Normalisation, parabolic scaling, heat equation and derivative bounds for the heat kernel

Work under [A1] and fix $t>0$. By [F1], $\mathrm{\Gamma }(x,t)=(4\pi t{)}^{-n\mathrm{/}2}\exp (-\mathrm{\mid }x{\mathrm{\mid }}^2\mathrm{/}(4t))$. Let $T(u)=(4t{)}^{1\mathrm{/}2}u$, an invertible linear self-map of ${\mathbb{R}}^n$ with $\det DT(u)=(4t{)}^{n\mathrm{/}2}>0$ in the sense of [F5], and put $\phi (x)=\exp (-\mathrm{\mid }x{\mathrm{\mid }}^2\mathrm{/}(4t))\ge 0$; since $T$ is a $C^1$ diffeomorphism and $\phi$ is nonnegative and measurable, [F6] applied with $U=V={\mathbb{R}}^n$ gives ${\int }_{{\mathbb{R}}^n}\mathrm{\Gamma }(x,t)dx=(4\pi t{)}^{-n\mathrm{/}2}{\int }_{{\mathbb{R}}^n}\exp (-\mathrm{\mid }u{\mathrm{\mid }}^2)(4t{)}^{n\mathrm{/}2}du={\pi }^{-n\mathrm{/}2}{\int }_{{\mathbb{R}}^n}{e}^{-\mathrm{\mid }u{\mathrm{\mid }}^2}du$. By [F3] the Lebesgue integral over ${\mathbb{R}}^n$ is the completed product integral, so Tonelli's theorem [F4] factorises ${\int }_{{\mathbb{R}}^n}{e}^{-\mathrm{\mid }u{\mathrm{\mid }}^2}du={\prod }_{i<n}{\int }_{\mathbb{R}}{e}^{-u_i^2}d{u}_i=(\sqrt{\pi }{)}^n$ by the one-dimensional Gaussian integral [F2]; hence ${\int }_{{\mathbb{R}}^n}\mathrm{\Gamma }(x,t)dx=1$, absolutely and as a nonnegative integral, and with the strict positivity recorded in [F1] this proves (i).

Scaling: for $\lambda >0$, [F1] gives $\mathrm{\Gamma }(\lambda x,{\lambda }^{2}t)=(4\pi {\lambda }^{2}t{)}^{-n\mathrm{/}2}\exp (-\mathrm{\mid }\lambda x{\mathrm{\mid }}^2\mathrm{/}(4{\lambda }^{2}t))={\lambda }^{-n}(4\pi t{)}^{-n\mathrm{/}2}\exp (-\mathrm{\mid }x{\mathrm{\mid }}^2\mathrm{/}(4t))={\lambda }^{-n}\mathrm{\Gamma }(x,t)$, using $\mathrm{\mid }\lambda x{\mathrm{\mid }}^2={\lambda }^2\mathrm{\mid }x{\mathrm{\mid }}^2$ and $(4\pi {\lambda }^{2}t{)}^{-n\mathrm{/}2}={\lambda }^{-n}(4\pi t{)}^{-n\mathrm{/}2}$, which is (ii).

Smoothness: the map $(x,t)\mapsto -\mathrm{\mid }x{\mathrm{\mid }}^2\mathrm{/}(4t)$ is a quotient of polynomials defined and smooth on the open set ${\mathbb{R}}^n\times (0,\mathrm{\infty })$, the map $\exp$ is $C^{\mathrm{\infty }}$ by [F12], and $t\mapsto (4\pi t{)}^{-n\mathrm{/}2}$ is a nonzero scalar multiple of $t^{-n\mathrm{/}2}$, smooth for $t>0$; closure under products, scalar multiples and composition [F11] makes $(x,t)\mapsto \mathrm{\Gamma }(x,t)$ $C^{\mathrm{\infty }}$ on ${\mathbb{R}}^n\times (0,\mathrm{\infty })$.

Derivative bound at $t=1$: by induction on $\mathrm{\mid }\alpha \mathrm{\mid }$ we show that $D^{\alpha }\mathrm{\Gamma }(x,1)=P_{\alpha }(x)e^{-\mathrm{\mid }x{\mathrm{\mid }}^2\mathrm{/}4}$ for a polynomial $P_{\alpha }$ with $K_{\alpha }:={\sup }_{x\in {\mathbb{R}}^n}\mathrm{\mid }{P}_{\alpha }(x)\mathrm{\mid }{e}^{-\mathrm{\mid }x{\mathrm{\mid }}^2\mathrm{/}8}<\mathrm{\infty }$. For $\alpha =0$, [F1] gives $P_0=(4\pi {)}^{-n\mathrm{/}2}$, and $K_0<\mathrm{\infty }$ because $e^{-\mathrm{\mid }x{\mathrm{\mid }}^2\mathrm{/}8}$ is bounded and tends to $0$ at infinity. If the claim holds for $\alpha$, then differentiating once more in a coordinate multiplies by a linear polynomial (the derivative of $P_{\alpha }$ plus $-(x_i\mathrm{/}2)P_{\alpha }$) and keeps the Gaussian factor, so the polynomial form is preserved; for the finiteness, each monomial $x^{\beta }$ of $\mathrm{\mid }{P}_{\alpha }\mathrm{\mid }$ satisfies $\mathrm{\mid }{x}^{\beta }\mathrm{\mid }{e}^{-\mathrm{\mid }x{\mathrm{\mid }}^2\mathrm{/}8}\le \mathrm{\mid }x{\mathrm{\mid }}^{\mathrm{\mid }\beta \mathrm{\mid }}{e}^{-\mathrm{\mid }x{\mathrm{\mid }}^2\mathrm{/}8}\to 0$ as $\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }$ by [F8] applied to the radial variable $\mathrm{\mid }x\mathrm{\mid }$, and a continuous function on ${\mathbb{R}}^n$ that tends to $0$ at infinity is bounded, so the supremum is finite. Hence $\mathrm{\mid }{D}^{\alpha }\mathrm{\Gamma }(x,1)\mathrm{\mid }\le K_{\alpha }{e}^{-\mathrm{\mid }x{\mathrm{\mid }}^2\mathrm{/}8}$ for every $x$.

