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Instantaneous smoothing of the Lp heat flow

Statement

Assume Countable Choice, let $n \geq 1$ , $1 \leq p \leq q \leq \infty$ and let $Q$ be the Young exponent of $L^{p}$ to $L^{q}$ smoothing estimate for the heat flow. For every $f \in L^{p} (R^{n})$ the integral representative $u (x, t) : = \int_{R^{n}} Γ (x - y, t) f (y) d y$ is $C^{\infty}$ on $R^{n} \times (0, \infty)$ and satisfies $u_{t} = Δ u$ there; moreover for every multi-index $α$ and every integer $m \geq 0$ , $D_{x}^{α} \partial_{t}^{m} u = (Δ_{x}^{m} D_{x}^{α} Γ_{t}) * f, ∥ D_{x}^{α} \partial_{t}^{m} u (\cdot, t) ∥_{q} \leq C_{n, α, m, Q} t^{- \frac{2 m + ∣ α ∣}{2} - \frac{n}{2} (\frac{1}{p} - \frac{1}{q})} ∥ f ∥_{p}$ for every $t > 0$ , with $C_{n, α, m, Q}$ the constant produced by the derivative bounds of Spatial and time derivatives pass through heat convolution for positive time and $L^{p}$ to $L^{q}$ smoothing estimate for the heat flow. No strong continuity of $H_{t} f$ at $t = 0$ is asserted for $q = \infty$ or $p = \infty$ .

Facts & Assumptions

Given: Countable Choice, $n \geq 1$ , $1 \leq p \leq q \leq \infty$ , the Young exponent $Q$ with $\frac{1}{Q} = 1 + \frac{1}{q} - \frac{1}{p}$ , $f \in L^{p} (R^{n})$ , a multi-index $α$ , an integer $m \geq 0$ , and $t > 0$ .

[A1]

Countable Choice is the ambient hypothesis (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

The absolutely convergent representative $u (x, t) = \int Γ (x - y, t) f (y) d y$ is $C^{\infty}$ for $t > 0$ , and $D_{x}^{α} \partial_{t}^{k} u = (D_{x}^{α} \partial_{t}^{k} Γ_{t}) * f$ , with the Gaussian bound $∣ D_{x}^{α} \partial_{t}^{k} Γ (x, t) ∣ \leq C_{n, α, k} t^{- (n + ∣ α ∣ + 2 k) / 2} e^{- ∣ x ∣^{2} / (8 t)}$ (Spatial and time derivatives pass through heat convolution for positive time).

[F2]

For the Young exponent $Q$ and every $t > 0$ , $∥ H_{t} f ∥_{q} \leq C_{n, p, q} t^{- \frac{n}{2} (\frac{1}{p} - \frac{1}{q})} ∥ f ∥_{p}$ , and for every multi-index $α$ , $∥ D^{α} H_{t} f ∥_{q} \leq C_{n, p, q, α} t^{- \frac{∣ α ∣}{2} - \frac{n}{2} (\frac{1}{p} - \frac{1}{q})} ∥ f ∥_{p}$ ( $L^{p}$ to $L^{q}$ smoothing estimate for the heat flow, Spatial derivative estimates for the heat flow).

[F3]

$Γ$ is $C^{\infty}$ on $R^{n} \times (0, \infty)$ , solves $\partial_{t} Γ = Δ_{x} Γ$ there, and satisfies the parabolic scaling $Γ (λ x, λ^{2} t) = λ^{- n} Γ (x, t)$ for $λ > 0$ (Normalisation, parabolic scaling, heat equation and derivative bounds for the heat kernel).

[F4]

Mixed partial derivatives of a sufficiently smooth function commute (Clairaut--Schwarz theorem for continuous second partial derivatives), and Young's convolution inequality $∥ K * f ∥_{q} \leq ∥ K ∥_{Q} ∥ f ∥_{p}$ holds when $\frac{1}{Q} = 1 + \frac{1}{q} - \frac{1}{p}$ (Young's convolution inequality under Countable Choice).

[F5]

The representative is the one defining $H_{t} f$ , and $u$ is written in the multi-index notation of $C^{k}$ maps and multi-index derivative notation in Euclidean space (The heat evolution $H_{t}$ of initial data).

