PSEUDOCONVEXITY AND THE ENVELOPE OF HOLOMORPHY FOR FUNCTIONS OF SEVERAL COMPLEX VARIABLES
by
LORNE BARRY MULLETT B.Sc, University of British Columbia, 1964
A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS
in the Department of MATHEMATICS
^ We accept this thesis as conforming to the required standard
THE UNIVERSITY OF BRITISH COLUMBIA August, 1966 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. 1 further agree that permission, for extensive copying of this thesis for scholarly purposes may be granted by the Head of my
Depa-rtment or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed
•without my written permission.
Department of MATHEMATICS The University of British Columbia Vancouver 8, Canada
Date October 17th, 1966 ii
ABSTRACT
We first handle some generalizations from the theory of functions of a single complex variable, including results regarding analytic continuation.
Several "theorems of continuity" are considered, along with the associated definitions of pseudoconvexity, and these are shown to be equivalent up to a special kind of trans• formation. By successively applying a form of analytic contin• uation to a function f , a set of pseudoconvex domains is constructed, and the union of these domains is shown to be the envelope of holomorphy of f . iii
TABLE OF CONTENTS
Page
I. Introduction 1
II. Pseudoconvexity and the Envelope of Holomorphy 7
IIT. Bibliography 27 iv
ACKNOWLEDGEMENT
I wish to thank Dr. A.H. Cayford for his aid both in the preparation of this thesis and in research in the field of complex analysis. I also with to thank Dr. CA. Swanson for hi s encouragement.
The financial support of the University of British
Columbia and the National Research Council is gratefully acknowledged. 1.
INTRODUCTION
The theory of several complex variables is about
seventy years old. A relatively dormant period was reached when scholars found that their generalizations from the theory of a single complex variable began to fail, notable exceptions being discovered principally by P. Hartogs about 1910. Some
striking differences in the theories are listed. Except for the case n=l , for f(Bva0,...^ ) in Dc$^ , we 'have in general
(i) holomorphic functions do not produce conformal maps (rather the result is a "pseudoconformal" map, in which only the projection onto $~ is necessarily conformal) .
(ii) holomorphic functions do not have isolated zeros (or singularities).
(iii) not every domain Bc(|Jn is a domain of holomorphy.
In particular, we also have except for n=l ,
(iv) the boundary of a domain of holomorphy is always connected. i (v) the exterior of a bounded set cannot be a domain of holomorphy.
About twenty years later, due to the work of H. Behnke,
H. Cartan, and P. Thullen on domains, and the contribution by S. Bergman of the invariant metric and the "kernel function", the theory "branched off in several new directions and became recognized as being independent. However, another period of dormancy was reached] this time the restriction was the old- fashioned machinery mathematicians were using in an attempt to
solve contemporary problems. The most recent breakthrough is the language of "fibre-bundles'1, which is now being applied successfully to the study of "complex spaces", which are the extensions to n complex dimensions of the familiar Riemann
surfaces.
z z We define f(a) =f( i> 2> ••• n) to be "holomorphic"
n in Dc$Y iff at all H^eD , exists, and f satisfies a — aak the Cauchy-Riemann equations
= 0 k = 1,2,...,n .
Observe that = + i|= ) . Let d 3X 3y 9ik k k
0(D) ={f : f(ja) is holomorphic in D}
Equivalently, fe<0>(D) <=> f can oe developed into a multiple power series
a (o) kl a (o) k2 a f a ^K,,k0,?..,k =o k,v..k (v i ) (v 2 ) ---(v 12 ' n 12 n (o) that converges uniformly in a neighbourhood of _av 1 for any choice of jJ^eD .
One of the early generalizations was Cauchy's integral formula, namely that if f(a) satisfies the Cauchy-Riemann equations for ae(jj , then f(a) = 75^- J' dt, where C is a 1 C ' 'contour" (simple, closed, rectifiable, oriented). The generalization yields the following:'
If f(ia) is holomorphic in a neighbourhood of the product domain D-^xDgX. . .xDn , then
aD^^DgX. . .xaDn (t1-a1)(t2-a2) .. -(tn-an)
We state without proof the following fundamental theorem, the proof of wnich depends basically on the above formula.
