Spatial and time derivatives pass through heat convolution for positive time

Work under [A1]; let $u(x,t)={\int }_{{\mathbb{R}}^n}\mathrm{\Gamma }(x-y,t)f(y)dy$ be the everywhere-defined representative of the evolution $H_{t}f$ of [F1]. Kernel derivatives and their $L^{p^{\mathrm{\prime }}}$ bounds: by [F2] the identity ${\mathrm{\partial }}_t\mathrm{\Gamma }={\mathrm{\Delta }}_x\mathrm{\Gamma }$ holds on ${\mathbb{R}}^n\times (0,\mathrm{\infty })$, and $\mathrm{\Gamma }$ is $C^{\mathrm{\infty }}$, so Clairaut–Schwarz [F3] lets the time and space derivatives be interchanged; iterating the heat equation gives ${\mathrm{\partial }}_t^k\mathrm{\Gamma }={\mathrm{\Delta }}_x^k\mathrm{\Gamma }$ and therefore $D_x^{\alpha }{\mathrm{\partial }}_t^k\mathrm{\Gamma }={\mathrm{\Delta }}_x^k{D}_x^{\alpha }\mathrm{\Gamma }$, a finite sum of terms $D^{\beta }\mathrm{\Gamma }$ with $\mathrm{\mid }\beta \mathrm{\mid }=\mathrm{\mid }\alpha \mathrm{\mid }+2k$, so [F2] yields $\mathrm{\mid }{D}_x^{\alpha }{\mathrm{\partial }}_t^k\mathrm{\Gamma }(x,t)\mathrm{\mid }\le C_{n,\alpha ,k}{t}^{-(n+\mathrm{\mid }\alpha \mathrm{\mid }+2k)\mathrm{/}2}{e}^{-\mathrm{\mid }x{\mathrm{\mid }}^2\mathrm{/}(8t)}$ for a finite constant. Writing $p^{\mathrm{\prime }}$ for the conjugate exponent and $m:=\mathrm{\mid }\alpha \mathrm{\mid }+2k$, for $1<p\le \mathrm{\infty }$ (so $p^{\mathrm{\prime }}<\mathrm{\infty }$) the function $z\mapsto e^{-p^{\mathrm{\prime }}\mathrm{\mid }z{\mathrm{\mid }}^2\mathrm{/}(8t)}$ is $e^{-\mathrm{\mid }z{\mathrm{\mid }}^2\mathrm{/}(4t^{\mathrm{\prime }})}=(4\pi t^{\mathrm{\prime }}{)}^{n\mathrm{/}2}\mathrm{\Gamma }(z,t^{\mathrm{\prime }})$ with $t^{\mathrm{\prime }}=2t\mathrm{/}{p}^{\mathrm{\prime }}$, so its integral is $(4\pi t^{\mathrm{\prime }}{)}^{n\mathrm{/}2}=(8\pi t\mathrm{/}{p}^{\mathrm{\prime }}{)}^{n\mathrm{/}2}$ by unit mass in [F2], while for $p=1$ the function $z\mapsto e^{-\mathrm{\mid }z{\mathrm{\mid }}^2\mathrm{/}(8t)}$ is bounded by $1$; in every case $D_x^{\alpha }{\mathrm{\partial }}_t^k\mathrm{\Gamma }(\cdot ,t)\in L^{p^{\mathrm{\prime }}}$ with norm at most $C_{n,\alpha ,k}{t}^{-(n+m)\mathrm{/}2}(8\pi t\mathrm{/}{p}^{\mathrm{\prime }}{)}^{n\mathrm{/}(2p^{\mathrm{\prime }})}$ when $p>1$, and at most $C_{n,\alpha ,k}{t}^{-(n+m)\mathrm{/}2}$ when $p=1$.

