Introduction to Holomorphic Functions of Several Variables, Volume I: Function Theory

(Z) = [Z,,+I - Pr(z I, ... , z,,)] e(Z) in that neighborhood. Since ae = 0 in the dense complement of the submanifold M, it follows from continuity that ae = 0 in a full open neighborhood of P (PI' ... , Pr-I; e) x K(O; e). This last set is, of course, a polynomial polyhedron defined by r - 1 polynomials; so by the induction hypothesis there is a Coo differential form n in a smaller open neighborhood of P(PI' ... , Pr-I; e) x K(O; e) such that e = an there. Now a[(Z) - [Z,,+I - Pr(z I' ... , z,,)]n(Z)] = 0 in this neighborhood. So by an additional application of the induction hypothesis there exists a Coo differential form 'P in an even smaller open neighborhood of P(PI' ... ' Pr-I; e) x K(O; e) such that (Z) - [Z,,+I - Pr(ZI, ... , z,,)]n(Z) = a'P(Z) there. Let t/! = E*('PIM) where E: en x C I -+ C" is the natural projection, or equivalently let t/!(z I, ... , z,,) = 'P(z I, ... , Z,,' Pr(z I' ... , z,,)). That defines a Coo form t/! of bidegree (p, q - 1) in an open neighborhood of P(PI' ... , Pr; e), and it follows immediately from the construction of this form that

If P is a polynomial polyhedron, then [1jJp =

(rJp.

Proof. The proof will be by induction on the number r of polynomials defining the polyhedron. When r = 0 the polyhedron is a polydisc, and the desired result holds in that case as already noted. For the induction step consider a polynomial polyhedron P = PUI, ... , f..; c5) for some r > O. In order to complete the proof, it clearly suffices merely to show that for any function f E (rJp, any compact subset K c: P, and any constant e > 0, there exists a polynomial P such that If(Z) - p(Z)1 < e whenever Z E K. Recall as before that P can be identified with the complex analytic submanifold M n {PUI' ... , f..-I; c5) x L\(O; c5)} where M = {Z E C"+I : Z"+I - f..(ZI' ... , z,,) = O}. With the choice of c5 1 < c5 so that K c: PUI' ... , f..; c5d, it follows from Theorem 10 and Corollary E8 that there exists

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a holomorphic function 9 in PUl, ... ,f,.-l; 01) x L\(O; od such that the restriction of 9 to M (1 {PUI' ... , f,.-I; 01) x L\(O; Ol )} coincides with the function f in that set, when f is viewed as a function on M. Now by the induction hypothesis there exists a polynomial P in n + 1 variables so that IP(Zl' ... ' zn+d - g(Zl' ... ' zn+l)1 < e whenever (z I, ... , Zn+1) represents a point in the set K -that is, whenever Zn+1 = f,.(Z) where Z=(ZI, ... ,Zn)EK; but then P(ZI, ... ,Zn)=P(ZI, ... ,Zn,f,.(Zl, ... ,Zn)) is also a polynomial and Ip(Z) - f(Z) I < e whenever Z E K. That suffices to complete the proof.

12. COROLLARY. Proof.

If D is a polynomially convex domain open subset ofcn, then

As usual it suffices to show that for any function f

E (!)D'

?/D = (!)D·

any compact subset

KeD, and any constant e > 0, there exists a polynomial P such that Ip(Z)-

f(Z) I < e whenever Z E K. For this purpose it is enough merely to find a polynomial polyhedron P such that K c Po c D, where Po is the union of some of the connected components of P. Then the function fo in P defined by if Z if Z

E E

Po P - Po

is holomorphic in P, and the desired result follows immediately from Theorem 11. Since D is polynomially convex by hypothesis, K ~ D, and K is also compact since it is clearly closed and the coordinate functions in C" are bounded on K by their bounds on K. Choose an open set U in cn such that [] is compact and K ~ u ~ [] ~ D. For each point A on the boundary au there exists a polynomial PA such that IPA(A)I > 1 > SUPZeK IPA(Z)I, since Art. K. The open sets {Z E C": IPA(Z)I > I} cover the compact boundary au, so a finite number of these sets also serve to cover au. If PI' ... , p,. is the corresponding finite set of polynomials, it is clear that the polynomial polyhedron P(Pl' ... , p,.; 1) is disjoint from an open neighborhood of the boundary of U, and that K ~ P(Pl, ... , p,.; 1). The union Po of those connected components of P(Pl, ... , p,.; 1) contained in U is then the desired set, and that suffices to conclude the proof of the theorem. It should be pointed out that by using Theorem 10 and Corollary 12 in place of the corresponding results for polydiscs, the proof of Theorem E5 can be carried over to give a proof of the assertion that £p,q(D) = 0 for any polynomially convex subset D and any index q > O. Since even more general results will be derived later, though, it is not worth pursuing this observation any further here.

