ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passaudited 2026-09-30

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.

Heat and Poisson semigroups as Fourier multipliers

Example

Assume Countable Choice, let $n \geq 1$ , and use the complex $L^{2}$ conventions of Complex Lp classes and Euclidean test-function conventions. For $t \geq 0$ define the bounded continuous frequency symbols $m_{t} (ξ) = e^{- 4 π^{2} t ∣ ξ ∣^{2}}, p_{t} (ξ) = e^{- 2 π t ∣ ξ ∣}, ξ \in R^{n},$ and let $H_{t} : = T_{m_{t}}, P_{t} : = T_{p_{t}}$ be the unique bounded extensions to $L^{2} (R^{n}; C)$ that Exact L2 Fourier multiplier norm supplies for the Schwartz-core multiplier of Translation-invariant Fourier multiplier on the Schwartz core; explicitly $H_{t} = F_{2}^{- 1} M_{m_{t}} F_{2}, P_{t} = F_{2}^{- 1} M_{p_{t}} F_{2} .$ These are the operators customarily written $e^{t Δ}$ and $e^{- t \sqrt{- Δ}}$ . They are defined here only through the bounded Fourier symbols: no spectral theorem, generator, or functional calculus is assumed. Then:

Both families are $L^{2}$ contractions, with operator norm exactly one: $∥ H_{t} f ∥_{2} \leq ∥ f ∥_{2}$ and $∥ P_{t} f ∥_{2} \leq ∥ f ∥_{2}$ for all $f \in L^{2} (R^{n}; C)$ and all $t \geq 0$ .
$H_{0} = P_{0} = {i d}_{L^{2} (R^{n}; C)}$ .
$H_{t} H_{r} = H_{t + r}$ and $P_{t} P_{r} = P_{t + r}$ for all $t, r \geq 0$ .
For every $f \in L^{2} (R^{n}; C)$ and every $t > 0$ the map $s \mapsto H_{s} f$ is differentiable from $(0, \infty)$ into $L^{2}$ with derivative $u^{'} (t) = F_{2}^{- 1} (- 4 π^{2} ∣ ξ ∣^{2} m_{t} (ξ) F_{2} f)$ , and the distributional Laplacian in $x$ of $u (t) = H_{t} f$ is the regular distribution of the $L^{2}$ class $u^{'} (t)$ ; that is, $u$ solves the heat equation $\partial_{t} u = Δ u$ for $t > 0$ . Likewise $s \mapsto P_{s} f$ is twice differentiable on $(0, \infty)$ and $w (t) = P_{t} f$ satisfies the upper-half-space Laplace equation $\partial_{t}^{2} w + Δ_{x} w = 0$ in $S^{'} (R^{n})$ for every $t > 0$ .

Facts & Assumptions

Given: Countable Choice, $n \geq 1$ , an $L^{2}$ class $f \in L^{2} (R^{n}; C)$ , and real parameters $t, r \geq 0$ and $h$ with $0 < ∣ h ∣ \leq t / 2$ wherever these appear.

[A1]

Countable Choice is the hypothesis carried by every cited Fourier and integration interface below (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

On $S (R^{n})$ the multiplier with symbol $m$ has domain $D_{m} = {f : m \hat{f} locally integrable with tempered regular distribution}$ and acts by $T_{m} f = F^{- 1} (u_{m \hat{f}})$ (Translation-invariant Fourier multiplier on the Schwartz core).

[F2]

If $m$ is measurable with $M = ∥ m ∥_{\infty} = ess sup ∣ m ∣ < \infty$ , then $S \subseteq D_{m}$ , $T_{m} f = F_{2}^{- 1} (m F_{2} f)$ as $L^{2}$ classes for Schwartz $f$ , and $T_{m}$ has a unique bounded $L^{2}$ extension $T_{m} = F_{2}^{- 1} M_{m} F_{2}$ whose operator norm is exactly $M$ (Exact L2 Fourier multiplier norm).

[F3]

Plancherel $F_{2}$ is a surjective complex-linear isometry of $L^{2} (R^{n}; C)$ , so $F_{2}^{- 1}$ exists, is complex-linear, and preserves norms (Plancherel theorem).

[F4]

For $u \in S^{'} (R^{n})$ and every multi-index $α$ , $F (\partial^{α} u) = (2 π i ξ)^{α} F u$ in $S^{'} (R^{n})$ (Fourier differentiation and multiplication identities on tempered distributions).

