The complex-time heat kernel is L1-differentiable in its parameter

For a compact $K⋐S_{\theta }$, choose $d>0$ such that its closed $d$-neighbourhood $K^{+}$ is compact and contained in $S_{\theta }$. For $z\in K$ and $0<\mathrm{\mid }h\mathrm{\mid }<d$, the segment $z+sh$ lies in $K^{+}$. By [F2] and the segment derivative in [F4], [F3] gives $({\mathrm{\Gamma }}_{z+h}(x)-{\mathrm{\Gamma }}_z(x))\mathrm{/}h={\int }_0^1{\mathrm{\partial }}_z{\mathrm{\Gamma }}_{z+sh}(x)ds$. Hence [F1] on $K^{+}$ bounds the quotient by $C(1+\mathrm{\mid }x{\mathrm{\mid }}^2)e^{-c\mathrm{\mid }x{\mathrm{\mid }}^2}$, an integrable function independent of $z$ and $h$.

For every fixed $x$, the definition's derivative formula of [F2] shows that the difference quotients converge to ${\mathrm{\partial }}_z{\mathrm{\Gamma }}_z(x)$ as $h\to 0$; for $z\in K$ and $0<\mathrm{\mid }h\mathrm{\mid }<d$ as in step 1.1 both the difference quotient and ${\mathrm{\partial }}_z{\mathrm{\Gamma }}_z(x)$ are bounded by the $L^1$ majorant of step 1.1, so the difference is bounded by $2C(1+\mathrm{\mid }x{\mathrm{\mid }}^2)e^{-c\mathrm{\mid }x{\mathrm{\mid }}^2}$ and converges pointwise to $0$; [F5] therefore gives $\parallel ({\mathrm{\Gamma }}_{z+h}-{\mathrm{\Gamma }}_z)\mathrm{/}h-{\mathrm{\partial }}_z{\mathrm{\Gamma }}_z{\parallel }_1\to 0$. Since $z$ was arbitrary, $z\mapsto {\mathrm{\Gamma }}_z$ is complex differentiable on $S_{\theta }$ with derivative ${\mathrm{\partial }}_z{\mathrm{\Gamma }}_z$, first as a limit in $L^1$.

For each fixed $x$, the explicit derivative ${\mathrm{\partial }}_z{\mathrm{\Gamma }}_z(x)$ is continuous and therefore uniformly continuous on $K^{+}$. The segment identity of step 1.1 consequently implies ${\sup }_{z\in K}\mathrm{\mid }({\mathrm{\Gamma }}_{z+h}(x)-{\mathrm{\Gamma }}_z(x))\mathrm{/}h-{\mathrm{\partial }}_z{\mathrm{\Gamma }}_z(x)\mathrm{\mid }\to 0$. This supremum is measurable in $x$: the integrand is jointly continuous in $(z,x)$ and a maximum over compact $K$ is continuous in $x$, as follows from uniform continuity on $K$ times a compact spatial neighbourhood. It is bounded by twice the integrable majorant of step 1.1. Dominated convergence [F5] gives convergence of its integral to zero, which bounds the supremum over $z\in K$ of the $L^1$ error. This proves uniform convergence on every compact $K$, as well as the asserted $L^1$ differentiability.