G Domains of Holomorphy and Holomorphic Convexity

One of the most interesting and characteristic phenomena observed in the study of holomorphic functions of several variables is the existence of pairs of open sets DeE ~ Cn such that every function that is holomorphic in D necessarily extends to a function that is holomorphic in the strictly larger set E; several examples of this have already been pointed out. It is clearly of some interest to determine those sets D for which no such extension is possible. As will later become apparent, such sets playa basic role in the function theory of several variables. 1. DEFINITION. A domain of holomorphy in Cn is an open subset D ~ en for which there exists at least one function f E (!)n that cannot be extended as a holomorphic function through any boundary point of D. Iff E (!)n, then f has a power series expansion about any point A ED, and that power series converges in any polydisc .1\(A; R) for which .1\(A; R) ~ D. The set D is a domain of holomorphy if there exists a function f E (!)n such that whenever the power series expansion of f about a point A ED converges in a polydisc .1\(A; R), then conversely .1\(A; R) ~ D. Note that if the power series expansion of a function f E (!)n about a point A ED converges in a polydisc .1\(A; R) not contained in D, this does not necessarily provide an extension of the function f to a function holomorphic in an open subset ofC" properly containing D, for D n .1\(A; R) may consist of several connected components and may be dense in .1\(A; R). Thus, the condition that D be a domain ofholomorphy is slightly stronger than the condition that there exists a function f E (!)n that cannot be extended to a holomorphic function in a properly larger domain. It is well known that the unit disc .1\(0; 1) c C 1 is a domain of holomorphy. The Weierstrass product formula can be used to construct a nontrivial holomorphic function f in .1\(0; 1) such that every boundary point of .1\(0; 1) is the limit of a sequence of zeros of f, or alternatively the explicit power series expansion f(z) = I.z·! is easily seen not to have any continuation beyond the unit circle. More generally, any open polydisc .1\(A; R) ~ C" is a domain of holomorphy, for if f is either of the functions of one variable just mentioned, then 01=1 f«zj - aj)/rj) is holomorphic in .1\(A; R) but cannot be continued beyond any boundary point. There is a useful general criterion for a set to be a domain of 61

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holomorphy, reminiscent of the condition of polynomial convexity introduced in the preceding section.

2. DEFINITION. An open subset D K ~ D the set

~

KD = {A ED: If(A)1 ~

en is holomorphically convex if for any compact subset IIfllK for all f

E

(DD}

is also compact. The set KD is the holomorphically convex hull of Kin D.

Note that the set KD is always bounded, since the coordinate functions in en are bounded on KD by their maximal values on K, and that the set KD is a relatively closed subset of D. Thus, the condition that KD be compact is really just that it be closed in en, or equivalently, that it be disjoint from an open neighborhood of aD. In studying holomorphically convex hulls, it is convenient to have a simple notation for the distance from a point inside D to the boundary of D.

3. DEFINITION.

For any open subset D

~

dD(Z) = sup{e E IR: B(Z; e)

en and any point ZED, let ~

D}

and (jD.R(Z) = sup{e

E

IR:

~(Z; R) ~

D}

where R = (rl' ... , rn ), rj > 0, is any fixed polyradius. For any compact subset K let

~

D,

dD(K) = inf dD(Z) ZEK

and

Thus, dD(Z) is the supremum of the radii of all balls centered at Z and contained in D, and hence is the usual Euclidean distance from Z to the boundary of D, and (jD,R(Z) is an analogous measure of the distance from Z to the boundary of D but defined by polydiscs having poly radii proportional to R rather than by balls. The distance function (jD,R is more useful in many contexts than is the usual Euclidean distance, since it reflects the analytic situation more closely. As a further simplification, the notation (jD will sometimes be used in place of (jD,I' where 1= (1, ... , 1). Both dD and OD,R are clearly continuous functions in the domain D, so the expressions dD(K) and OD,R(K) have well-defined finite values. Note that dD(K) is the distance from the subset K to the boundary of D in the usual sense, while

G Domains of Holomorphy and Holomorphic Convexity

63

bv,R(K) is the distance measured by polydiscs having polyradii proportional to R. Note also that if Kl ~ K 2, then dv(Kt> ~ dv (K2) and bv,R(K 1 ) ~ bv ,R(K 2).

4. LEMMA. If K is a compact subset of an open set D ~ en and if 15 = bv,R(K) for some polyradius R, then any function f E &v extends to a holomorphic function in ~(A; bR) for any point A E iv.

UA

Proof. Whenever 0 < 6 < 15, the set K. = eK K(A; 6R) is clearly a compact subset of D. Iff E &v, then it follows from Cauchy's inequalities, Theorem A5, that

for all multi-indices I whenever A E K. Since the partial derivatives of fare holomorphic in D, the same inequalities also hold for all points A E iv, by definition of the holomorphically convex hull. That clearly implies that the Taylor expansion of the functionf at any point A E iv must converge in the polydisc ~(A; 6R), and since that holds for all 6 < 15, the proof is thereby concluded.