[F5]

For an $L^{2}$ class $h$ with regular distribution $u_{h}$ , $F u_{h} = u_{F_{2} h}$ in $S^{'} (R^{n})$ (Fourier transform agrees with l one and plancherel transforms).

[F6]

$F$ is a topological automorphism of $S^{'} (R^{n})$ , in particular injective with inverse $F^{- 1}$ (Fourier transform is a topological automorphism of tempered distributions).

[F7]

A smooth function $a$ whose derivatives are all polynomially bounded multiplies $S^{'}$ by $⟨ a u, φ ⟩ = ⟨ u, a φ ⟩$ (Smooth polynomially bounded multipliers on schwartz space); if $g, a g \in L^{2}$ , their regular distributions are tempered by [F5], and for every Schwartz test $φ$ the integrals $⟨ a u_{g}, φ ⟩ = \int g (a φ) = \int (a g) φ = ⟨ u_{a g}, φ ⟩$ converge by Cauchy–Schwarz, since $a φ \in S \subseteq L^{2}$ . Thus $a u_{g} = u_{a g}$ in the case used below. The underlying compact-test regular functional is that of Regular distribution from a locally integrable function.

[F8]

Dominated convergence: if measurable $F_{j}$ satisfy $F_{j} \to F$ almost everywhere and $∣ F_{j} ∣ \leq G$ almost everywhere for one nonnegative integrable $G$ , then $\int ∣ F_{j} - F ∣ \to 0$ (Dominated convergence).

[F9]

$\exp (x + y) = \exp (x) \exp (y)$ for real $x, y$ , and the real exponential is the power series of The real exponential function and the number $e$ by a power series, so $\exp (0) = 1$ (The exponential addition formula $\exp (x + y) = \exp (x) \exp (y)$ ).

[F10]

$1 + x \leq \exp (x)$ for every real $x$ ( $1 + x \leq \exp (x)$ for every real $x$ , hence $(1 - p)^{m} \leq \exp (- m p)$ ).

[F11]

If $g$ is continuous on $[a, b]$ and differentiable on $(a, b)$ , there is $c \in (a, b)$ with $g (b) - g (a) = g^{'} (c) (b - a)$ (The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ ).

[F12]

$(\exp)^{'} = \exp$ , and the chain rule computes $(f \circ g)^{'} (s) = f^{'} (g (s)) g^{'} (s)$ (The exponential function is smooth and $(\exp)^{'} = \exp$ , The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ ).

[F13]

The essential supremum of a measurable function is the least essential bound: if $∣ a ∣ \leq L$ almost everywhere then $∥ a ∥_{\infty} \leq L$ (The essential supremum is attained as the least essential bound).

[F14]

Every nonempty Euclidean ball has positive finite Lebesgue measure (Euclidean balls have positive finite Lebesgue measure).

Verification

Given: The data and conventions of the Example above and of [F1]-[F14].

1.1F9F10F13F14algebra

Each map $ξ \mapsto m_{t} (ξ) = e^{- 4 π^{2} t ∣ ξ ∣^{2}}$ is continuous, and $∣ m_{t} ∣ \leq 1$ everywhere because $- 4 π^{2} t ∣ ξ ∣^{2} \leq 0$ ; the same holds for $p_{t}$ with $∣ p_{t} ∣ \leq 1$ . At $ξ = 0$ both symbols equal $1$ by [F9]. For $ε > 0$ , continuity at $0$ gives a ball on which $∣ m_{t} ∣ > 1 - ε$ and $∣ p_{t} ∣ > 1 - ε$ , these balls have positive measure by [F14], so no number below $1$ is an essential bound; with [F13] this gives $∥ m_{t} ∥_{\infty} = ∥ p_{t} ∥_{\infty} = 1$ for every $t \geq 0$ .

