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The heat Cauchy problem for $L^{p}$ data

Statement

Assume Countable Choice, let $n \geq 1$ and $1 \leq p < \infty$ . For every $f \in L^{p} (R^{n})$ the heat evolution of The heat evolution $H_{t}$ of initial data satisfies: (i) $∥ H_{t} f - f ∥_{p} \to 0$ as $t ↓ 0^{+}$ ; (ii) $∥ H_{t} f ∥_{p} \leq ∥ f ∥_{p}$ for every $t > 0$ ; (iii) the semigroup law $H_{t + s} = H_{t} H_{s}$ on $L^{p}$ for all $s, t \geq 0$ , with $H_{0}$ the identity.

Facts & Assumptions

Given: Countable Choice, $n \geq 1$ , $1 \leq p < \infty$ , $f \in L^{p} (R^{n})$ , and $s, t > 0$ .

[A1]

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

[F1]

For $1 \leq p \leq \infty$ and $g \in L^{p} (R^{n})$ , $H_{t} g$ is the $L^{p}$ class of $x \mapsto \int Γ (x - y, t) g (y) d y$ , defined almost everywhere, each $H_{t}$ is complex-linear, and $H_{0}$ is the identity on $L^{p}$ (The heat evolution $H_{t}$ of initial data).

[F2]

For every $t > 0$ the kernel satisfies $∥ Γ_{t} ∥_{1} = 1$ and the family $(Γ_{t})_{t > 0}$ is an $L^{1}$ approximate identity (Normalisation, parabolic scaling, heat equation and derivative bounds for the heat kernel).

[F3]

For all $s, t > 0$ , $Γ_{t} * Γ_{s} = Γ_{t + s}$ as functions on $R^{n}$ , the convolution converging absolutely everywhere (The heat kernel semigroup identity $Γ_{t} * Γ_{s} = Γ_{t + s}$ ).

[F4]

Assume countable choice; if $(K_{ε})$ is an $L^{1}$ approximate identity and $f \in L^{p}$ with $1 \leq p < \infty$ , then $∥ f * K_{ε} - f ∥_{p} \to 0$ as $ε \to 0^{+}$ (Every $L^{1}$ approximate identity converges to the identity in $L^{p}$ for $1 \leq p < \infty$ ).

[F5]

Assume Countable Choice; for $1 \leq p, q, r \leq \infty$ with $1 / r = 1 / p + 1 / q - 1$ , convolution gives $∥ f * g ∥_{r} \leq ∥ f ∥_{p} ∥ g ∥_{q}$ (Young's convolution inequality under Countable Choice).

[F6]

On sigma-finite product spaces, Tonelli's theorem holds for nonnegative product-measurable functions and Fubini's theorem for $L^{1}$ functions (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product, Fubini's theorem for L^1 functions on a sigma-finite product); the Lebesgue measure on $R^{2 n}$ is the completed product of two copies of $λ_{n}$ (The Euclidean Lebesgue measure is the completion of the product of the factor Lebesgue measures).

Proof

technique · direct

1.1A1F1F2F5givenalgebra

Contraction: by [F1] the class $H_{t} f$ is represented by the convolution $Γ_{t} * f$ , so Young's inequality [F5] with the exponent triple $(p, 1, p)$ , which satisfies $1 / p = 1 / p + 1 / 1 - 1$ , gives $∥ H_{t} f ∥_{p} \leq ∥ Γ_{t} ∥_{1} ∥ f ∥_{p} = ∥ f ∥_{p}$ by [F2]; at $t = 0$ , $H_{0} f = f$ has the same norm.

2.1step 1.1F1F2F4given

Strong convergence at time zero: by [F2] the family $(Γ_{t})_{t > 0}$ is an $L^{1}$ approximate identity, so the published $L^{p}$ approximate-identity theorem [F4] gives $∥ f * Γ_{t} - f ∥_{p} \to 0$ as $t ↓ 0^{+}$ ; by [F1] the class $H_{t} f$ is exactly the class of $f * Γ_{t}$ , so $∥ H_{t} f - f ∥_{p} \to 0$ , which is (i).

2.2step 1.1F1F3F5F6givenalgebra

Semigroup law: assume first $s, t > 0$ and put $g (z) : = \int Γ_{s} (z - y) f (y) d y$ , a representative of $H_{s} f$ defined for almost every $z$ by [F1]. For fixed $x$ consider the nonnegative function $(y, z) \mapsto ∣ Γ_{t} (x - z) ∣ Γ_{s} (z - y) ∣ f (y) ∣$ on $R^{n} \times R^{n}$ ; by the identification of [F6] and Tonelli's theorem, its iterated integral equals $\int_{R^{n}} ∣ f (y) ∣ (\int_{R^{n}} Γ_{t} (x - z) Γ_{s} (z - y) d z) d y$ , and the inner integral is $(Γ_{t} * Γ_{s}) (x - y) = Γ_{t + s} (x - y)$ by [F3], so the whole integral is $(Γ_{t + s} * ∣ f ∣) (x)$ , finite for almost every $x$ because $Γ_{t + s} \in L^{1}$ and Young's inequality [F5] applies. Hence the signed double integral is absolutely convergent, Fubini [F6] applies, and $\int Γ_{t} (x - z) g (z) d z = \int f (y) Γ_{t + s} (x - y) d y$ for almost every $x$ ; that is, $H_{t} (H_{s} f) = H_{t + s} f$ . If $s = 0$ or $t = 0$ the identity is the operator identity $H_{0} = i d$ of [F1], so the semigroup law holds for all $s, t \geq 0$ , which is (iii).

3.1step 1.1step 2.1step 2.2given∎

Steps 1.1, 2.1 and 2.2 prove the contraction bound (ii), the strong convergence (i) and the semigroup law (iii) for every $f \in L^{p} (R^{n})$ , $1 \leq p < \infty$ , which is the whole statement.

Depends on

Used by

Monotonicity and Lᵖ contractivity of the heat flow Corollary
The heat flow need not converge in supremum norm Counterexample
The heat flow of an interval indicator is a difference of Gaussian tails Example
Heat generator at zero on compactly supported smooth data Lemma
Uniqueness of strongly continuous mild heat solutions Theorem

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Sources

John K. Hunter, Notes on Partial Differential Equations (revised 18 June 2014, UC Davis) (standard reference, not scraped)
Gerald Teschl, Partial Differential Equations: From Classical to Modern (2025 archived author manuscript, AMS Graduate Studies in Mathematics) (standard reference, not scraped)
Victor Ivrii, Partial Differential Equations (University of Toronto, 2018, CC BY-SA) (standard reference, not scraped)

The heat Cauchy problem for LpLp data