5. THEOREM. An open subset D morphically convex.

~

en is a domain of holomorphy if and only if it is holo-

Proof. First suppose that D is holomorphically convex. Choose a sequence of compact subsets Kv c D such that Kv c interior K v+1 and Uv Kv = D; also choose a countable dense sequence of points Av E D, and for each point A v, let ~(Av; Rv) be a maximal polydisc centered at Av and contained in D. For each index v there exists a point Zv E ~(Av; Rv) such that Zv ¢ iv, since D is holomorphically convex and hence iv is compact. Since Zv ¢ iv, there exists a holomorphic function J. E &v such that IJ.(Z,) I > II fv 11K . After multiplying this function by a constant and then raising it to a sufficiently large power, it can even be assumed that J.(Zv) = 1 and that Ifv(Z)1 ~ v- 12- v whenever Z E Kv' The series Lv vfv(Z) is thus absolutely and uniformly convergent on any compact subset of D. Consequently the infinite product f(Z) = flv[l - fv(Z»)" converges to a nontrivial holomorphic functionf E &v, and by construction this function is of total order at least v at the point ZV' It will be demonstrated that this function f cannot be extended as a holomorphic function across any boundary point of D, and hence that D is a domain of holomorphy. Suppose contrariwise that for some point A E D the function f extends to a holomorphic function in a polydisc ~(A; R) not contained in D. There is a subsequence ZVj such that the points ZVj lie in the connected component of D (\ ~(A; R) containing A and converge to a point Zo E aD (\ ~(A; R). The function f is of total order at least Vj at the point Zv.; hence, for any multi-index I, J

since this partial derivative is zero at ZVj whenever

Vj>

In That means that the

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function f is identically zero, which is a contradiction; hence, every holomorphically convex domain is a domain of holomorphy. Next suppose that D is a domain ofholomorphy, and choose a function f E (9D that cannot be extended as a hoi om orphic function across any boundary point of D. If K £; D is a compact subset of D and (j = (jD R(K), it follows from Lemma 4 that A(A; (jR) £; D whenever A E K D; for f is holom'orphic in A(A; (jR) by that lemma, but f cannot be extended as a holomorphic function beyond any boundary point of D. Thus KD must be disjoint from an open neighborhood of aD; that suffices to show that D is holomorphically convex and therefore to conclude the proof of the theorem. Actually the second part of the proof of the preceding theorem demonstrated a somewhat more precise assertion than merely that a domain of holomorphy is holomorphically convex. If D £; en is a domain of holomorphy, then for any compact subset K and any polyradius R,

6. COROLLARY.

£;

D

(1)

and

(2) Proof. Since K £; K D, the inequalities dD(K) ~ dD(K D) and (jD,R(K) ~ (jD,R(K D) are trivial. On the other hand, in the second part of the proof of Theorem 5 it was demonstrated that A(A; (jD R(K)· R) £; D whenever A E K D, so that (jD R(K) ~ (jD,R(K D) and hence (jD,R(K) = bD,R(K D), demonstrating (2). To derive the ~quality (1) from this one, note that

dD(Z)=sup{eEIR:Z+eWEDforall WE Cn with IIWII ~ I} =

inf dD , w(Z) IIWII=l

where dD,w(Z) = sup{e E IR: Z + rW E D for all r E IR with Irl ~ e}

Hence it is sufficient to show that inf dD,w(Z) = in[ dD,w(Z) ZeK

ZeK D

for any fixed W E c n with II W II = 1. After a nonsingular linear change of coordinates in en it can be assumed that W = (1, 0, ... ,0). The line segment from Z - W to Z + W is then just the intersection of the monotonically decreasing sequence of

G Domains of Holomorphy and Holomorphic Convexity

65

polydiscs .1(Z; RJ, where R. = (1 + (1/v), 1/v, ... , 1/v), and consequently it is clear that dD.w(Z) = lim. bD.R,(Z). Actually since .1(Z; R.) ;2 .1(Z; R.+ 1 ), the functions bD • R are a monotonically increasing sequence of continuous functions in D, and hence these functions converge uniformly on any compact subset of D. From this observation and the already demonstrated equality (2) it follows that inf dD • w(Z) = lim lim bD,R'(Z)

ZEK

•

=

ZEK

lim inf bD R (Z) = inf dD w(Z) v

ZeK D

."

ZeK D

'

and that suffices to conclude the proof. Theorem 5 is perhaps the most useful criterion that an open subset D ~ a domain of holomorphy and easily leads to still other criteria as follows.

en be

7. THEOREM. An open subset D ~ en is a domain of holomorphy if and only if for any discrete sequence of distinct points A. E D there exists a function f E (!)D such that lim sup.lf(A.)1 = 00. Proof. First suppose that D is a domain of holomorphy and consider a discrete sequence of distinct point A. E D. Choose a sequence of compact subsets K. c D such that K. c interior K.+1 and U. K. = D. Since D is holomorphically convex by Theorem 5, the holomorphically convex hulls K•. D are also compact subsets of D and hence cannot contain the entire sequence {A.}; so by passing to subsequences of the sequences of points {A.} and subsets {K.} it can be assumed that A. rf= K. D but A. E K.+ 1 • D • From the definition ofthe holomorphically convex hull of a set,'it follows that for each v there exists a function f. E (!)D such that If.(A.)1 > II f.IIK' After multiplying this function by a constant and then raising it to a sufficiently large power, it can even be assumed that (3)

and .-1

1f.(A.)1 > v +

L

,,=1

IfiA.)1

(4)