1.2F10algebra

For all $t > 0$ and $r \geq 0$ : (i) $4 π^{2} r^{2} e^{- 4 π^{2} t r^{2}} \leq \frac{1}{e t}$ , (ii) $2 π r e^{- 2 π t r} \leq \frac{1}{e t}$ , (iii) $4 π^{2} r^{2} e^{- 2 π t r} \leq \frac{4}{e^{2} t^{2}}$ . Indeed [F10] with $x = s - 1$ gives $0 < s e^{- s} \leq e^{- 1}$ for every real $s > 0$ ; substituting $s = 4 π^{2} t r^{2}$ proves (i), substituting $s = 2 π t r$ proves (ii), and writing $s^{2} e^{- s} = (2 \cdot \frac{s}{2} e^{- s / 2})^{2} \leq (2 e^{- 1})^{2}$ with the same substitution proves (iii).

2.1F1F2F3F9step 1.1

Step 1.1 bounds the symbols, so the operators under discussion are the domain-qualified Schwartz multipliers $T_{m_{t}}, T_{p_{t}}$ of [F1], and [F2] gives $S \subseteq D_{m_{t}} \cap D_{p_{t}}$ together with, for every $t \geq 0$ , the unique bounded extensions $H_{t} = T_{m_{t}}$ and $P_{t} = T_{p_{t}}$ on $L^{2} (R^{n}; C)$ with $H_{t} = F_{2}^{- 1} M_{m_{t}} F_{2}$ , $P_{t} = F_{2}^{- 1} M_{p_{t}} F_{2}$ and operator norm $∥ m_{t} ∥_{\infty} = ∥ p_{t} ∥_{\infty} = 1$ ; in particular $∥ H_{t} f ∥_{2} \leq ∥ f ∥_{2}$ and $∥ P_{t} f ∥_{2} \leq ∥ f ∥_{2}$ for every $f \in L^{2}$ . Since $m_{0} = p_{0} = 1$ by [F9], the same formulas and [F3] give $H_{0} = P_{0} = F_{2}^{- 1} F_{2} = {i d}_{L^{2}}$ .

2.2F3step 1.2algebra

Fix $f \in L^{2}$ and $t > 0$ and put $g_{t} = - 4 π^{2} ∣ ξ ∣^{2} m_{t} F_{2} f$ , $q_{t} = - 2 π ∣ ξ ∣ p_{t} F_{2} f$ and ${\tilde{g}}_{t} = 4 π^{2} ∣ ξ ∣^{2} p_{t} F_{2} f$ . Each is a measurable function of $ξ$ and step 1.2 bounds its modulus by $\frac{1}{e t} ∣ F_{2} f ∣$ , $\frac{1}{e t} ∣ F_{2} f ∣$ and $\frac{4}{e^{2} t^{2}} ∣ F_{2} f ∣$ respectively; hence by [F3], $g_{t}, q_{t}, {\tilde{g}}_{t} \in L^{2} (R^{n}; C)$ with $∥ g_{t} ∥_{2} \leq \frac{1}{e t} ∥ f ∥_{2}$ , $∥ q_{t} ∥_{2} \leq \frac{1}{e t} ∥ f ∥_{2}$ and $∥ {\tilde{g}}_{t} ∥_{2} \leq \frac{4}{e^{2} t^{2}} ∥ f ∥_{2}$ .

3.1F3F9step 2.1

For all $t, r \geq 0$ the addition law [F9] gives the pointwise symbol identities $m_{t} m_{r} = m_{t + r}$ and $p_{t} p_{r} = p_{t + r}$ . Substituting the explicit formulas of step 2.1 and using that composition of multiplication operators multiplies symbols, $H_{t} H_{r} = F_{2}^{- 1} M_{m_{t}} F_{2} F_{2}^{- 1} M_{m_{r}} F_{2} = F_{2}^{- 1} M_{m_{t} m_{r}} F_{2} = H_{t + r},$ and identically $P_{t} P_{r} = P_{t + r}$ ; the case $t = r = 0$ reduces to $H_{0}^{2} = H_{0}$ , consistent with step 2.1.