Theorem A: If f(z)=f(z^yz2,...,an) e o(P) , where
0 0 P- {|a1-a1( )| I3 23 3 n 12 n uniquely, for all aeP , where 4. Moreover, f(js) converges absolutely and uniformly in P . An important and very useful tool is the maximum principle. Theorem B: Let f(j3)'€<>(D) > not constant. Then |f(is)| does not assume its maximum in D . Proof: Assume the theorem is false. Let eS°}<= intD be a maximal point of f(js) , and let N(^°^;r_) be a (polycylindrical) neighbourhood of such that NeD . Wow the holomorphic functions ffA Ba a a (°h fCa (°) a a (°' a (°h £(*^9 a^a ) \ p°2 n 1 23 »•••» °n /*»*«y-V'2_ **"n-l,n of a single complex variable must then be constant, for they assume their maxima at the centres of the composite disks of N . But this implies that f(js) is constant in N with respect B to all of a^Bg, •' • > n > ^ Theorem A . We assert that f(_a) must have been constant, contrary to our assumption. Por if there were a point ^^^B^0'' in D , by the connectivity of D we can construct an arc from a^0^to a/1/1 and cover it with polycylindrical neighbourhoods {IL } . By the Heine-Borel A. theorem there exists a finite subcovering {.Nj} which we may choose to overlap in such a way that each neighbourhood contains the centre of the previous. Prom our earlier result, f(js) is constant in all these neighbourhoods by construction, and 5. thus f(a/°)) = f(B^) . Since our choice of a/1) was arbi• trary, the contradiction follows. Corollary: |f(a)| assumes its maximum on dD , the boundary of D , provided the function is continuous in D , the closure of D . Proof: Weierstrass' theorem on the maximum of a continuous function. This thesis is concerned with the "envelope of holomorphy" of a given (univalent) domain D , which is the smallest domain of holomorphy containing D . The generalization of the convex envelope from n real dimensions leads us to con• sider non-univalent regions. This is analogous to continuing a holomorphic function of one complex variable and getting a multiple-valued function as the result. We can make the contin• uation single-valued by considering the associated Riemann surface. We may allow complex manifolds to be envelopes, but examples show they will often be non-univalent. Hence there are no norms, and we must find a more general definition for a pseudoconvex region (equivalent to a region of holomorphy), preferably not de• pendent on the concept of "plurisubharmonic function", to be de• fined later. From the familiar unit disk of complex analysis, we can branch off in our generalization in one of two ways. We could take as our basic domain the polycylinder, which in fact b. we do in this paper for some of the results, or we could choose our domain to he the n - hall. Modern theories have extended the polycylinder to the analytic polyhedron, and the n - hall to the pseudoconvex domain. The latter parts of this thesis are concerned with the second extension. 7. PSEUDOCONVEXITY AND THE ENVELOPE OP HOLOMORPHY i We begin by recalling some definitions from complex analysis, and by stating some results and definitions which are directly relevant to the topic. The space under consideration is ^n=^ , consisting of n-tuples of complex variables a=:(Bj,a2,...,an) . The topology will be that generated by the euclidean norm; all topologies generated by arbitrary norms are equivalent. A "region" is an open set, and a "domain" is a connected region. We let Q(D) denote the convex hull, and E(D) the "envelope of holomorphy" i of the domain D . E(D) • is the largest domain to which every function'holomorphic in D may be continued. The envelope of holomorphy has become useful to theoretical physicists in the derivation of dispersion relations of quantum field theory. We will call D a "domain of holomorphy" iff D=E(D), and some equivalent definitions are clear. H is a "region of , holomorphy" iff there exists a function holomorphic in H but not in any larger region. A "Riemann domain" is a pair E(D) has been determined only for certain special domains, the definitions of which follow. I. The "polycylinder" is probably the most commonly en• countered domain in dealing with functions of several complex variables. It is represented by {a_: | a1| product of n disks {a1: |a1| II. The "hypersphere" is well known. It can be represented as {a^a-J +|a2l + ...