Lemma: Boundary-value calculus on Ω

This lemma concerns the model family $T_m$ itself; it does not yet identify $T_m$ with derivatives of the predecessor kernel $S$. For $m\ge 1$ let $T_m:=(Q-i0{)}^{-m}:={\lim }_{\tau \downarrow 0}(Q-i\tau {)}^{-m}$ in ${\mathcal{D}}^{\prime }(\Omega )$. The limit exists, and:

1. (a)

   $Q{T}_{m+1}=T_m$;

2. (b)

   ${\partial }_{\rho }{T}_m=-m({\partial }_{\rho }Q)T_{m+1}$;

3. (c)

   $T_m{T}_n=T_{m+n}$, where the product is the existing Hörmander product on $\Omega$ (wavefront criterion).

Proof. {Points with $Q(z_0)\ne 0$.} Near such a point $Q\ne 0$ and $(Q-i\tau {)}^{-m}\to Q^{-m}$ in $C^{\infty }$, so $T_m=Q^{-m}$ there and (a)–(c) are identities of smooth functions.

Nonzero null points. Let $Q(z_0)=0$ with $z_0\ne 0$. Then $dQ(z_0)=2\eta (z_0,\cdot )\ne 0$, because $\eta$ is nondegenerate and $z_0\ne 0$. Hence $Q$ is a local coordinate: there is a neighbourhood $U$ and a diffeomorphism $\Phi :U\to (-\delta ,\delta )\times V^{\prime }$ with ${\Phi }_1=Q$. On $(-\delta ,\delta )$, $(s-i\tau {)}^{-m}\to (s-i0{)}^{-m}$ in ${\mathcal{D}}^{\prime }$. This is the boundary value from $\{\mathrm{Im}\zeta <0\}$ of the holomorphic function ${\zeta }^{-m}$, and $(s-i0{)}^{-m}=\frac{(-1{)}^{m-1}}{(m-1)!}{{\partial }_s}^{m-1}(s-i0{)}^{-1}$. Pullback by the submersion $\Phi$ (Hörmander, Thm. 6.1.2) is continuous, so $T_m{|}_U={\Phi }^{\ast }((s-i0{)}^{-m}\otimes 1)$ and the limit exists on $U$.

Its Fourier transform satisfies $\widehat{(s-i0{)}^{-m}}(\sigma )=\frac{2\pi i(-i\sigma {)}^{m-1}}{(m-1)!}\theta (-\sigma )$ for $\hat{u}(\sigma )=\int e^{-is\sigma }u(s)ds$. This transform is supported in the single half-line $\sigma \le 0$, the same for every $m$. Hence $\mathrm{WF}((s-i0{)}^{-m})\subset \{(0,\sigma ):\sigma <0\}$, and by the pullback theorem (Hörmander, Thm. 8.2.4)

with one sign for all $m$.

Compatibility. All $T_m$ are defined by the same limit $\tau \downarrow 0$ from the same side, globally on $\Omega$. The local descriptions therefore agree on overlaps, and with $Q^{-m}$ off the cone.

(a) $Q(Q-i\tau {)}^{-m-1}=(Q-i\tau {)}^{-m}+i\tau (Q-i\tau {)}^{-m-1}$. In the local coordinate, $\tau (s-i\tau {)}^{-m-1}$ is a constant times ${{\partial }_s}^m(\tau (s-i\tau {)}^{-1})$, and $\tau (s-i\tau {)}^{-1}={\partial }_s(\tau \log (s-i\tau ))$ with $\tau \log (s-i\tau )\to 0$ in ${L^1}_{\mathrm{loc}}$; so the remainder tends to $0$ in ${\mathcal{D}}^{\prime }$.

(b) Derivatives commute with limits in ${\mathcal{D}}^{\prime }$, and ${\partial }_{\rho }(Q-i\tau {)}^{-m}=-m({\partial }_{\rho }Q)(Q-i\tau {)}^{-m-1}$.

(c) Same-sign boundary-value product. Since $\mathrm{WF}(T_m),\mathrm{WF}(T_n)\subset {\Gamma }_{-}$ and ${\Gamma }_{-}$ contains no opposite covectors $(z,\xi ),(z,-\xi )$, Hörmander's criterion (Thm. 8.2.10) defines the product $T_m{T}_n$ in ${\mathcal{D}}^{\prime }(\Omega )$. It remains to identify that product with $T_{m+n}$.