3.2F3F8F11F12step 1.2step 2.1step 2.2

(Heat, first time derivative.) Fix $f \in L^{2}$ and $t > 0$ . For $0 < ∣ h ∣ \leq t / 2$ , step 2.1 and linearity of $F_{2}^{- 1}$ write the difference quotient as $\frac{H_{t + h} f - H_{t} f}{h} = F_{2}^{- 1} (\frac{m_{t + h} - m_{t}}{h} F_{2} f)$ . For fixed $ξ$ the function $s \mapsto m_{s} (ξ) = e^{- 4 π^{2} s ∣ ξ ∣^{2}}$ is differentiable on the interval with endpoints $t + h$ and $t$ (both positive), with derivative $s \mapsto - 4 π^{2} ∣ ξ ∣^{2} e^{- 4 π^{2} s ∣ ξ ∣^{2}}$ by [F12]; [F11] therefore gives a point $s_{h} (ξ)$ between $t + h$ and $t$ with $\frac{m_{t + h} (ξ) - m_{t} (ξ)}{h} = - 4 π^{2} ∣ ξ ∣^{2} e^{- 4 π^{2} s_{h} (ξ) ∣ ξ ∣^{2}}$ . As $h \to 0$ one has $s_{h} (ξ) \to t$ , so the quotient tends to $- 4 π^{2} ∣ ξ ∣^{2} m_{t} (ξ)$ pointwise. Since $s_{h} (ξ) \geq t / 2$ , step 1.2(i) with $t / 2$ bounds the quotient by $\frac{2}{e t}$ , while step 1.2(i) also gives $∣ 4 π^{2} ∣ ξ ∣^{2} m_{t} (ξ) ∣ \leq \frac{1}{e t}$ . Hence $F_{h} = ∣ \frac{m_{t + h} - m_{t}}{h} + 4 π^{2} ∣ ξ ∣^{2} m_{t} ∣^{2} ∣ F_{2} f ∣^{2}$ tends to $0$ pointwise and is dominated by $\frac{9}{e^{2} t^{2}} ∣ F_{2} f ∣^{2}$ , which is integrable; for every sequence $h_{j} \to 0$ with $∣ h_{j} ∣ \leq t / 2$ , [F8] gives $\int F_{h_{j}} \to 0$ , and the isometry [F3] converts this into $∥ \frac{H_{t + h_{j}} f - H_{t} f}{h_{j}} - F_{2}^{- 1} (g_{t}) ∥_{2} \to 0$ with $g_{t}$ from step 2.2, which is the two-sided limit statement. Thus $s \mapsto H_{s} f$ is differentiable on $(0, \infty)$ with $u^{'} (t) = F_{2}^{- 1} (g_{t}) = F_{2}^{- 1} (- 4 π^{2} ∣ ξ ∣^{2} m_{t} F_{2} f)$ .

3.3F3F8F11F12step 1.2step 2.1step 2.2

(Poisson, first time derivative.) The same computation with $p_{t}$ in place of $m_{t}$ : for fixed $ξ$ the function $s \mapsto p_{s} (ξ) = e^{- 2 π s ∣ ξ ∣}$ has derivative $- 2 π ∣ ξ ∣ e^{- 2 π s ∣ ξ ∣}$ by [F12], so [F11] gives points with quotient tending pointwise to $- 2 π ∣ ξ ∣ p_{t} (ξ)$ ; step 1.2(ii) bounds the quotient by $\frac{2}{e t}$ and the limit symbol by $\frac{1}{e t}$ , so [F8] and [F3] give that $s \mapsto P_{s} f$ is differentiable on $(0, \infty)$ with $w^{'} (t) = F_{2}^{- 1} (q_{t})$ , $q_{t}$ as in step 2.2.

4.1F4F5F6F7step 2.1step 2.2step 3.2

(Heat equation.) Let $u (t) = H_{t} f$ . By step 2.1, $F_{2} u (t) = m_{t} F_{2} f$ , so [F5] gives $F (u_{u (t)}) = u_{m_{t} F_{2} f}$ . Summing the coordinate identities of [F4] with $∣ α ∣ = 2$ gives $F (Δ u_{u (t)}) = \sum_{j} (2 π i ξ_{j})^{2} F (u_{u (t)}) = - 4 π^{2} ∣ ξ ∣^{2} u_{m_{t} F_{2} f}$ , and the polynomial $- 4 π^{2} ∣ ξ ∣^{2}$ together with [F7] identifies this as the regular distribution $u_{g_{t}}$ of the $L^{2}$ class $g_{t}$ of step 2.2. Since [F5] applied to $h = F_{2}^{- 1} g_{t}$ gives $F (u_{F_{2}^{- 1} g_{t}}) = u_{g_{t}}$ , injectivity of $F$ on $S^{'}$ [F6] yields $Δ u_{u (t)} = u_{F_{2}^{- 1} g_{t}}$ . By step 3.2 the class $u^{'} (t)$ is exactly $F_{2}^{- 1} g_{t}$ , so for every $t > 0$ the distributional Laplacian of $u (t) = H_{t} f$ is the regular distribution of the strong $L^{2}$ derivative $u^{'} (t)$ : the solution satisfies $\partial_{t} u = Δ u$ for $t > 0$ .