Whenever v ~ J-l it follows from (3) that 11f.IIK. ~ Ilf.k < 2-'; hence, the series f = L. f. converges in the topology of (!)D to a function f E (!)D' For this function, (5)

If J-l ~ v + 1, note that. A. E

K.+ 1 • D~ K".D; hence, by (3) necessarily If,,(A.)1 ;;:;; 1I.t;.IIK. < 2-", and by thiS and (4), it follows from (5) that

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and hence that lim vIf(AJI = 00. Next suppose that D has the property that for any discrete sequence of distinct points Av E D there exists a function f E (YD such that lim supv If(Av)1 = 00. If Dis not a domain of holomorphy and hence by Theorem 5 is not holomorphically convex, there must be a compact subset KeD such that KD is not compact. For any discrete sequence of distinct points Av E K D' there is a function f E (YD such that lim sup vIf(Av)1 = 00; but then If(AJI > IlfilK for some values of v, contradicting the assumption that Av E KD • Thus, D must be a domain of holomorphy, and that concludes the proof. It should perhaps be noted that if D is a domain of holomorphy and {Av} is a discrete sequence of distinct points of D, there actually exists a function f E (YD such that limy If(Av)1 = 00, a rather stronger assertion than lim sup vIf(Av)1 = 00. This stronger assertion is not that much more useful, though, and is a special case of a more general result which will be demonstrated later; so this point will not be pursued further here.

8. THEOREM. If an open subset D ~ en is not a domain of holomorphy, there exist a point A ED and a polydisc L\(A; r) not contained in D such that for every function f E (YD the power series expansion off about the point A converges throughout L\(A; r). Proof. If D is not a domain of holomorphy, then by Theorem 5 it is not holomorphically convex, so there exists a compact subset KeD such that KD is not compact and hence such that bD(K D) = O. For a point A E KD such that bD(A) < bD(K) = r, the polydisc L\(A; r) is not contained in D. It follows from Lemma 4 that for every function f E (YD the power series expansion of f about the point A converges throughout L\(A; r), which was to be proved. Although the preceding theorem is a simple consequence of already established results, it is nonetheless worth stating separately. On the one hand, it shows that if an open subset D ~ en is not a domain of holomorphy, then all functions f E (YD can simultaneously be extended to a properly larger manifold; but, as will be observed in the next section, this larger manifold may not be an open subset of en. On the other hand, by the same light if for each point A E iJD there is at least one function fA E (YD that cannot be extended through A, then D is necessarily a domain of holomorphy. This provides an apparently simpler characterization of domains of holomorphy, one that is sometimes used as the definition of domains of holomorphy. Examples. The criterion provided by Theorem 5 can be used to show that a number of standard domains D ~ en are domains of holomorphy, as follows.

9. THEOREM.

Any open subset D ~

e 1 is a domain of holomorphy.

Proof. It is sufficient. to show that any open subset D ~ e 1 is holomorphically convex. For any compact subset K ~ D and any point a E iJD, the function (z - a)-l

G Domains of Holomorphy and Holomorphic Convexity

67

is holomorphic in D and bounded on K, but in modulus tends uniformly to 00 as Z approaches a; hence, Kn is disjoint from an open neighborhood of aD and is consequently compact. That shows that D is holomorphically convex as desired.

10. THEOREM.

Any open subset D domain of holomorphy.

£;

en

that is convex in the usual geometric sense is a

Proof. If D is an open convex set in en, then for any compact subset K £; D and any point A E aD there is a real-valued linear function tA(Z) = tA(X, Y) of the real coordinates in en = 1R 2 n such that tA(A) = 0 and tA(Z) < 0 whenever Z E K. When expressed in terms of Zj and Zj, this linear function can be written in the form tA(Z) = C + t Lj(CjZj + cjz) for some complex constants c, Cj, and cj. The condition that tA(Z) be real for all Z implies that C = c and cj = Cj; hence, tA(Z) = Re[c + LjCjzJ is the real part of the complex linear function C + LjCjZj' Now £(Z) = exp(c + LjCjz) is holomorphic in D, and fA(A) = 1 while IfA(Z) I = exp(tA(Z» < 1 for all Z E K and hence for all Z E Kn. Thus, again Kn is disjoint from an open neighborhood of aD, so Kn is compact and D is holomorphically convex. That suffices to conclude the proof.

11. THEOREM.

A complete Reinhardt domain having a logarithmically convex base is a domain of holomorphy.

