The complex-time heat kernel on a proper sector

Assume Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Fix $n\ge 1$ and $\theta \in (0,\pi \mathrm{/}2)$ and put $S_{\theta }:=\{z\in \mathbb{C}\setminus \{0\}:\mathrm{\mid }\arg z\mathrm{\mid }<\theta \}$. For $z\in S_{\theta }$ and $x\in {\mathbb{R}}^n$ define the complex-time heat kernel ${\mathrm{\Gamma }}_z(x):=(4\pi z{)}^{-n\mathrm{/}2}\exp (-\frac{\mathrm{\mid }x{\mathrm{\mid }}^2}{4z}),$ where $(4\pi z{)}^{-n\mathrm{/}2}:=\exp (-\frac{n}{2}\mathrm{Log}(4\pi z))$ with the principal logarithm. This is legitimate: $z\mathrm{\ne }0$ and $\mathrm{Re}(4\pi z)=4\pi \mathrm{\mid }z\mathrm{\mid }\cos (\arg z)>0$ because $\mathrm{\mid }\arg z\mathrm{\mid }<\theta <\pi \mathrm{/}2$, so $4\pi z$ lies in the slit plane on which the principal logarithm is holomorphic (The principal logarithm is the normalised holomorphic branch on the slit plane, Complex powers defined from a holomorphic logarithm branch) and the power is the complex exponential of The complex exponential by its power series. Then:

(i) ${\mathrm{\Gamma }}_z\in C^{\mathrm{\infty }}({\mathbb{R}}^n)\cap L^1({\mathbb{R}}^n)$ and ${\int }_{{\mathbb{R}}^n}{\mathrm{\Gamma }}_z(x)dx=1$;

(ii) for every $\sigma \in (0,\theta )$ and every $z$ with $\mathrm{\mid }\arg z\mathrm{\mid }\le \sigma$, $\mathrm{\parallel }{\mathrm{\Gamma }}_z{\mathrm{\parallel }}_1\le (\cos \sigma {)}^{-n\mathrm{/}2}$;

(iii) for real $z=t>0$, ${\mathrm{\Gamma }}_t$ is the heat kernel of The heat kernel on ${\mathbb{R}}^n$ and its causal extension;

(iv) for every fixed $x$ the map $z\mapsto {\mathrm{\Gamma }}_z(x)$ is holomorphic on $S_{\theta }$ with ${\mathrm{\partial }}_z{\mathrm{\Gamma }}_z(x)=(-\frac{n}{2z}+\frac{\mathrm{\mid }x{\mathrm{\mid }}^2}{4z^2}){\mathrm{\Gamma }}_z(x)$;

(v) for every compact $K⋐S_{\theta }$ there are constants $c_K,C_K>0$ with $\mathrm{\mid }{\mathrm{\Gamma }}_z(x)\mathrm{\mid }\le C_K{e}^{-c_K\mathrm{\mid }x{\mathrm{\mid }}^2}$ and $\mathrm{\mid }{\mathrm{\partial }}_z{\mathrm{\Gamma }}_z(x)\mathrm{\mid }\le C_K(1+\mathrm{\mid }x{\mathrm{\mid }}^2)e^{-c_K\mathrm{\mid }x{\mathrm{\mid }}^2}$ for all $z\in K$, $x\in {\mathbb{R}}^n$.

The complex exponential is entire with derivative itself (The complex exponential is entire and its complex derivative is itself), and the complex chain rule is The chain rule for complex derivatives. Its modulus is $\mathrm{\mid }{e}^w\mathrm{\mid }=e^{\mathrm{Re}w}$ ($\exp (x+iy)=e^x(\cos y+i\sin y)$, $\mathrm{\mid }\exp (x+iy)\mathrm{\mid }=e^x$, and $e^{i\pi }+1=0$). Smoothness in (i) follows by repeated coordinate differentiation of the exponential and power on ${\mathbb{R}}^n$ (Linearity, product, reciprocal, and quotient rules for complex derivatives, Sums, scalar multiples, products and quotients: $(f+g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)+g^{\mathrm{\prime }}(c)$, $(\alpha f{)}^{\mathrm{\prime }}(c)=\alpha f^{\mathrm{\prime }}(c)$, $(fg{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)g(c)+f(c)g^{\mathrm{\prime }}(c)$, and $(f\mathrm{/}g{)}^{\mathrm{\prime }}(c)=(f^{\mathrm{\prime }}(c)g(c)-f(c)g^{\mathrm{\prime }}(c))\mathrm{/}g(c{)}^2$ when $g(c)\mathrm{\ne }0$, The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g(c)$, then $f\circ g$ is differentiable at $c$ with $(f\circ g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(g(c))g^{\mathrm{\prime }}(c)$, Complex differentiability at a point, the complex derivative, holomorphic functions, and entire functions); square-integrability and the exponential bound in (v) follow from $\mathrm{Re}(1\mathrm{/}z)=\cos (\arg z)\mathrm{/}\mathrm{\mid }z\mathrm{\mid }>0$ together with the compactness of $K$ and the growth of the exponential against polynomials (The exponential dominates every fixed nonnegative integer power at $+\mathrm{\infty }$, Heine-Borel in ${\mathbb{R}}^n$: with the Euclidean metric a subset of ${\mathbb{R}}^n$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line, Open cover, subcover, compact metric space, and compact subset of a metric space). The remaining assertions are justified in the reminders below.