All $T_m$ are boundary values from the same lower half-plane. Locally near a nonzero null point, in the coordinate $s=Q$ of the first paragraph,

The one-variable distributions $(s-i0{)}^{-m}$ and $(s-i0{)}^{-n}$ therefore arise as boundary values of the holomorphic functions $\zeta \mapsto {\zeta }^{-m}$ and $\zeta \mapsto {\zeta }^{-n}$ from $\{\mathrm{Im}\zeta <0\}$. We identify their Hörmander product by an explicit Fourier computation. Write $u_m:=(s-i0{)}^{-m}$ and use the one-variable convention $\hat{u}(\sigma )=\int e^{-is\sigma }u(s)\mathrm{d}s$, with inverse $u(s)=\frac{1}{2\pi }\int e^{is\sigma }\hat{u}(\sigma )\mathrm{d}\sigma$; then $\widehat{uv}=\frac{1}{2\pi }\hat{u}\ast \hat{v}$ (Hörmander's $(2\pi {)}^{-n}$ with $n=1$). As above, ${\hat{u}}_m(\sigma )=\frac{2\pi i(-i\sigma {)}^{m-1}}{(m-1)!}\theta (-\sigma )$ (for $m=1$: $\widehat{\mathrm{PV}\frac{1}{s}}=-i\pi \mathrm{sgn}\sigma$ and $\widehat{i\pi \delta }=i\pi$; then $\widehat{{\partial }_{s}u}=i\sigma \hat{u}$).

Beta integral. For $\sigma <0$ the convolution runs over the finite interval $[\sigma ,0]$, where both factors are supported; with ${\sigma }^{\prime }=\sigma x$, $x\in [0,1]$, $\mathrm{d}{\sigma }^{\prime }$ over $[\sigma ,0]$ becomes $|\sigma |\mathrm{d}x$, and

Since $B(m,n)=\frac{(m-1)!(n-1)!}{(m+n-1)!}$ and $i|\sigma |=-i\sigma$ for $\sigma <0$, this equals $\frac{2\pi i(-i\sigma {)}^{m+n-1}}{(m+n-1)!}={\hat{u}}_{m+n}(\sigma )$; for $\sigma >0$ the integrand vanishes. So $\frac{1}{2\pi }{\hat{u}}_m\ast {\hat{u}}_n={\hat{u}}_{m+n}$, the integral being absolutely convergent.

Identification with the Hörmander product. For $\phi \in {C^{\infty }}_c(\mathbb{R})$ the Hörmander product is characterized locally by $\widehat{{\phi }^{2}uv}=\frac{1}{2\pi }\widehat{\phi u}\ast \widehat{\phi v}$, the integral converging absolutely under the wavefront condition (Hörmander, proof of Thm. 8.2.10). Here $\widehat{\phi u_m}=\frac{1}{2\pi }\hat{\phi }\ast {\hat{u}}_m$. Because $\hat{\phi }$ is Schwartz while ${\hat{u}}_m,{\hat{u}}_n$ are polynomially bounded and supported in $\sigma \le 0$, the iterated integral $\int \int \int |\hat{\phi }(a)\hat{\phi }(b){\hat{u}}_m(\sigma -a-b-c){\hat{u}}_n(c)|$ runs in $c$ over an interval of length at most $|\sigma -a-b|$ and converges absolutely; Fubini allows regrouping:

Hence ${\phi }^2(u_m{u}_n)={\phi }^2{u}_{m+n}$ for every $\phi$, i.e.

Transfer to the chart. With $p(s,y^{\prime })=s$, $u_m\otimes 1=p^{\ast }{u}_m$, and pullback is multiplicative on distributions satisfying the product condition (Hörmander, Thms. 8.2.4 and 8.2.10: the product is the pullback of the tensor product by the diagonal map, and pullbacks compose); thus $p^{\ast }{u}_m\cdot p^{\ast }{u}_n=p^{\ast }{u}_{m+n}$, and, pulling back further by the diffeomorphism $\Phi$, $T_m{T}_n=T_{m+n}$ on $U$. Off the cone the identity is the smooth identity $Q^{-m}{Q}^{-n}=Q^{-(m+n)}$. On overlaps the two local descriptions agree by the global same-side limit $\tau \downarrow 0$. Thus $T_m{T}_n=T_{m+n}$ on $\Omega$.