Proof. Let D be a complete Reinhardt domain having a logarithmically convex base B, and let K be a compact subset of D, which can always be assumed to contain A(O; R.) such that the origin. There is a finite union of open polydiscs U' = K £; U' £; D; the set U' is also a complete Reinhardt domain, and the logarithmically convex hull Uo of its base U~ is the base of a complete Reinhardt domain U such that K £; U £; D. This domain U has the property that there exists a constant e > 0 such that whenever A = (aI' ... , an) E au and aj = 0 for some index j, then Ax = (aI' ... , Xj' .•• , an) E au whenever 0 ~ Xj < e. By iterating this observation, it follows that to every point A E au there can be associated a family of points Ax E au for some parameter values 0 ~ xi!' ... , x jk < e, such that Ax = A when xi! = ... = x jk = 0 and no coordinate of Ax is zero when Xj, ... Xjk > O. The point 10glAxi =(loglall, ... ,loglxjl, ... ) then lies on the boundary of the convex set log Uo = {(log Xl"'" log Xn): X E Uo and Xl'" Xn =F O}. The projection KB = {IZI=(lzII, ... ,lznl):ZEK} of the compact subset KeD to the base B is a compact subset KB £; B, and log KB = {loglZI = (lOgIZII, ... , loglznl): Z E K and Zl ... Zn =F O} is a closed set in the interior ofthe convex set log Uo. For any point A E au there is a real linear function tA(S) = v + LjVjSj such that tA(loglAxl) = 0 and tA(S) < 0 whenever S E log K B • Note that it can be assumed that tA is independent of the variable Sj whenever aj = O. Since the coefficients of the linear function tA can be varied slightly so that VI' ... , Vn become rational, and tA can then be multiplied by any positive integer, it can also be assumed that VI' ... , Vj are integers. Since K contains the origin and tA(S) < 0 for all S E log K B , all the integers Vl , ... , Vj are necessarily nonnegative. The function fA(Z) = e z~"" is then holomorphic in e" and hence in D, and I£(Z)I = exp tA(S) where Sj = loglzjl. Therefore, I£(A)I = 1 and IfA(Z) 1 < 1 whenever Z E K and Zl ... z" =F 0, and the same

Uv

V •

z:n

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inequality holds for all points Z E KD by definition of the holomorphically convex hull K D • That means that KD is disjoint from an open neighborhood of au and hence that KD is compact. Therefore, D is holomorphically convex, which suffices to conclude the proof. As demonstrated in Theorem B2, the domain of convergence of any power series expansion about the origin in en is a complete Reinhardt domain having a logarithmically convex base. The last result shows that for each such domain there is a function that is holomorphic in that domain but not in any properly larger domain and hence that there is a power series having that domain as precisely its domain of convergence. That completes the description of the domains of convergence of power series in several complex variables.

12. THEOREM.

The intersection of any two domains of holomorphy is also a domain of holomorphy.

Proof. If Dl and D2 are domains of holomorphy and K s;;; Dl n D2 is a compact subset, then KDlnD2~S;;; KDI n K D2 , since there are more holomorphic functions on a smaller set. Thus, KD I nD 2 is disjoint from an open neighborhood of aD l U aD2 and hence KDlnD2 is compact and Dl n D2 is holomorphically convex. That suffices to complete the proof. ~

13. DEFINITION. An open subset D s;;; en is locally a domain of holomorphy if each point A E aD has an open neighborhood U such that Un D is a domain of holomorphy. If D is a domain ofholomorphy and A E aD, then any ball B(A; 6) is a domain ofholomorphy by Theorem 10, and hence any intersection B(A; 6) n D is a domain ofholomorphy by Theorem 12; thus, any domain of holomorphy is locally a domain of holomorphy. It would clearly be very useful if the converse were also true, since that would show that the property of being a domain of holomorphy is a purely local property of the boundary of a domain. Actually the converse assertion is true, but that is a rather deeper result and will not be proved until some further machinery has been developed. The problem of whether an open subset of en that is locally a domain of holomorphy is actually a domain of holomorphy is often called the Levi problem in en, since it first arose in about 1910 in connection with an investigation by E. E. Levi oflocal conditions for a subset of en having a smooth boundary to be a domain of holomorphy. The study of this problem and of its natural generalizations has been one of the principal themes in the development of the theory of hoI om orphic functions of several variables. There are still other criteria for an open subset of en to be a domain of holomorphy, one of which involves the Dolbeault cohomology groups introduced in section E. This criterion provides another example, to add to those discussed in the preceding two sections, of the usefulness of the operator in a variety of problems in complex analysis.

a

G Domains of Holomorphy and Holomorphic Convexity

14. THEOREM. If D is an open subset of en for which Ye°·q(D) = D is a domain of holomorphy.

°for 1

~ q ~ n-

69

1, then

Proof. The proof will be by induction on the complex dimension n. When n = 1 the hypothesis involving the Dolbeault cohomology groups Ye°,q(D) is vacuous and hence is automatically satisfied for any open subset D s; e 1. But by Theorem 9, any open subset D s; e 1 is a domain ofholomorphy, so the present theorem holds in this case. Assume that the theorem has been proved to hold for open subsets of en- 1 for n ;?; 2, and consider an open subset D s; en for which Ye°·q(D) = for 1 ~ q ~ n - 1. As a useful preliminary observation it will first be demonstrated that for any complex linear submanifold L s; en of dimension n - 1, each connected component of the intersection D n L, viewed as an open subset of L = en-I, is a domain of holomorphy. In view of the inductive hypothesis, it is enough for this purpose just to show that Ye°·q(D n L) = for 1 ~ q ~ n - 2. So consider a a-closed coo differential form t/J of bidegree (0, q) on D n L, where 1 ~ q ~ n - 2. To simplify notation perform a translation and a nonsingular linear change of coordinates in en so that L = {Z E en: Zn = o}. It is clear that the differential form t/J can be extended to a a-closed coo differential form of bidegree (0, q) in an open neighborhood U of the closed subset D n L s; D merely by viewing t/J as a differential form independent of the variable Zn' For a Coo function p in D, such that p is identically equal to one in an open neighborhood of D n L in D and that the support of p is contained in U, introduce the differential form of bidegree (0, q + 1) in D defined by