• The complex Gaussian and the total mass (i). The complex Gaussian identity ${\int }_{{\mathbb{R}}^n}{e}^{-a\mathrm{\mid }x{\mathrm{\mid }}^2}dx=(\pi \mathrm{/}a{)}^{n\mathrm{/}2}$ for $\mathrm{Re}a>0$ follows from the real Gaussian integral The Gaussian integral ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-x^2}dx=\sqrt{\pi }$ by the scalar identity theorem in $a$: truncating to $[-R,R]$, the finite-interval holomorphic parameter-integral theorem A jointly continuous finite-interval parameter integral of holomorphic functions is holomorphic makes $F_R(a)={\int }_{-R}^R{e}^{-a{x}^2}dx$ holomorphic on $\mathrm{Re}a>0$; on a compact parameter set $\mathrm{Re}a\ge c>0$ the tails ${\int }_{\mathrm{\mid }x\mathrm{\mid }>R}{e}^{-c\mathrm{\mid }x{\mathrm{\mid }}^2}dx$ tend to $0$ uniformly, so $F_R\to F$ locally uniformly and Holomorphic functions form a closed subspace for locally uniform convergence makes $F$ holomorphic; on $(0,\mathrm{\infty })$ the real Gaussian identity and the substitution $x\mapsto x\mathrm{/}\sqrt{a}$ give $F(a)=\sqrt{\pi }\exp (-\frac{1}{2}\mathrm{Log}a)$, so Identity theorem for holomorphic functions extends this formula to all of $\mathrm{Re}a>0$. Fubini for the absolutely convergent $n$-dimensional product integral Tonelli and Fubini for the completed product, with only almost-everywhere section measurability gives ${\int }_{{\mathbb{R}}^n}{e}^{-a\mathrm{\mid }x{\mathrm{\mid }}^2}dx=F(a{)}^n=(\pi \mathrm{/}a{)}^{n\mathrm{/}2}$; substituting $a=1\mathrm{/}(4z)$, and comparing principal branches on the right half-plane, yields $\int {\mathrm{\Gamma }}_z=1$. This route uses the published holomorphy inputs listed in the dependencies and not the later semigroup law.

• The $L^1$ bound (ii) and the derivative formula (iv). Writing $\mathrm{Re}(1\mathrm{/}z)=\cos (\arg z)\mathrm{/}\mathrm{\mid }z\mathrm{\mid }$ gives $\mathrm{\mid }{\mathrm{\Gamma }}_z(x)\mathrm{\mid }=(4\pi \mathrm{\mid }z\mathrm{\mid }{)}^{-n\mathrm{/}2}{e}^{-\cos (\arg z)\mathrm{\mid }x{\mathrm{\mid }}^2\mathrm{/}(4\mathrm{\mid }z\mathrm{\mid })}$, and integrating the Gaussian yields exactly $(\cos \arg z{)}^{-n\mathrm{/}2}$, which is at most $(\cos \sigma {)}^{-n\mathrm{/}2}$ when $\mathrm{\mid }\arg z\mathrm{\mid }\le \sigma$. The formula in (iv) is the product, chain and quotient rule for the holomorphic factors $z\mapsto \exp (-\frac{n}{2}\mathrm{Log}(4\pi z))$ and $z\mapsto e^{-\mathrm{\mid }x{\mathrm{\mid }}^2\mathrm{/}(4z)}$ on the slit plane, where $d\log (4\pi z)\mathrm{/}dz=1\mathrm{/}z$.

• Relation to the real kernel (iii). For real $z=t>0$ the principal logarithm is the real logarithm, $(4\pi t{)}^{-n\mathrm{/}2}$ is the usual positive power and ${\mathrm{\Gamma }}_t(x)=(4\pi t{)}^{-n\mathrm{/}2}{e}^{-\mathrm{\mid }x{\mathrm{\mid }}^2\mathrm{/}(4t)}$ is exactly the heat kernel of The heat kernel on ${\mathbb{R}}^n$ and its causal extension; the compatibility of the real normalisation with Normalisation, parabolic scaling, heat equation and derivative bounds for the heat kernel is what makes (iii) a consistency statement rather than a new definition.