+|an| III. A "circular" domain Dd^ with centre (a^a^...,an) contains with each a^^e D the points 19 i9 (o) i9 (a^a/^-a^e ! , a2+( a^ °) -a2) e 2 , ... *n -aj e n) e D, where Q\>Q2''' ',Q n are real and °£ek£.2'ir ' k=l,2,...,n. Prom the definition we observe that this domain admits the automorphisms i€ ak=(ak^°^-ak)e k +ak 0<9k<2Tr > l IV. The "Hartogs" domain is given by {(ja,w) : aeD , r(a_) < |w-w | < R(aJ } where De$*} , we({! , r(a), R(B) > 0 . In its most general' O """" —• form this domain admits the automorphisms 9. .a = _a * / iQ w = (w-w )e +w 9 real. V. An open set Md$n is called a "tube" if there is an open set «,cRn , called the"base"of K , such that K= {ja: Re _a e K } . VI. "Product" domains are well known also, and included here for completeness. They are simply a generaliz• ation of tne polycylinder and are given by {a : a1eD1,a2eD2, . .. ,aneDn} where D1,D2,...,Dn are plane domsins. The plurisubharmonic functionals are valuable tools in E on a tne theory. A real-valued function V( a^, .. >n) -V( a.) region Tc<)J^ will be called plurisubharmonic iff (i) - a <_ V(B) < » in D (ii) V(a_) is upper semi continuous in D (iii) V(a_) is subharmonic in the intersection of every analytic plane [z_ : z = + Xa} with D, where ae(j;n , \e$ or (ill)' V(a_^°^+ \a) is subharmonic in \ on the open set of \e(jj such that a^°) + \a e D . There is a close relation between the familiar convex I 10. functions and domains and the above concepts. If we replace "straight line" by"analytic plane", "linear majorant" by "harmonic majorant", "Rn" by "(|Jn" , then the plurisubharmonic functions have the same definition as convex functions. Many theorems hold for both types. We define k1n k,2 n €N 6 the origin. It is usual also to define a distance functional dj^(j3) , which measures the distance of a_ from the boundary dD of D with respect to the norm N . A more rigorous definition is d^^(a)ssup r , where the supremum is taken over all r such that {js1 : iia.'-ajj^ < r} e D . Where the norm is understood, we drop the superscript (N) . Thus a necessary and sufficient condition for a Riemann domain D to be a domain of holomorphy is the existence of fe<>0) such that &(f;a_) = dD(a_) for all aeD . A domain D will be called "pseudoconvex" iff -log d^a) is plurisubharmonic. Proposition 1; d^N/1(a_) is continuous with respect to the topology generated by a norm N provided D^n . Proof: Since the topology generated by the euclidean norm is equi ralent to the topology generated by N , we can employ the usual continuity argument. By definition, if \ 11. 1 (l) 1 Ila-JB^ ^! < dD(a ) , then aeD . If lla^ )- a^U < € , then by the triangle inequality we have ||ja-a^1 ^ || < ||a-a_(2/,|| + e . Tnu s for all ja sucn that !|a_-a_^2^|| < ^(a^1/1)-s > we have 2/1 1/1 aeD , and then by definition dD(a/ ) >_ d (a^ )-e . Also , (3 I / by symmetry, D(a^" " ') 2. ^(B^^ )-€ . Therefore (l) (2) (l) (2) |dD(a ) - dn(B )l < e ^ !!l -i l! then dD(_a) =oo . Another important functional is that representing the distance d^Nl (a.) of the point a on IT : {a':a'=a+\a} from the boundary of irHD . That is, d^^ssup r where the supremum is taken over {a». a/sa+xa > llalljj=1 » I x| < r}cD . It is clear that d^ (a) = inf (js) } wnere ||a!lN=l . ae([!n Tnis brings us to a fundamental theorem for contin• uation of functionals . Proposition 2: Let fe<>(D) , f a functional. Then f can be continued from D into D*5{z:a=i^+aa + pb , aeS , |p| < p|h(a)|_1} , where Serr is simply connected, ScDftfr , and h is a nonzero functional defined and continuous on the projection of S and holomorphic in the projection of S in the a-plane and satisfying the 12. boundary condition |h(a)! da D (a) > p > 0 a£0S . ' IT Proof: The continuation of f as a function of S is straight• forward; the