Absolute convergence and compact domination: fix $x\in {\mathbb{R}}^n$ and $t>0$; by Hölder [F5] and step 1.1, $\int \mathrm{\mid }{D}_x^{\alpha }{\mathrm{\partial }}_t^k\mathrm{\Gamma }(x-y,t)f(y)\mathrm{\mid }dy\le \mathrm{\parallel }{D}_x^{\alpha }{\mathrm{\partial }}_t^k\mathrm{\Gamma }(\cdot ,t){\mathrm{\parallel }}_{p^{\mathrm{\prime }}}\mathrm{\parallel }f{\mathrm{\parallel }}_p<\mathrm{\infty }$, so every derivative integral converges absolutely and defines $u_{\alpha }^k(x,t):=\int D_x^{\alpha }{\mathrm{\partial }}_t^k\mathrm{\Gamma }(x-y,t)f(y)dy$. If $K\subseteq B_R$ and $0<\tau \le t\le T$, then $\mathrm{\mid }x-y{\mathrm{\mid }}^2\ge \mathrm{\mid }y{\mathrm{\mid }}^2\mathrm{/}2-R^2$ gives $\mathrm{\mid }{D}_x^{\alpha }{\mathrm{\partial }}_t^k\mathrm{\Gamma }(x-y,t)\mathrm{\mid }\le C_{n,\alpha ,k}{\tau }^{-(n+m)\mathrm{/}2}{e}^{R^2\mathrm{/}(8\tau )}{e}^{-\mathrm{\mid }y{\mathrm{\mid }}^2\mathrm{/}(16T)}$ for every $x\in K$ and $t\in [\tau ,T]$, an $x$-independent, $t$-independent bound whose product with $\mathrm{\mid }f\mathrm{\mid }$ is integrable by [F5] because the Gaussian $e^{-\mathrm{\mid }y{\mathrm{\mid }}^2\mathrm{/}(16T)}$ is in every $L^q$.

Differentiation under the integral sign: for fixed $x$ apply [F6] on any open parameter interval $I=(\tau ,T)$ with $0<\tau <T$ to $F(y,t)=\mathrm{\Gamma }(x-y,t)f(y)$, whose $t$-derivative is dominated uniformly on $I$ by the integrable function from step 2.1 with $\alpha =0,k=1$; this gives ${\mathrm{\partial }}_{t}u(x,t)=\int {\mathrm{\partial }}_t\mathrm{\Gamma }(x-y,t)f(y)dy$, and the same argument with ${\mathrm{\partial }}_{x_i}$ in place of ${\mathrm{\partial }}_t$ gives ${\mathrm{\partial }}_{x_i}u(x,t)=\int {\mathrm{\partial }}_{x_i}\mathrm{\Gamma }(x-y,t)f(y)dy$. Reapplying [F6] to the resulting integral representations, whose integrands are dominated on each compact set exactly as in step 2.1 for the next multi-index, produces every mixed derivative $D_x^{\alpha }{\mathrm{\partial }}_t^{k}u$ as the corresponding convolution integral; the domination of step 2.1 is uniform in $(x,t)$ on $K\times [\tau ,T]$, and dominated convergence [F4] makes each such integral a continuous function of $(x,t)$ there, so $u$ is $C^{\mathrm{\infty }}$ on ${\mathbb{R}}^n\times (0,\mathrm{\infty })$ and $D_x^{\alpha }{\mathrm{\partial }}_t^{k}u=(D_x^{\alpha }{\mathrm{\partial }}_t^k{\mathrm{\Gamma }}_t)\ast f$.

Heat equation: by step 3.1 with $\alpha =0$, $k=1$ and with the spatial derivatives, ${\mathrm{\partial }}_{t}u=({\mathrm{\partial }}_t{\mathrm{\Gamma }}_t)\ast f$ and ${\mathrm{\Delta }}_{x}u=({\mathrm{\Delta }}_x{\mathrm{\Gamma }}_t)\ast f$; the kernel identity ${\mathrm{\partial }}_t\mathrm{\Gamma }={\mathrm{\Delta }}_x\mathrm{\Gamma }$ of [F2] makes the two convolution integrands equal, so ${\mathrm{\partial }}_{t}u={\mathrm{\Delta }}_{x}u$ on ${\mathbb{R}}^n\times (0,\mathrm{\infty })$.

Steps 1.1 and 2.1 give the derivative bounds of the kernel, absolute convergence of the representative and the uniform compact domination; steps 3.1 and 4.1 give the $C^{\mathrm{\infty }}$ property, the derivative identity $D_x^{\alpha }{\mathrm{\partial }}_t^{k}u=(D_x^{\alpha }{\mathrm{\partial }}_t^k{\mathrm{\Gamma }}_t)\ast f$ and the heat equation $u_t=\mathrm{\Delta }u$, which is the whole statement.