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$L^{p}$ to $L^{q}$ smoothing estimate for the heat flow

Statement

Assume Countable Choice, let $n \geq 1$ and $1 \leq p \leq q \leq \infty$ , and let $Q \in [1, \infty]$ be determined by $\frac{1}{Q} = 1 + \frac{1}{q} - \frac{1}{p}$ . Then for every $f \in L^{p} (R^{n})$ and every $t > 0$ , $∥ H_{t} f ∥_{q} \leq C_{n, p, q} t^{- \frac{n}{2} (\frac{1}{p} - \frac{1}{q})} ∥ f ∥_{p}, C_{n, p, q} : = (4 π)^{- \frac{n}{2} (\frac{1}{p} - \frac{1}{q})} Q^{- \frac{n}{2 Q}},$ with the endpoint $Q = \infty$ (which occurs exactly at $p = 1$ , $q = \infty$ ) read as $Q^{- n / (2 Q)} \to 1$ ; for $p = q$ the constant is $1$ and the estimate is the contraction clause.

Facts & Assumptions

Given: Countable Choice, $n \geq 1$ , $1 \leq p \leq q \leq \infty$ , the exponent $Q$ with $1 / Q = 1 + 1 / q - 1 / p$ , $f \in L^{p} (R^{n})$ 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 $1 \leq p \leq \infty$ and $f \in L^{p} (R^{n})$ , $H_{t} f$ is the $L^{p}$ class of the convolution $Γ_{t} * f$ (The heat evolution $H_{t}$ of initial data).

[F2]

For every $s > 0$ the kernel satisfies $Γ (x, s) = (4 π s)^{- n / 2} e^{- ∣ x ∣^{2} / (4 s)}$ , the scaling identity $Γ (λ x, λ^{2} s) = λ^{- n} Γ (x, s)$ , and unit mass $\int_{R^{n}} Γ (x, s) d x = 1$ (Normalisation, parabolic scaling, heat equation and derivative bounds for the heat kernel).

[F3]

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

Proof

technique · direct

1.1A1F1F3given

Young estimate: the triple $(p, Q, q)$ is admissible because $1 / Q = 1 + 1 / q - 1 / p$ means exactly $1 / q = 1 / p + 1 / Q - 1$ , and $Q \in [1, \infty]$ lies in the Young range since $0 \leq 1 / p - 1 / q \leq 1$ ; hence, by [F3] applied to $f$ and $Γ_{t} \in L^{Q}$ , $∥ H_{t} f ∥_{q} \leq ∥ Γ_{t} ∥_{Q} ∥ f ∥_{p}$ for every $t > 0$ .

2.1step 1.1F2givenalgebra

Scaling of the kernel norm: the scaling identity of [F2] with $λ = \sqrt{t}$ gives $Γ_{t} (x) = t^{- n / 2} Γ_{1} (x / \sqrt{t})$ , so the substitution $x = \sqrt{t} z$ yields $∥ Γ_{t} ∥_{Q} = t^{- n / 2} t^{n / (2 Q)} ∥ Γ_{1} ∥_{Q} = t^{- \frac{n}{2} (1 - \frac{1}{Q})} ∥ Γ_{1} ∥_{Q}$ for $Q < \infty$ , and for $Q = \infty$ the same substitution gives $∥ Γ_{t} ∥_{\infty} = t^{- n / 2} ∥ Γ_{1} ∥_{\infty}$ .

3.1step 2.1F2givenalgebra

Gaussian norm: by [F2] with $s = 1$ , $Γ_{1} (x) = (4 π)^{- n / 2} e^{- ∣ x ∣^{2} / 4}$ , and for $Q < \infty$ the identity $e^{- Q ∣ x ∣^{2} / 4} = (4 π / Q)^{n / 2} Γ (x, 1 / Q)$ holds by the explicit formula, so unit mass gives $∥ Γ_{1} ∥_{Q} = (4 π)^{- n / 2} (\int e^{- Q ∣ x ∣^{2} / 4} d x)^{1 / Q} = (4 π)^{- n / 2} (4 π / Q)^{n / (2 Q)}$ ; for $Q = \infty$ the same formula is read as $∥ Γ_{1} ∥_{\infty} = (4 π)^{- n / 2}$ .

4.1step 1.1step 2.1step 3.1givenalgebra

Assembling the estimate: since $1 - \frac{1}{Q} = \frac{1}{p} - \frac{1}{q}$ by the definition of $Q$ , steps 1.1, 2.1 and 3.1 give $∥ H_{t} f ∥_{q} \leq t^{- \frac{n}{2} (\frac{1}{p} - \frac{1}{q})} (4 π)^{- n / 2} (4 π / Q)^{n / (2 Q)} ∥ f ∥_{p}$ , and $(4 π)^{- n / 2} (4 π / Q)^{n / (2 Q)} = (4 π)^{- \frac{n}{2} (1 - \frac{1}{Q})} Q^{- n / (2 Q)} = (4 π)^{- \frac{n}{2} (\frac{1}{p} - \frac{1}{q})} Q^{- n / (2 Q)} = C_{n, p, q}$ , which is the displayed estimate.

5.1step 4.1givenalgebra

Endpoints: if $p = q$ then $1 / Q = 1$ and $Q = 1$ , so $C_{n, p, q} = 1$ and the estimate reads $∥ H_{t} f ∥_{p} \leq ∥ f ∥_{p}$ ; if $Q = \infty$ then $1 / p - 1 / q = 1$ , which forces $p = 1$ and $q = \infty$ , and the factor $Q^{- n / (2 Q)}$ tends to $1$ while the exponent is $t^{- n / 2}$ , so the estimate reads $∥ H_{t} f ∥_{\infty} \leq (4 π t)^{- n / 2} ∥ f ∥_{1}$ .

6.1step 4.1step 5.1given∎

Steps 1.1, 2.1, 3.1, 4.1 and 5.1 prove the displayed $L^{p}$ to $L^{q}$ estimate with the stated constant, including the endpoint conventions and the $p = q$ contraction case.

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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)

LpLp to LqLq smoothing estimate for the heat flow