difficulty is the extension to f as a function of a_ . Por we can expand f ( a_^ ° ^ + a_a+ 3b) in powers of S if we fix a . Now from our corollary to the maximum principle we see that f converges in D and the expansion represents our continuation. Choose c 'so that ||c||=l , let y be a complex parameter, and let E= {^:^==^0^4aa+pb+yc} . Consider _1 the domain D1=[z:a_eB , |B| <_ ( p-€) |h(a)! , ae^S , |Y|<.6] • Then given e>o there is a 6>o such that D^dD. Moreover, D-, is relatively compact in B (we will write D^cdB) , and fl is holomorphic with respect to a,B,v, and by Hartogs' theorem (Caratneodory [1]) , it is bounded by some number M(c) Expanding f| in a power series in B and y , we have ,:pl »1 J,k=o I 33JoY ^=o subjecc to the restrictions on D, . Applying Cauchy's in- equality, a straightforward generalization of the formula for functions of a single complex variable, we deduce k -I a Y h(a) -1| J+K'^f 'p, f(&\) pO M( c) ae0' j»M o3°'oYk The left-hand side assumes its maximum with respect to S on SS and thus the inequality is valid for aeS . Hence we have uniform convergence in every compact subset of Dg , wnich is with the condition ae^S replaced by aeS , and we have continued fl^ (a) into Dg . Since our choice of c was arbitrary up to its norm, we let c range through all values of $n , and e 0 . Thus we have imposed holomorphy on f in any neighbourhood of D* . (A routine calculation shows our new definition of f(_z) is consistent) . f is single-valued in D since D is simply-connected, and the theorem is proved, f must be holo• morphic at z_^°^eD* since it is holomorphic in a neighbourhood of 2^°^ xwien restricted to an arbitrary analytic plane through 0/" . Corollary: f a functional, fc<>(D) => f€o(D») > where D' = {a_: ||a.^°^+aa-_a||< p' i(cx) | ~1 , aeS] and the conditions of the theorem hold with replacement of d^ ^'a_) by o^/jl) • We say D is "holomorph-convex" if DqX as a com• pact subset, where < X , h > is a Riemann domain, and there exists fe.o(X) such that | f(a) j >|jf 1^ =^P| f (z) | for all zjJ . If X= Q D , D is holomorph-convex for all n , and ~~ h=l n n DncDr+-j_ » we say X is holomorph-convex. A holomorph-convex Riemann domain is a domain of holomorphy. 14. An Important result of relatively recent date is the following: H is a (univalent) region of holomorphy iff it is pseudoconvex. The more difficult necessary condition was proved for n complex dimensions by K. Oka in 1954. It is this equivalence theorem which underlies and motivates this paper. Closely related and also fundamental to the theory is the Kontinuitatssata, the name of which is not translated to pre• vent confusion with other "theorems of continuity." The proof of tne Kontinuitatssata is given in the appendix for completeness and as an Illustration of techniques encountered. Each of the various continuity theorems have proved valuable in certain applications. We will show that they are often convenient rephrasings of the same theorem. Tnese theorems are important to us because our definition of a pseudoconvex domain depends directly upon them. In the following, when we speak generally of a "projection", we mean onto some fixed subspace of $n . Domains D-^Dg will be called "equivalent", Dn ~ Dg, if there-is a -one- one bicontinuous mapping between the points of D^ and Dg such trat the corresponding points have the same projection on a space $m . Dp will be said to be "contained in D^ in the sense of Behnke-Thullen", 32 of D^ which has the same projection as D2 . We will assume our domains are bounded and contain no singular points. 