Reports of the Academy of Sciences of the USSR
1960. Volume 134, No. 2

MATHEMATICS

V. S. VLADIMIROV

ON THE CONSTRUCTION OF ENVELOPES OF HOLOMORPHY FOR DOMAINS OF A SPECIAL TYPE

(Presented by Academician N. N. Bogolyubov, 3 V 1960)

We shall denote by (R^{n+1}) the real space of dimension (n+1), and by
(x=(x_0,x_1,\ldots,x_n)=(x_0,\mathbf{x})), (p,q,\ldots) its points; by
(C^{n+1}=R^{n+1}+iR^{n+1}) the complex space consisting of points
(\zeta=(\zeta_0,\vec{\zeta})=p+iq); the scalar products
(\zeta x=\zeta_0x_0-\vec{\zeta}\mathbf{x}),
(\vec{\zeta}\mathbf{x}=\sum_{1\le j\le n}\zeta_jx_j),
(x^2=x_0^2-\mathbf{x}^2); the Euclidean distances
(|\mathbf{x}|=\sqrt{\mathbf{x}^2}), (|x|=\sqrt{x_0^2+\mathbf{x}^2}).
The future ((x_0>|\mathbf{x}|)) and past ((x_0<-|\mathbf{x}|)) light cones will be denoted respectively by (\Gamma^+) and (\Gamma^-),
(\Gamma=\Gamma^+\cup\Gamma^-). The tube domains in (C^{n+1}) of the form
(R^{n+1}+i\Gamma^+) and (R^{n+1}+i\Gamma^-) will be denoted respectively by
(T^+) and (T^-), (T=T^+\cup T^-).

A smooth curve will be called timelike if its tangent vector belongs to (\Gamma); a smooth surface will be called spacelike if its normal belongs to (\Gamma).

By a generalized function we mean any linear continuous functional on the Schwartz space (S) ((1), Ch. VII). The space of generalized functions (conjugate to (S)) will be denoted by (S^). Generalized functions (F^+(p)) and (F^-(p)) will be called retarded and advanced*, respectively, if their Fourier transforms

[
\widetilde{F}^{\pm}(x)=\int F^{\pm}(p)e^{-ipx}\,dp,\qquad
px=p_0x_0-\mathbf{p}\mathbf{x}
]

vanish respectively outside (\overline{\Gamma}^{+}) and (\overline{\Gamma}^{-}). A generalized function (f(p)) will be called a commutator if (\widetilde{f}(x)=0) for (x^2<0).

We shall say that a function (F(\zeta)), holomorphic in the domain (T^+) (respectively (T^-)), belongs to the class (N^+) (respectively (N^-)) if:

1) for any (q\in\Gamma^+) (respectively (\Gamma^-)), (\varepsilon>0), and (t\ge t_0>0), the estimate

[
|F(p+itq)|<c(q,\varepsilon,t_0)e^{\varepsilon t}(1+|p|)^l
]

holds for some real (l), depending only on (F);

2) in the space (S^*) there exists a boundary value (F(p_0+i0,\mathbf{p})) (respectively (F(p_0-i0,\mathbf{p}))) of the function (F(p+iq)) as (q\to0), (q\in\Gamma^+) (respectively (\Gamma^-)).

We shall denote by (N) the intersection of the classes (N^+) and (N^-). A function (F(\zeta)) of class (N^+) (respectively (N^-)) is called an analytic continuation of the generalized function
(F(p_0+i0,\mathbf{p})) (respectively (F(p_0-i0,\mathbf{p}))).

Let (G) be an arbitrary open set in (R^{n+1}), and let (\widetilde{G}) be a complex neighborhood of (G) such that, if a ball of radius (\eta) belongs to (G), then the corresponding complex ball of radius (0.1\eta) belongs to (\widetilde{G}).

Theorem 1. In order that there exist a function (F(\zeta)) of class (N), holomorphic in the domain (T\cup\widetilde{G}), it is necessary and sufficient that its boundary values (F(p_0+i0,\mathbf{p})) and (F(p_0-i0,\mathbf{p})): 1) be respectively retarded and advanced functions, and 2) coincide in (G).

The necessity of condition 1) follows from the results of Schwarz (²), Lions (³), and N. N. Bogolyubov and O. S. Parasyuk (⁴) (for (n=0)). The necessity of condition 2) is obvious.

Sufficiency. From condition 1) it follows (²–⁴) that the functions (F(p_0 \pm 0,\mathbf p)) admit analytic continuations (F^{\pm}(\zeta)) of class (N^{\pm}), respectively, in the domains (T^{\pm}). From condition 2), by virtue of the “edge of the wedge” theorem*, it follows that the functions (F^{\pm}(\zeta)) are holomorphic in (\widetilde G) and, consequently, determine a single function (F(\zeta)), holomorphic in the domain (T\cup \widetilde G), of class (N).