Derivatives: differentiating the formula of [F1] in the coordinate $x_i$ with the one-variable chain and product rules [F9, F10] and ${\exp }^{\mathrm{\prime }}=\exp$ [F12] gives ${\mathrm{\partial }}_{x_i}\mathrm{\Gamma }=-(x_i\mathrm{/}(2t))\mathrm{\Gamma }$ and, differentiating once more, ${\mathrm{\partial }}_{x_i}{\mathrm{\partial }}_{x_i}\mathrm{\Gamma }=-(1\mathrm{/}(2t))\mathrm{\Gamma }+(x_i^2\mathrm{/}(4t^2))\mathrm{\Gamma }$; differentiating in $t$ gives ${\mathrm{\partial }}_t\mathrm{\Gamma }=(-n\mathrm{/}(2t)+\mathrm{\mid }x{\mathrm{\mid }}^2\mathrm{/}(4t^2))\mathrm{\Gamma }$. Summing the spatial identities over $i$ yields ${\mathrm{\Delta }}_x\mathrm{\Gamma }={\sum }_{i<n}{\mathrm{\partial }}_{x_i}{\mathrm{\partial }}_{x_i}\mathrm{\Gamma }=(-n\mathrm{/}(2t)+\mathrm{\mid }x{\mathrm{\mid }}^2\mathrm{/}(4t^2))\mathrm{\Gamma }={\mathrm{\partial }}_t\mathrm{\Gamma }$ on ${\mathbb{R}}^n\times (0,\mathrm{\infty })$, which together with step 1.3 is (iii).

Derivative bound at general $t$: by step 1.2 applied with $\lambda =\sqrt{t}$ and with $x$ replaced by $x\mathrm{/}\sqrt{t}$, $\mathrm{\Gamma }(x,t)=t^{-n\mathrm{/}2}\mathrm{\Gamma }(x\mathrm{/}\sqrt{t},1)$ for every $x$ and $t>0$; differentiating this identity $\alpha$ times in $x$, the chain rule [F9] contributes one factor $t^{-1\mathrm{/}2}$ for each spatial derivative, so $D^{\alpha }\mathrm{\Gamma }(x,t)=t^{-(n+\mathrm{\mid }\alpha \mathrm{\mid })\mathrm{/}2}(D^{\alpha }\mathrm{\Gamma })(x\mathrm{/}\sqrt{t},1)$, and step 1.4 yields $\mathrm{\mid }{D}^{\alpha }\mathrm{\Gamma }(x,t)\mathrm{\mid }\le K_{\alpha }{t}^{-(n+\mathrm{\mid }\alpha \mathrm{\mid })\mathrm{/}2}{e}^{-\mathrm{\mid }x{\mathrm{\mid }}^2\mathrm{/}(8t)}$, which is (iv) with $C_{n,\alpha }=K_{\alpha }$.

Tail estimate: by step 1.2, $\mathrm{\Gamma }(x,t)=t^{-n\mathrm{/}2}\mathrm{\Gamma }(x\mathrm{/}\sqrt{t},1)$; applying the diffeomorphism substitution [F6] to $x=\sqrt{t}z$ gives ${\int }_{\mathrm{\mid }x\mathrm{\mid }>\delta }\mathrm{\Gamma }(x,t)dx={\int }_{\mathrm{\mid }z\mathrm{\mid }>\delta \mathrm{/}\sqrt{t}}\mathrm{\Gamma }(z,1)dz$ for every $\delta >0$. As $t\downarrow 0^{+}$ the integrands $1_{\{\mathrm{\mid }z\mathrm{\mid }>\delta \mathrm{/}\sqrt{t}\}}\mathrm{\Gamma }(z,1)$ are dominated by the fixed integrable function $\mathrm{\Gamma }(\cdot ,1)$ from step 1.1 and converge pointwise to $0$ at every $z$, including $z=0$, so dominated convergence [F7] gives ${\int }_{\mathrm{\mid }z\mathrm{\mid }>\delta \mathrm{/}\sqrt{t}}\mathrm{\Gamma }(z,1)dz\to 0$; with unit mass and positivity from step 1.1 and [F1], the three defining clauses of [F13] hold for the family $K_{\epsilon }:=\mathrm{\Gamma }(\cdot ,\epsilon )$, so $(\mathrm{\Gamma }(\cdot ,t){)}_{t>0}$ is an $L^1$ approximate identity.

Steps 1.1, 1.2, 1.3, 2.1, 1.4, 2.2 and 2.3 prove (i) unit mass and positivity, (ii) parabolic scaling, (iii) smoothness and the heat equation, (iv) the derivative bounds with finite constants $C_{n,\alpha }=K_{\alpha }$, and the unit $L^1$ norm together with the approximate-identity property of [F13]; this is the whole statement.