15. Property 1; Divide $a so that (a-^Bg,...) becomes (w^,w2,...,w^_1,w) , and let (x,y) be the coordinates of PSQD . Then to P there' corresponds p>0 so that if S=={(w,w) : w^=x^ i=l,2,..., n-l , 0<|w-y| there exists r>0 such that if A=(a_ : Iw^-x^^r i=l,2>.../i-l, |w-y|=6% then there exists A contained in a unique univalent subdomain of D such that p(A )=A , and S OA =}= 0 . Property 2; To the point P there corresponds a pair of positive numbers 6,6' where 6 |w-y|=6' satisfying tne following condition: Let P(x,y) be the point of A* in C^t ~, , let s be the segment ^joining the points w' and y,, and let L be the segment in tip ~ of the form (w'.s) . Then if we trace a curve ° Tw jW in P from P and such that the projection is always on L , we must necessarily come to a point of . In the literature, a domain has been said to satisfy the theorem-of continuity if it satisfied one of the following conditions. lb. Continuity Condition 1 (CC1) : Let Dcf , Pe}3. Then D satisfies CC1 , DeCCl, if whenever all points of <*D have Property 1 they have Property 2 . If Ddfe" , DeCCl , and DeCCl after any one-one w pseudoconformal transformation of ^n into the above-mentioned c w neighbourhood of P , then we will say D is "pseudoconvex". Thus for n=l , all domains are pseudoconvex. Continuity Condition 2 (CC2) : Consider the following geometrical configuration. Let DdfjJ , P=(x,y_) € 3D , where (w^w^w-^Wg,... ,wn_2) =(JjJJ ~ is a new representation of $a . Let Sd$w be a hypersphere with centre a^x such that ^SH £x} 4= 0 > let H be another hypeispiare witn centre x , let a be a linearly simply-connected domain, namely that part of H outside S , and let 0=(a,y_)e$^ ~ . Then we will say D satisfies Continuity Condition 2 , DeCC2 , if for arbitrary a and H and for all P£9D , there does not exist a point set in the neighbourhood of P having projection 8 . Again we have the corresponding definition, of pseudo- convexity and the observation that Dc^ => DeCC2 . 17. Continuity Condition 5 (CC3) : D=$" , n>l, satisfies CC? ,' DeCC3 , if for all domains D» such that D,~,&, D'eD , there exists , with the property <£ ~, 0 £f1 : Iw^-ff^ ) | 0 &2 : Iw^-wJ ) | 0 i=l,2,...,n-l , and ^={(w,w) : | w^-wj ^ | As usual we have made a variable modification, from (a^, z^, . . ., ) to (w,w^,w2,...,wn-1) = (w,w) . For n=l we consider all domains to satisfy CC3 . By means of CC3 , our definition of pseudoconvex!ty assumes a slightly different form. Let > n>l > and Pe^D . Let <£> be a polycylinder about P of radius p , and let A be the component Dfl «C such that Pe&A • If for all P there is a number p >0 such that for all p< p we have ro — o AeCC3 and A^C33 for all one-one pseudoconformal transformations of £ , we say D is "pseudoconvex" . Note that CC1, CC2 are local conditions and CC3 is global. 16. Domains satisfying the above continuity conditions can be shown to satisfy the Kontinuitatssata. We will show that the three definitions of pseudoconvexity are equivalent. Theorem: The three continuity conditions' are equivalent up to a one-one pseudoconformal transformation. Proposition 3: CC1 => GC2 . a ,w v M,,w Proof: Let Ddfg and DeCCl . Let (ja,w) = (a1> 2 r 2' n-2 ^ be a new representation of variables-. In the geometrical con• figuration of CC2 , assume the existence of a set Batf*} contained in some neighbourhood of Pe^D , and such that p(B)=f3 . By means of suitable translations aiid,, rotations map a-»(OjO) and x-(R,Q) , where R is the radius of S, and the images of the other point sets involved will be designated by the same symbols. In the new space, (jj^ , except for the point (R,0) , any point of Uj.=R satisfies Re(u1) n for Dc*tT we cannot have DeCCl . Since this is ,a contradic- z,w tion, we have DeCC2 . D Lemma: Let ^>D2 eCC2 , and let DQdD1nD2 be a (connected) component. Then DQeCC2 . Proof: Let t be a one-one pseudoconformal mapping of a neighbourhood of PsfcD into . Assume there exists a point 19. set with the property mentioned in CC2 , with respect to t(P) and t(DQ) . Note that t(DQ), t(D1), "t(D2) are only defined in a neighbourhood of t(P) . Now , 0 t(D1), t(Dp) € CC2 => t(P) € int t(DQ) , which is a contradiction. Proposition 4; CC2 m> CCJ5 . Proof; Let (a, ,a0, ...,a_ ,, w) be a new representation of . -——— J. cL TX" J- j a Consider the geometrical configuration of CC3 with DeCC2 . If there exists a domain D'feD such that D'~,#, we must show the existence of a domain with the property cC1 ~& and D'cofcD . We first perform a translation so that (a<°),wo) -.