The function (F(\zeta)) is the simultaneous analytic continuation of the generalized functions (F(p_0 \pm i0,\mathbf p)).

The domain (T\cup \widetilde G) in Theorem 1 is not a domain of holomorphy. Therefore the function (F) is holomorphic in a wider domain—in the envelope of holomorphy (\mathscr E(T\cup \widetilde G)). Hence, in particular, it follows that the set on which the functions (F(p_0 \pm i0,\mathbf p)) coincide cannot be arbitrary: it must be a real section of a certain domain of holomorphy. Theorems 2 and 3 are closely connected with this fact.

Theorem 2. If the commutator (f(p)) vanishes in the domain (G), then it also vanishes in the least convex envelope (B_0(G)) of the domain (G) with respect to timelike curves.

Proof. Construct the advanced and retarded functions**

[
F^{+}(p)=\int \theta(x_0)\widetilde f(x)e^{ipx}\,dx,\qquad
F^{-}(p)=-\int \theta(-x_0)\widetilde f(x)e^{ipx}\,dx,
]

where (\theta(\xi)=\frac12(1+\operatorname{sign}\xi)). Then

[
F^{+}(p)-F^{-}(p)=(2\pi)^{n+1}f(p).
]

By Theorem 1 there exists a holomorphic function (F(\zeta)) of class (N) in (T\cup \widetilde G) such that, in the space (S^*), the limiting relations

[
F(p_0+i\varepsilon,\mathbf p)\to F^{+}(p),\qquad
F(p_0-i\varepsilon,\mathbf p)\to F^{-}(p)\quad \text{as } \varepsilon\to +0
]

hold.

It is therefore enough to prove that

[
B_0(G)\subset \operatorname{Re}\mathscr E(T\cup \widetilde G).
]

Let (p_1) and (p_2) be arbitrary points of the domain (G) such that there exists a timelike curve connecting them.

First suppose that this curve lies entirely in (G), and let us prove that all timelike curves connecting the points (p_1) and (p_2) lie entirely in (\mathscr E(T\cup \widetilde G)). It may happen that all timelike curves connecting (p_1) and (p_2) lie in (G). Allowing for the other possibility, construct a family of timelike curves (C_\alpha), depending continuously on the parameter (\alpha) ((0\le \alpha\le \alpha_0)) and connecting the points (p_1) and (p_2), with (C_\alpha) ((\alpha>0)) lying entirely inside (G), while (C_0) contains boundary points of the domain (G) forming a compact set (K). The curves (C_\alpha) may be assumed analytic, given locally by the equations

[
p_j=p_{j,\alpha}(\xi),\qquad j=0,1,\ldots,n,\qquad 0\le \alpha\le \alpha_0,\qquad 0\le \xi\le 1.
]

Continue the functions (p_{j,\alpha}(\xi)) analytically to complex values (\lambda=\xi+i\eta). By the Heine–Borel lemma, one may assume that the continued functions (p_{j,\alpha}(\lambda)) are holomorphic in the rectangle

[
\pi_\delta:\quad |\eta|\le \delta,\ 0\le \xi\le 1
]

for sufficiently small (\delta). Thus a family of two-dimensional analytic surfaces (F_\alpha) has been constructed,

[
\zeta_j=p_{j,\alpha}(\lambda),\qquad j=0,1,\ldots,n,\qquad 0\le \alpha\le \alpha_0,\qquad \lambda\in \pi_\delta,
]

* This theorem was first proved by N. N. Bogolyubov (⁵), p. 152. For other proofs see (⁶–⁸).

** The operation of multiplication by (\theta(x_0)) of a function of type (\widetilde f(x)) can be defined up to a finite linear combination of (\delta^{(k)}(x)). Therefore the functions (F^{\pm}(p)) are defined up to a polynomial.

passing through the curves (C_\alpha). For sufficiently small (\delta), all points (\zeta) on (F_\alpha) for which (\lambda\in \pi_\delta) and (\eta\ne 0) belong to (T). Indeed, expanding (p_{j,\alpha}(\lambda)) in a Taylor series in a neighborhood of the point (\lambda=\xi\in C_\alpha), we obtain

[
\zeta_j=p_{j,\alpha}(\xi)+p'_{j,\alpha}(\xi)(\lambda-\xi)+O[(\lambda-\xi)^2],
]

whence (\operatorname{Im}\zeta_j=\eta p'{j,\alpha}(\xi)+O(\eta\delta)). But the curves (C\alpha) are time-like. Therefore (\operatorname{Im}\zeta\in \Gamma) if (\eta\ne 0). If, however, (\eta=0), then, by construction, (C_\alpha\subset G) for (\alpha>0) and (C_0\setminus K\subset G). By the continuity theorem (see ((^9))) (K\subset \mathscr E(T\cup \widetilde G)), and, consequently, the curve (C_0) is entirely contained in (\operatorname{Re}\mathscr E(T\cup \widetilde G)).

