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Monotonicity and $L^{p}$ contractivity of the heat flow

Statement

Assume Countable Choice, let $n \geq 1$ and $1 \leq p \leq \infty$ . If $f, g \in L^{p} (R^{n})$ satisfy $f \leq g$ almost everywhere, then $H_{t} f \leq H_{t} g$ almost everywhere for every $t > 0$ ; and $∥ H_{t} f ∥_{p} \leq ∥ f ∥_{p}$ for every $f \in L^{p} (R^{n})$ and every $t \geq 0$ .

Facts & Assumptions

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

[A1]

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

[F1]

Each $H_{t}$ is a complex-linear operator on $L^{p} (R^{n})$ , $H_{0}$ is the identity, and $H_{t} f$ is the class of $Γ_{t} * f$ (The heat evolution $H_{t}$ of initial data).

[F2]

If $f \in L^{p}$ satisfies $f \geq 0$ almost everywhere, then $H_{t} f \geq 0$ almost everywhere (Mass conservation and positivity of the heat flow).

[F3]

For $1 \leq p < \infty$ and $f \in L^{p}$ , $∥ H_{t} f ∥_{p} \leq ∥ f ∥_{p}$ (The heat Cauchy problem for $L^{p}$ data).

[F4]

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

Proof

technique · direct

1.1A1F1F2given

Order preservation: assume $f \leq g$ almost everywhere and fix $t > 0$ . The difference $g - f$ is a class in $L^{p}$ with $g - f \geq 0$ almost everywhere, so $H_{t} (g - f) \geq 0$ almost everywhere by [F2]; by linearity of $H_{t}$ in [F1], $H_{t} g - H_{t} f = H_{t} (g - f)$ as classes, so any representatives satisfy $H_{t} f \leq H_{t} g$ almost everywhere, the comparison being independent of representatives because changing them on null sets does not affect an almost-everywhere inequality.

2.1step 1.1F1F3F4givenalgebra

Contractivity: for $1 \leq p < \infty$ the bound $∥ H_{t} f ∥_{p} \leq ∥ f ∥_{p}$ for every $t > 0$ is [F3], while at $t = 0$ it is the identity case of [F1]; for $p = \infty$ Young's inequality [F4] with the exponent triple $(\infty, 1, \infty)$ , which satisfies $1 / \infty = 1 / \infty + 1 - 1$ , gives $∥ H_{t} f ∥_{\infty} \leq ∥ Γ_{t} ∥_{1} ∥ f ∥_{\infty} = ∥ f ∥_{\infty}$ for every $t > 0$ , and again $∥ H_{0} f ∥_{\infty} = ∥ f ∥_{\infty}$ .

3.1step 1.1step 2.1given∎

Steps 1.1 and 2.1 prove the almost-everywhere monotonicity for $f \leq g$ and the $L^{p}$ contraction for all $t \geq 0$ and all $1 \leq p \leq \infty$ .

Depends on

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Sources

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

Monotonicity and LpLp contractivity of the heat flow