Proof

Given: Countable Choice, $n \geq 1$ , $1 \leq p \leq q \leq \infty$ , the Young exponent $Q$ , $f \in L^{p} (R^{n})$ , a multi-index $α$ , an integer $m \geq 0$ , and $t > 0$ .

1.1A1F1F5given

By [F1] the representative $u$ is $C^{\infty}$ on $R^{n} \times (0, \infty)$ and $D_{x}^{α} \partial_{t}^{m} u = (D_{x}^{α} \partial_{t}^{m} Γ_{t}) * f$ for every $t > 0$ , the integral being absolutely convergent.

2.1step 1.1F3F4given

Since $\partial_{t} Γ = Δ_{x} Γ$ on $R^{n} \times (0, \infty)$ by [F3] and all partial derivatives of the $C^{\infty}$ function $Γ$ commute by [F4], induction on $m$ gives $\partial_{t}^{m} Γ_{t} = Δ_{x}^{m} Γ_{t}$ ; substituting into step 1.1 gives $D_{x}^{α} \partial_{t}^{m} u = (Δ_{x}^{m} D_{x}^{α} Γ_{t}) * f$ , and taking $α = 0$ , $m = 1$ gives $u_{t} = Δ u$ .

3.1step 2.1F1F3given

For every $t > 0$ the scaling identity of [F3], differentiated $∣ α ∣$ times in space and $2 m$ times in space (equivalently $m$ times in time through the equation), gives $Δ_{x}^{m} D_{x}^{α} Γ (y, t) = t^{- \frac{n}{2} - m - \frac{∣ α ∣}{2}} (Δ_{x}^{m} D_{x}^{α} Γ) (t^{- 1 / 2} y, 1)$ ; substituting $y = t^{1 / 2} z$ in the $L^{Q}$ integral and using $\frac{1}{Q} = 1 + \frac{1}{q} - \frac{1}{p}$ yields $∥ Δ_{x}^{m} D_{x}^{α} Γ (\cdot, t) ∥_{Q} = t^{- \frac{2 m + ∣ α ∣}{2} - \frac{n}{2} (\frac{1}{p} - \frac{1}{q})} C_{n, α, m, Q}$ with $C_{n, α, m, Q} : = ∥ Δ_{x}^{m} D_{x}^{α} Γ (\cdot, 1) ∥_{Q}$ , which is finite because the bound of [F1] at $t = 1$ majorises the integrand by a multiple of $e^{- ∣ y ∣^{2} / 8}$ .

4.1step 2.1step 3.1F4given

Applying Young's inequality [F4] with the kernel $K = Δ_{x}^{m} D_{x}^{α} Γ_{t}$ and the exponent relation $\frac{1}{Q} = 1 + \frac{1}{q} - \frac{1}{p}$ , and inserting the norm identity of step 3.1, gives $∥ D_{x}^{α} \partial_{t}^{m} u (\cdot, t) ∥_{q} = ∥ (Δ_{x}^{m} D_{x}^{α} Γ_{t}) * f ∥_{q} \leq ∥ Δ_{x}^{m} D_{x}^{α} Γ_{t} ∥_{Q} ∥ f ∥_{p} \leq C_{n, α, m, Q} t^{- \frac{2 m + ∣ α ∣}{2} - \frac{n}{2} (\frac{1}{p} - \frac{1}{q})} ∥ f ∥_{p}$ .

5.1step 4.1F2given∎

Steps 1.1 and 2.1 give the $C^{\infty}$ smoothness, the equation $u_{t} = Δ u$ and the representation $D_{x}^{α} \partial_{t}^{m} u = (Δ_{x}^{m} D_{x}^{α} Γ_{t}) * f$ , while step 4.1 gives the displayed $L^{q}$ bound with constant $C_{n, α, m, Q}$ built from the derivative bounds of [F1] and [F2]; nothing in the argument uses or asserts continuity of $H_{t} f$ at $t = 0$ , so the caveat for $q = \infty$ or $p = \infty$ is preserved.

Depends on

Used by

A mild heat solution need not be classical at the initial time Counterexample

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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)
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)