°

°

if ZE U if ZED-U It is easily seen that is a a-closed Coo differential form in all of D, since ap is identically equal to zero in open neighborhoods of D n iJU and of D n L = {Z ED: zn = O}. Since Ye 0 .q+l(D) = 0, it follows that there is a Coo differential form 'iI of bidegree (0, q) in D such that = a'il. Then since = zn( - 'iI) = ap 1\ - zna'il = a(p - zn - zn'il = a'l'. Upon restricting this last identity to L and setting t/I = 'I'lL, it is evident that t/I is a Coo differential form of bidegree (0, q - 1) on L such that t/J = at/l,

°

°

thus showing that Ye°·q(D n L) = as desired. Now to conclude the proof of the inductive step, if D is not a domain of holomorphy it follows from Theorem 8 that there exists a point A E D and an open polydisc .1(A; r) not contained in D such that for every function g E (rJD the power series expansion of g about the point A converges in .1(A; r). Choose a complex linear submanifold L £ en of dimension n - 1 passing through A and such that there is a point B E iJ(D n L) n .1(A; r). Since each connected component of D n L has been shown to be a domain of holomorphy in L = en-I, there is a function f that is holomorphic in D n L but that cannot be extended as a holomorphic function '

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through the point B. However, since J{'o,l(D) = 0, it follows from Theorem E7 that this function f is the restriction to D (") L of a function 9 holomorphic in D. But the power series expansion of 9 about the point A converges in an open neighborhood of B, which leads to an extension of the function f through B and thereby to a contradiction. Consequently, D must be a domain ofholomorphy after all, and the proof is thereby concluded. It was demonstrated in Theorem E5 that J{'o,q(D) = 0 for all q ~ 1 when D is an open polydisc in the extended sense in en, and it was mentioned in section F that the proof of Theorem E5 can be carried over to show that J{'°,q(D) = 0 for all q ~ 1 when D is any polynomially convex open subset of en. Thus, there are cases in which Theorem 14 is applicable, although in both of these cases it is directly obvious that D is holomorphically convex and hence is a domain of holomorphy. Actually the proof of Theorem E5 can be extended to show that J{'o,q(D) = 0 for all q ~ 1 when D is any domain of holomorphy in en, so that Theorem 14 really characterizes domains ofholomorphy. There are complications in attempting to extend the proof of Theorem E5 directly, however, so the further discussion of this topic will be left until sufficient machinery has been developed.

H Envelopes of Holomorphy and Riemann Domains

en is not a domain ofholomorphy, then by Theorem G7 there is at least one point A E oD through which all functions f E (!)D can simultaneously be extended. One might expect that there is a maximal open subset E £; en containing D and such that every f E (!)D extends to a holomorphic function in E. For

If an open subset D

£;

various special classes of subsets D there do exist easily described maximal sets E for which such extension is possible. Examples.

If D £;

en is a complete Reinhardt domain, then by Corollary B3 every

function f E (!)D extends to a holomorphic function in the Reinhardt hull E of D; E is a domain of holomorphy by Theorem G 11, so E is the maximal set for which such extension is possible. If D £; en is a tube domain with connected base B £; [Rn, then by Theorem D 11 every function f E (!)D extends to a holomorphic function in the tube domain E with base the convex hull of B. E is itself a convex subset of en so is a domain of holomorphy by Theorem GI0; hence, E is the maximal set for which such extension is possible. In general, if D £; en is not a domain of holomorphy, all functions f E (!)D can be extended to holomorphic functions in a larger subset of en, but these extended functions may no longer be single-valued. So the attempt to find a maximal set to which all holomorphic functions can be extended leads naturally to analytic entities that cannot be realized as open subsets of en. Example. Let D1 and D2 be the Reinhardt domains in respective sets

e2

B1

= {R E [R2: 0 ~ r1 < 1, 0 ~ r2 < t} u {R E [R2: 1 <

B2

= {R E [R2: 0 ~ r1 < t, ~ <

having as bases the

r1 < 1, 0 ~ r2 < I}

r2 < I}

as sketched in Figure 7. It follows from the continuity theorem, Theorem D6, that every function holomorphic in D1 can be extended to a holomorphic function in the Reinhardt domain E1 having as base the convex hull of the set B 1 • Thus, any 71