4.2F3F8F11F12step 1.2step 2.2step 3.3

(Poisson, second time derivative.) Apply the argument of step 3.2 to the family the scalar symbols $b_{s} (ξ) = - 2 π ∣ ξ ∣ p_{s} (ξ)$ , so that $q_{s} = b_{s} F_{2} f$ is the $L^{2}$ class of step 2.2: for fixed $ξ$ , the map $s \mapsto - 2 π ∣ ξ ∣ e^{- 2 π s ∣ ξ ∣}$ has derivative $4 π^{2} ∣ ξ ∣^{2} p_{s} (ξ)$ by [F12], so [F11] gives points $σ_{h} (ξ) \geq t / 2$ with $\frac{b_{t + h} (ξ) - b_{t} (ξ)}{h} = 4 π^{2} ∣ ξ ∣^{2} e^{- 2 π σ_{h} (ξ) ∣ ξ ∣}$ tending to $4 π^{2} ∣ ξ ∣^{2} p_{t} (ξ)$ pointwise, and step 1.2(iii) with $t / 2$ in place of $t$ bounds these quotients by $\frac{16}{e^{2} t^{2}}$ while step 1.2(iii) bounds $∣ 4 π^{2} ∣ ξ ∣^{2} p_{t} (ξ) ∣$ by $\frac{4}{e^{2} t^{2}}$ . Hence $F_{h} = ∣ \frac{b_{t + h} - b_{t}}{h} - 4 π^{2} ∣ ξ ∣^{2} p_{t} ∣^{2} ∣ F_{2} f ∣^{2} \to 0$ pointwise, dominated by $(\frac{20}{e^{2} t^{2}})^{2} ∣ F_{2} f ∣^{2}$ ; [F8] and [F3] give $∥ \frac{w^{'} (t + h) - w^{'} (t)}{h} - F_{2}^{- 1} ({\tilde{g}}_{t}) ∥_{2} \to 0$ with ${\tilde{g}}_{t}$ of step 2.2, so $s \mapsto P_{s} f$ is twice differentiable on $(0, \infty)$ with $w^{''} (t) = F_{2}^{- 1} ({\tilde{g}}_{t})$ .

5.1F4F5F6F7step 2.1step 4.2

(Upper-half-space Laplace equation.) Let $w (t) = P_{t} f$ . By step 2.1, $F_{2} w (t) = p_{t} F_{2} f$ , so [F5] and [F4] with $∣ α ∣ = 2$ give $F (Δ_{x} w (t)) = - 4 π^{2} ∣ ξ ∣^{2} u_{p_{t} F_{2} f} = u_{- {\tilde{g}}_{t}}$ by [F7]. Step 4.2 gives $F (u_{w^{''} (t)}) = u_{{\tilde{g}}_{t}}$ by [F5], so by linearity of $F$ the distribution $u_{w^{''} (t)} + Δ_{x} w (t)$ has Fourier transform $u_{{\tilde{g}}_{t}} + u_{- {\tilde{g}}_{t}} = 0$ ; injectivity of $F$ [F6] gives $w^{''} (t) + Δ_{x} w (t) = 0$ in $S^{'} (R^{n})$ for every $t > 0$ , the upper-half-space Laplace equation with boundary control left entirely to the symbol $e^{- 2 π t ∣ ξ ∣}$ .

6.1A1step 2.1step 3.1step 4.1step 5.1∎

The operators $H_{t}, P_{t}$ are defined only through the bounded symbols $m_{t}, p_{t}$ by [F2] (step 2.1): step 2.1 gives the contraction and identity claims, step 3.1 the semigroup laws, step 4.1 the heat equation, and step 5.1 the upper-half-space Laplace equation, which are exactly the four asserted properties; Countable Choice enters only through the cited published interfaces of [A1], and no spectral theorem is used.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

107 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

Mark Williams, Notes on Harmonic Analysis (standard reference, not scraped)
Loukas Grafakos, Classical Fourier Analysis, 3rd ed. (standard reference, not scraped)