(0,0) Now consider the following construction. Let _a e {a_: | a^^ | on L . If this path intersects SD , let P=(a_ ,w) be the point of intersection. If no intersection occurs, then regard- less of w , a_ lies in 3 ^ : | | < r*^ 1=1,2, ... ,n-l} . Suppose now there is a point ae^:{_a: |a1| for b€9 there corresponds PCa,"^) . Define ^ c qjs(a: |B1| Also define the metric d(a_,w) s vik/B^ +|w| > k>0 . If, P»(a*,w* ) corresponds to (a*,w*) , we set up another cor- respondence between d(P') and (B_ ,w ); otherwise we make no correspondence. The set {d) has an upper bound, say d , since no intersection occurs between the constructed path and dD for all a. e&. 1 * 2 2 Now d(a,b1)>d1 = Jwr^) +( p") when 0 (when d exists) , for all (.a ,w ) € (cp,9) such that r2<|a1 | corresponding to the boundary point P0=(x,y ) , where x^=a^ , i=2,3, • • •, ri—1 • Let D , P be the images of D,P^ under the o o transformation of to (fc^ „ given by a u, v u =a 3 v w Let S e T : u^ak/a^, u2=a2, u^=a^,...> n_x n-l ~ * ^ a hypersphere in $ about (0,0) such that U-,, v P1(k/x1,y ) e 5? , let h be a small hypersphere centered at P.j , let n» = C(S)nn (c indicates complement) , and in n $Y let u, v l G=(u,v) : (u1,v)eh , VL±=X± , 1=2,3, . . . ,n-l} . 21. Now there exists G-,c$:: „ such that G,cH(P^ ; 6) for some l Tu,v l x o ' 6>0 , and p(G1)=G (see Property 1) . Then we have arrived at a contradiction since PQ e^D , and D €CC2 with respect to the neighbourhood of PQ . Thus there corresponds no boundary point of D for any choice of (a ,w ) € (5^ ,©) . Similar reasoning applied to $j : {a_: | a11 to 3 : [a: | a. | namely the existence of a domain ,£•' such that ~„C> D'c^t'c D. Proposition 5: CCJ5 => CC1 . Proof: This is clear. Corollary: The three definitions of pseudoconvexity are equivalent. All holomorphic functions can be continued into their pseudoconvex envelope. In particular, the tube domain is easily handled. The following statements are equivalent. For the tube T— , the pseudoconvex envelope is that tube with the convex hull of x as its base. A tube domain is pseudoconvex iff it is convex. Or, the envelope of holomorphy of T— is ^(T— ) , its convex hull. The accepted proof of this last fact is due to S. Bochner, and is based on the expansion of the an• alytic function in T— in multiple Legendre polynomials. It would seem that the following lemmas lead to a more direct 22. proof. Lemma: T— is a tube <=> T— admits the automorphisms I • X X (*) Hk=Viak k=l,2,...,n , a^ real constants. Lemma: E(T—) is a tube. Proof: Let G be an automorphism of the type (*) . Now whenever G(T-)=T— , we have E(T—) =3G(E(T—)) [Cartan-Thullen [1] , Theorem 3, Corollary 1] . But this means E(T—) is a X tube, by the first lemma. Since oC (T-) is a domain of holomorphy, [Bochner- Martin [l], p. 91] , we have £ (T—) o E(T—) . The converse X ~" X follows from the next lemma. Lemma: The base of E(T—) is convex. II 1 • X Another approach to the proof is to observe that the conformal map w. =e takes T— into a covering surface over Jx X a Reinhardt domain in (fc?; . And in the logarithmic sense, the wk Reinhardt domain of holomorphy Is convex. G. Kalien and A.S. Wightman have been working on a means of applying the Kontinuitatssata directly to determine the envelope of holomorphy of a given domain. Bremermann ([4]) has found an iterative procedure which is the best method to date. This is the most recent of the approximation theorems; the envelope 23. is determined as the limit' of a sequence of pseudoconvex domains. Proposition 6: Let {Dn} be a sequence of pseudoconvex domains lim in dT j . Then i D„ is cpseudoconve x if it exists and is ineri n-» n finite. Proof: Assume without loss of generality that the domains D are pseudoconvex in the sense of CC3 • Assume also that n n>2 , and consider the sequence c^1^^,l^^1nD2,...,J^~DLnD2n...nDn,. Then since the intersection of pseudoconvex domains is pseudoconvex, .