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I

Bl 2/3

/3

B1 1/3

2/3

I

Figure 7

function hoi om orphic in D1 u D2 can be extended to a holomorphic function in E 1, but the extended function may be double-valued in the subset D2 eEl' The disconnectedness of D1 u D2 is not really the difficulty here. Indeed to modify this example so as to avoid disconnectedness, connect the sets D1 and D2 by a small open neighborhood U of a path outside D1 u D2 joining a point on the boundary of D1 to a point on the boundary of D2. Any function that is holomorphic in D1 u D2 U U extends to a holomorphic function in E1 u U, but the extended function may be double-valued in the subset D2 c E1 U U. For example, if the set U is· chosen appropriately, any branch of the function ..;z:=2 in D1 will continue along U to a different branch of..;z:=2 in D2 ; hence, the single-valued function ..;z:=2 in D1 u D2 U U extends to a double-valued function in E1 u U. Of course, this extended function can be considered as a single-valued function on the disjoint union ofthe sets E1 u U and D2 • Although this union cannot naturally be viewed as a subset ofC 2 , it can be considered as a covering space over a portion ofC 2 • That suggests that in the analysis of the process of analytic extension, it is necessary to introduce such covering spaces. 1. DEFINITION.

A Riemann domain of dimension n is a complex manifold M together with a nonsingular holomorphic mapping P: M --+ cn, called the projection of M to en.

The terminology is, of course, motivated by Riemann's concept of the Riemann surface of an algebraic function as an abstract space realized by a branched covering space of the projective line; but here Riemann domains are unbranched covering spaces in a neighborhood of each of their points, since a nonsingular holomorphic mapping is locally biholomorphic and hence is locally a homeomorphism. If M is any Hausdorff topological space that admits a local homeomorphism P: M --+ en, then P induces on M the structure of a complex manifold, the mapping P itself providing local coordinates on M. With this complex structure, M is evidently a Riemann domain. Any subset of en is a Riemann domain, with P the identity mapping, and the examples discussed above can be considered as Riemann domains, showing that such domains do arise naturally when considering problems of analytic contipuation.

H Envelopes of Holomorphy and Riemann Domains

73

For any point A in a Riemann domain M with projection P: M ~ en, there is an open neighborhood U of A that is mapped homeomorphically to P(U) ~ en. When P(U) is a polydisc A(P(A); R) for some poly radius R, the neighborhood U can be viewed as a polydisc in M and hence will be denoted by AM(A; R). Similarly, if P(U) is a ball B(P(A); r), the neighborhood U can be viewed as a ball in M and will be denoted by BM(A; r). Measures of the distance to the boundary of a general Riemann domain can then be introduced, paralleling the measures of distance introduced in Definition G3 for the special case of open subsets of en. 2. DEFINITION.

For any Riemann domain M and any point Z

E

M, let

and

where R = (rl' ... , rn), rj > 0, is any polyradius. Furthermore, for any compact subset K ~ M, let dM(K)

= inf dM(Z) ZeK

and

Here dM and (jM,R are continuous functions on M, so that dM(K) and (jM,R(K) are well-defined positive numbers for any compact subset K c: M. Note that if r < dM(A), then BM(A; r) c: BM(A; dM(A)), and consequently the closure BM(A; r) is a compact subset of M. It is useful to observe the following topological property of Riemann domains.

3. THEOREM.

Any connected Riemann domain satisfies the second axiom of countability.

Proof. Fix a point Ao in the connected Riemann domain M and a constant r < dM(Ao), and for any positive integers Jl, v let D/J .• = {Z EM: dM(Z) > r/Jl, and there are points AI' ... , A. E M with A. = Z, dM(A) > r/Jl, and BM(Aj- l ; r/2Jl) n BM(Aj; r/2Jl) #- ,p for j = 1, ... , v}. The sets D/J,. are clearly open in M. Furthermore, since M is connected, necessarily M = U/J,.D/J,., for if A~ E M is not contained in this union, then the same construction based at A~ yields an open subset of M containing A~ and disjoint from this union, so the complement is also open. In order to complete the proof it is enough to show that D/J,. is compact, since that implies that D/J,. is second c~untable. Since D/J,l s;; BM(Ao; r/Jl) and r/u < dM(A o), the set D/J,l is compact. If D/J,. is compact, then choose finitely many points Zj E D/J,. such that D/J .• ~ UjBM(Zj; r/2Jl). Then clearly D/J,.+l ~ D/J,. u UiB(Zj; r/Jl), so that

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15,..• +1 is compact. That shows by induction on v that all the sets 15,..• are compact and concludes the proof. Since a Riemann domain M is second countable, the space (!}M of all holomorphic functions on M is a Frechet space with the natural topology as described in Definition Fl. The space (!}M is actually quite large, for the projection mapping P: M -+ (;n induces an injective homomorphism P*: (!}cn -+ (!}M' However, the functions in (!}M need not separate points, in the sense that there may be distinct points Al and A2 of M such that f(Ad = f(A 2) for all f E (!}M; but of course in this case necessarily P(Ad = P(A 2). To see a simple example, consider the domains Dl = {Z E (;2: 1 < IIZII < 3} and D2 = {Z E (;2: -2 < Xl < 2, Ihl < e, IZ21 < e}, noting that Dl n D2 consists of two connected components D~2' D~2' and that D2 is contained in the convex hull of Dl whenever e is sufficiently small. Let D be the Riemann domain consisting of the disjoint union of Dl and D2 , with points of Dl and D2 identified if they are contained in D~ 2 but not if they are contained in D~2' Thus, D is a Riemann domain, where P-l(Z) consists of two points whenever Z E D~2 but otherwise of one point, as indicated schematically in Figure 8. Every function f E (!}D viewed as a function in (!}D, extends to a unique holomorphic function in the convex hull of D 1 , and hence must have the same value at the two points P-l(Z) whenever Z E D~2'