Let us consider the other possibility, when there is no time-like curve connecting the points (p_1) and (p_2) and lying entirely in the domain (G). Connect these points by some curve (C) lying entirely in (G), and for all pairs of points on (C) that can be connected by a time-like curve lying entirely in (G), carry out the extension described above. It is not hard to see that, in the end, the original points (p_1) and (p_2) will also be included among these pairs. The theorem is proved.

Thus it has been proved that (B_0(G)\subset \operatorname{Re}\mathscr E(T\cup \widetilde G)).

We give three examples of applications of Theorem 2.

1) If (G) is (p^2<-b^2), then (B_0(G)) is (p^2<0); this result was established by another method in ((^{10})).

2) If (G) is (p^2>-b^2), then (B_0(G)=R^{n+1}) and the function (F(\zeta)) is a polynomial.

3) There are no commutators with compact support (except zero).

Theorem 3. Every function (F(\zeta)) of class (N), holomorphic in the domain (T\cup \widetilde G), where (G) is an arbitrary domain, is holomorphic in the domain (H(T\cup \widetilde G)), consisting of those and only those points (\zeta) for which every complex hyperboloid ((\zeta'-u)^2=s) ((u,s\geq 0) are real parameters), passing through (\zeta), has with (G) at least one common interior point.

For the proof we introduce the function (cf. ((^{11},{}^{12})))

[
u(p,k)=\frac{1}{(2\pi)^{n+1}}\int \widetilde f(x)\cos(k\sqrt{x^2})\,e^{ipx}\,dx.
]

Then (u\in S^*),

[
\left(\frac{\partial^2}{\partial p_0^2}
-\frac{\partial^2}{\partial p_1^2}
-\cdots
-\frac{\partial^2}{\partial p_n^2}
-\frac{\partial^2}{\partial k^2}\right)u=0,\qquad
u\bigg|{k=0}=f(p),\quad
\frac{\partial u}{\partial k}\bigg|=0.
\tag{1}
]

“Smoothing” (u(p,k)) with respect to (p), we obtain an infinitely differentiable function satisfying the relations (1) with the “smoothed” function (f(p)). Using Holmgren’s theorem and applying Theorem 2, we conclude that

[
u(p,k)=0\quad \text{for } \quad (p,k)\in B_0(G'),
]

where (G') is some ((n+2))-dimensional neighborhood of the domain (G) lying in the plane (k=0). With the aid of (2) we determine the position of the support of the weight function in the integral representation of Jost—Lehmann—Dyson (see ((^8,{}^{11},{}^{12}))) for the commutator (F(p_0+i0,p)-F(p_0-i0,p)), and then use known techniques ((^8,{}^{13})).

Corollary. If the commutator (f(p)) vanishes in the domain (G), then it also vanishes in the domain (B(G)) obtained by bending (G) by the hyperboloids ((p-u)^2=s,\ s\geq 0), i.e.

[
B(G)=\operatorname{Re}H(T\cup \widetilde G).
]

The domain (H(T\cup \widetilde G)) is pseudoconvex.

Remark. Theorem 3 is also valid for such open sets (G) in which all points of distinct connected components (G_j) are dis-

are separated by space-like intervals, i.e., ((x_i-x_j)^2<0), if (x_i\in G_i,\ x_j\in G_j,\ j\ne i).

Theorem 3 generalizes previously known results ((^8,\ ^{11-13})) for domains (G) bounded by two space-like surfaces to the case of arbitrary domains. Dyson ((^{11})) stated a conjecture equivalent to the assertion that Theorem 3 is valid for any open set (G).

The inclusion (\mathscr E(T\cup \widetilde G)\subset H(T\cup \widetilde G)) is valid, since (H(T\cup \widetilde G)) is the envelope of holomorphy of (T\cup \widetilde G) with respect to holomorphic functions of class (N), whereas (\mathscr E(T\cup \widetilde G)) is the envelope of holomorphy of (T\cup \widetilde G) with respect to all holomorphic functions. It is not clear whether (\mathscr E(T\cup \widetilde G)=H(T\cup \widetilde G)). If, for example, (G) is the strip (|p_0|