©^ is pseudoconvex for all n . By CC3 and D is observing that Sn<^_-^ for all n , we have * n oo 1 D if i-t exists pseudoconvex. Let An~k0n k « Then An_1 t o e a and we define Ao' ° domain satisfying An called the "limit" of the sequence C^n) > unique up to the equivalence class. Let o^= [(w,w) : |w1-w[°^| 1=1,2,... ,n-l be a polycylinder in (JJ^ ^ , where ( a^Bg, . . . ,aR) -» (w1,Wg,. .. ,wn_1,w) , and let «t' he the subset of points of (1) r2<|fr-w-o| $S~£? , and^c Ao • By the existence of Aq and 24. "by its first property, there corresponds a polycylinder % around any PeAQ and a positive integer n^ such that if P €A correspondr s to PSA , then P_ is contained in some n n o n subdomain of A which is equivalent to e for 'all n>n' . n ^ — o Let n Hminfn' 1 . o o Assume now that o& is relatively compact in AQ > ^OTAQ . The above reasoning and Borel's lemma imply the existence of a domain <£ such that oC ~°t, and &C£CL0 . Thus AQ€CC3 . € It remains to check that A0 CC3 after an arbitrary one-one pseudoconformal transformation. Suppose such a transformation T is made on a n polycylinder $ . Consider T( cprtD ) . The limit of the new sequence is T(CP("IA0) • Since T( cpnD^) is pseudoconvex for all n , tnen T( cpnA )eCCj5 . Therefore An is pseudoconvex. o o The above proposition has baen proved by H. Behnke and K. Stein for regions of holomorphy; that is, the limit of a sequence of regions of holomorphy is also a region of holomorphy. Lemma; Let V(a_) be the largest plurisubharmonic minorant in the domain D of the function -log dD(a_) , let D 5 U {2/ : ||j5' -ja|| D into D* . t Proof: We assume D^$n (otherwise the statement is trivial) . Then -log dD(a_)>-oo for all aeD . Let S={f:f is plurisub• harmonic in D , |f(j5)|<_ -log dD(z_), z€D) . Bremermann ([7]) has proved the plurisubharmonicity of the upper envelope U(a)=lim "„S.UP sup{U (a')l of a family {U } of functions plurisubnarmonic in a region D and bounded in every closed subregion of D . In our case we have, then, by our definition n of V , V(a) = af" g f(a<) . Let us indirectly define d(z_) by V(_z) = ~-log d(a_) . As shown in the above-mentioned article, the function V (a)= -log ^^-^(aj ^s plurisubharmonic in D . Now we have V (a_)<_ -log d-^ a_) since d d THEREFORE E(D^-^- D^-^ " V*(J5) Thus if D*= u {£': ||a.'-aj| Our next result shows in fact that the envelope of holomorphy is the limit of an increasing sequence of domains. One way to get such a sequence is by repeated application of the lemma. The sequence is D, D = D^,(D ) = DgjD-^Dj,, . . . • This construction is called "Bremermann1s i-process" . 2b. The proof of the next theorem illustrates that when we deal with the union of the domains of such a sequence, the limit (the envelope of holomorphy) need not even be finite. Proposition 7: u D =:s(D) . n=l n Proof; Let U= Q D . As usual, to insure -log dTT(a)>-oo, we n=.l n u ~ assume Uf(J! , the result being trivial otherwise. Let u^eU . Choose nQ a positive integer and a neighbourhood ^(u^ 6) so that WcDn for all n>n . Now -H (a) D , = u [a': ||z'-a|| But this implies -log d_. (a) u n,m n+1 - ~ n _aeN , n>n . Also, n_^>\(.?_) = "log dU^ for a11 -eN ' "°Y lim virtue of ( -log d~ (a))= -log d,T(z) . Furthermore, n-*» u — u — 1S Nn+1(z_) -log d^(a_) is plurisubharmonic in N . But by the generality of our choice of \i and N in U , we have that -log dTJ(z_) is plurisubharmonic for all aeU . Therefore U is pseudoconvex. But DncE(D) for all n (see proof of lemma). Therefore UdE(D) , and since E(D) is the smallest pseudo- convex region containing D , E(D)dJ . Thus U=E(D) and the proposition is proved. 27. BIBLIOGRAPHY 1. Behnke, H., and Sommer, F., [1] Analytische Funktionen mehrerer komplexer Veranderlichen. Uber die Voraussetaungen des Kontinuitatssatzes, Math. Ann. v. 121 (1950) , pp. 356-375 . 2. Behnke, H., and Stein, K., [1] Die Konvexitat in der Funktionentheorie mehrerer komplexer Veranderlichen, Mitt. Math. Ges. Hamburg, v. 8(1940) . 3. Behnke, H.,"and Thullen, P., [1] Theorie der Funktionen mehrerer komplexer Veranderlichen, Ergebn. d. Math., v. 3, no. 3 (1934). 4. 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