < \infty$ , and $∥ f_{λ} ∥_{\infty} = ∥ f ∥_{\infty} = 1$ for $p = \infty$ ; in both cases $∥ f_{λ} ∥_{p} = λ^{- n / p} ∥ f ∥_{p}$ with $1 / \infty = 0$ .

2.2step 1.1F1F2F6givenalgebra

Parabolic scaling of the flow: substituting $z = λ y$ in the defining convolution of [F1] and using the kernel scaling identity of [F2] with $λ$ replaced by $λ^{- 1}$ , $Γ (x - λ^{- 1} z, λ^{- 2}) = λ^{n} Γ (λ x - z, 1)$ , gives $H_{λ^{- 2}} f_{λ} (x) = \int Γ (x - λ^{- 1} z, λ^{- 2}) f (z) λ^{- n} d z = \int Γ (λ x - z, 1) f (z) d z = H_{1} f (λ x)$ for every $x$ .

3.1step 1.1step 2.2F3F4givenalgebra

Norm of the scaled flow: substituting $w = λ x$ in the defining integral of $∥ H_{λ^{- 2}} f_{λ} ∥_{q}^{q}$ through [F6] and using step 2.2 gives $∥ H_{λ^{- 2}} f_{λ} ∥_{q} = λ^{- n / q} ∥ H_{1} f ∥_{q}$ for $1 \leq q < \infty$ , and step 2.2 directly gives $∥ H_{λ^{- 2}} f_{λ} ∥_{\infty} = ∥ H_{1} f ∥_{\infty}$ ; moreover $0 < ∥ H_{1} f ∥_{q} < \infty$ , because the finiteness is [F3] with $g = f \in L^{p}$ , and the strict positivity follows from $H_{1} f > 0$ everywhere (the integrand $Γ (x - y, 1) f (y)$ is positive on the positive-measure set $B_{1}$ ) together with [F4] applied to $H_{1} f \geq 0$ when $q < \infty$ and with the fact that a zero essential supremum would force $H_{1} f = 0$ almost everywhere, contradicting positivity everywhere when $q = \infty$ .

4.1step 3.1givenalgebra

Forcing the exponent: apply the hypothesised estimate to $f_{λ}$ at time $t = λ^{- 2}$ : by steps 2.1 and 3.1, $λ^{- n / q} ∥ H_{1} f ∥_{q} \leq C λ^{2 β} λ^{- n / p} ∥ f ∥_{p}$ , that is, $0 < \frac{∥ H_{1} f ∥_{q}}{∥ f ∥_{p}} \leq C λ^{2 β - n (1 / p - 1 / q)}$ for every $λ > 0$ . If $2 β - n (\frac{1}{p} - \frac{1}{q})$ were positive, letting $λ ↓ 0$ would give the contradiction $0 < L \leq 0$ ; if it were negative, letting $λ \to \infty$ would give the same contradiction; hence $2 β = n (\frac{1}{p} - \frac{1}{q})$ and $β = \frac{n}{2} (\frac{1}{p} - \frac{1}{q})$ .

5.1step 1.1step 2.1step 2.2step 3.1step 4.1F3given∎

Steps 1.1, 2.1, 2.2, 3.1 and 4.1 exhibit a single nonzero nonnegative datum whose parabolic dilates force the time exponent to equal $\frac{n}{2} (\frac{1}{p} - \frac{1}{q})$ in any estimate of the stated form; this determines the necessary power and says nothing about the optimal constant, whose optimality is not asserted by [F3].

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

61 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John K. Hunter, Notes on Partial Differential Equations (revised 18 June 2014, UC Davis) (standard reference, not scraped)
Jared Speck, MIT 18.152 Introduction to Partial Differential Equations, Class Meeting #5: The Fundamental Solution for the Heat Equation (Fall 2011) (standard reference, not scraped)