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Spatial derivative estimates for the heat flow

Statement

Assume Countable Choice, let $n \geq 1$ , $1 \leq p \leq q \leq \infty$ , and let $Q$ and the constants be as in $L^{p}$ to $L^{q}$ smoothing estimate for the heat flow. For every multi-index $α$ , every $f \in L^{p} (R^{n})$ and every $t > 0$ , the function $x \mapsto \int_{R^{n}} Γ (x - y, t) f (y) d y$ is $C^{\infty}$ on $R^{n}$ , with $D^{α} H_{t} f = (D^{α} Γ (\cdot, t)) * f$ as an absolutely convergent integral, and $∥ D^{α} H_{t} f ∥_{q} \leq C_{n, p, q, α} t^{- \frac{∣ α ∣}{2} - \frac{n}{2} (\frac{1}{p} - \frac{1}{q})} ∥ f ∥_{p}, C_{n, p, q, α} : = ∥ D^{α} Γ (\cdot, 1) ∥_{Q} < \infty .$

Facts & Assumptions

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

[A1]

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

[F1]

For every $s > 0$ : $Γ (\cdot, s)$ is smooth, unit mass holds, $Γ (λ x, λ^{2} s) = λ^{- n} Γ (x, s)$ for every $λ > 0$ , and $D^{β} Γ (x, s) = s^{- (n + ∣ β ∣) / 2} (D^{β} Γ) (x / \sqrt{s}, 1)$ for every multi-index $β$ ; moreover $∣ D^{β} Γ (x, 1) ∣ \leq C_{n, β} e^{- ∣ x ∣^{2} / 8}$ (Normalisation, parabolic scaling, heat equation and derivative bounds for the heat kernel).

[F2]

$H_{t} f$ has the everywhere-defined absolutely convergent representative $u (x, t) = \int_{R^{n}} Γ (x - y, t) f (y) d y$ , which is $C^{\infty}$ in $x$ , and $D^{α} H_{t} f = (D^{α} Γ (\cdot, t)) * f$ for every multi-index $α$ (Spatial and time derivatives pass through heat convolution for positive time).

[F3]

For $1 \leq p, q, r \leq \infty$ with $1 / r = 1 / p + 1 / q - 1$ and $f \in L^{p}$ , $g \in L^{q}$ , $∥ f * g ∥_{r} \leq ∥ f ∥_{p} ∥ g ∥_{q}$ (Young's convolution inequality under Countable Choice).

[F4]

With $Q$ as in the statement, $1 / Q = 1 + 1 / q - 1 / p$ and the kernel estimate $∥ H_{t} f ∥_{q} \leq C_{n, p, q} t^{- \frac{n}{2} (\frac{1}{p} - \frac{1}{q})} ∥ f ∥_{p}$ holds ( $L^{p}$ to $L^{q}$ smoothing estimate for the heat flow); the exponent range $1 \leq p \leq q \leq \infty$ makes $Q \in [1, \infty]$ .

Proof

technique · direct

1.1A1F1F2given

Smoothness and derivative identity: by [F2] the representative $u$ is $C^{\infty}$ in $x$ for $t > 0$ and $D^{α} H_{t} f$ is the class of $(D^{α} Γ (\cdot, t)) * f$ , an absolutely convergent integral by the derivative bounds of [F1] together with Hölder and $f \in L^{p}$ .

2.1step 1.1F1givenalgebra

Scaling of the derivative kernel norms: by the derivative scaling identity of [F1], $D^{α} Γ (x, t) = t^{- (n + ∣ α ∣) / 2} (D^{α} Γ) (x / \sqrt{t}, 1)$ ; for $Q < \infty$ , substituting $x = \sqrt{t} z$ gives $∥ D^{α} Γ (\cdot, t) ∥_{Q}^{Q} = t^{- (n + ∣ α ∣) Q / 2} t^{n / 2} ∥ D^{α} Γ (\cdot, 1) ∥_{Q}^{Q}$ , hence $∥ D^{α} Γ (\cdot, t) ∥_{Q} = t^{- ∣ α ∣ / 2} t^{- \frac{n}{2} (1 - \frac{1}{Q})} ∥ D^{α} Γ (\cdot, 1) ∥_{Q}$ . For $Q = \infty$ , taking essential suprema in the same scaling identity gives $∥ D^{α} Γ (\cdot, t) ∥_{\infty} = t^{- (n + ∣ α ∣) / 2} ∥ D^{α} Γ (\cdot, 1) ∥_{\infty}$ , the same formula with $1 / Q = 0$ . The constant $C_{n, p, q, α} = ∥ D^{α} Γ (\cdot, 1) ∥_{Q}$ is finite because $∣ D^{α} Γ (x, 1) ∣ \leq C_{n, α} e^{- ∣ x ∣^{2} / 8}$ by [F1] and the Gaussian $e^{- ∣ \cdot ∣^{2} / 8}$ lies in every $L^{Q}$ , $1 \leq Q \leq \infty$ .

3.1step 1.1step 2.1F3F4givenalgebra

Young estimate for the derivative: applying Young's convolution inequality [F3] with the exponent triple $(p, Q, q)$ , admissible because $1 / Q = 1 + 1 / q - 1 / p$ , gives $∥ D^{α} H_{t} f ∥_{q} \leq ∥ D^{α} Γ (\cdot, t) ∥_{Q} ∥ f ∥_{p}$ , and step 2.1 together with the identity $1 - \frac{1}{Q} = \frac{1}{p} - \frac{1}{q}$ from [F4] turns this into $∥ D^{α} H_{t} f ∥_{q} \leq C_{n, p, q, α} t^{- \frac{∣ α ∣}{2} - \frac{n}{2} (\frac{1}{p} - \frac{1}{q})} ∥ f ∥_{p}$ , which is the stated estimate.

4.1step 1.1step 3.1given∎

Steps 1.1, 2.1 and 3.1 prove the smoothness of the representative, the absolutely convergent convolution formula for $D^{α} H_{t} f$ and the displayed derivative estimate with finite constant $C_{n, p, q, α} = ∥ D^{α} Γ (\cdot, 1) ∥_{Q}$ .

Depends on

Used by

The heat flow need not converge in supremum norm Counterexample

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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)
Gerald Teschl, Partial Differential Equations: From Classical to Modern (2025 archived author manuscript, AMS Graduate Studies in Mathematics) (standard reference, not scraped)