t

Figure 8

4. THEOREM. For any Riemann domain M with projection P there exist a Riemann domain Ml with projection P1 and a nonsingular holomorphic mapping F: M -+ Ml such that P1 0 F = P, that F induces an isomorphism F*: (!}M, -+ (!}M, and that functions in (!}M, separate points in M 1 • Proof. Introduce an equivalence relation on M by setting A ~ B whenever f(A) = f(B) for all functions f E (!)M' noting that this is indeed an equivalence relation in the usual sense. Let Ml = M/~ be the quotient space of M under this equivalence relation, and F: M -+ M 1 be the natural mapping that associates to any point A E M its equivalence class. It is evident that A ~ B implies that P(A) = P(B); hence, P

H Envelopes of Holomorphy and Riemann Domains

75

induces a natural mapping Pl : Ml -+ en such that P = Pl 0 F. In order to complete the proof ofthe theorem, it suffices to show that whenever A '" Band BM(A; e) S; M, BM(B; e) S; M, then A' '" B', for A' E BM(A; e), B' E BM(B; e) if and only if P(A') = P(B'). Now whenever f E (9M' then al11j"IazI E (9M for any multi-index I, where differentiation is with respect to the natural coordinates imposed on M by the projection mapping P. If A '" B, then

for any multi-index I. Thus, the functions f 0 (PIBM(B; en-l and f 0 (PIBM(B; en- 1 have the same Taylor expansions about P(A) = P(B), so that f(A') = f(B') whenever P(A') = P(B'). That suffices to conclude the proof. As a consequence of this observation, when the primary interest is in the holomorphic functions on a Riemann domain, there is no loss of generality in restricting attention to those Riemann domains for which holomorphic functions separate points, and that is frequently done. Next for the problem of the extension of hoI om orphic functions, it is convenient to introduce the following definition.

5. DEFINITION. A complex manifold E is a holomorphic extension of a complex manifold M if M is an open subset of E with the induced complex structure and the natural restriction mapping r: (9E -+ (9M is an isomorphism. If E is a holomorphic extension of M, then each connected component Ev of E must contain a connected component Mv of M, since otherwise the restriction mapping r cannot be an isomorphism, and Ev is clearly a holomorphic extension of Mv. Thus, in discussing holomorphic extensions, it is sufficient to consider only connected complex manifolds. If E is any connected complex manifold containing M as an open subset, then the restriction mapping r is necessarily injective, by the identity theorem. Hence, E is a holomorphic extension of M precisely when the restriction mapping r is surjective, or equivalently, precisely when every holomorphic function on M extends to a holomorphic function on E. As noted earlier, an open subset of en may have a Riemann domain as a holomorphic extension, but holomorphic extensions of Riemann domains are necessarily Riemann domains, as is clear from the following observation.

6. THEOREM. If M is a Riemann domain with projection P and E is a holomorphic extension of M, then E is also a Riemann domain, with a projection p* such that p* IM = P. Proof. If M is a Riemann domain with projection mapping P: M -+ en, then the coordinate functions of the mapping P extend to holomorphic functions on E, so the mapping P extends to a holomorphic mapping P*: E -+ e. The determinant of the Jacobian of this mapping p* is a holomorphic function p* = det Jp • on E, and. it is nowhere zero on M. That implies that p* is also nowhere zero on E and hence that P*: E -+ en is nonsingular, for if p*(A) = 0 for some point A E E - M, then IIp* is holomorphic on M but does not extend to a holomorphic function on E,

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contradicting the assumption that E is a holomorphic extension of M. That suffices to conclude the proof. If holomorphic functions separate points of a Riemann domain M, that is not necessarily true for all holomorphic extensions of M. The example of Figure 8 illustrates this, since the Riemann domain D is a holomorphic extension of the open subset Dl c 1C 2 • However, in any such case the reduction described in Theorem 4 can be applied to the holomorphic extension E without altering the subset M c E, yielding a hoi om orphic extension El for which the holomorphic functions do separate points. It is thus evident that it is quite reasonable to restrict attention to the category of connected Riemann domains for which the holomorphic functions do separate points; within this category there does always exist a maximal holomorphic extension. This maximal holomorphic extension can be constructed quite explicitly in terms of the Frechet algebra of holomorphic functions on the given Riemann domain. To carry out the construction, consider an arbitrary connected Riemann domain M with projection P: M -+ en, not necessarily assuming that holomorphic functions separate points of M. The set of all continuous algebra homomorphisms T: (9M -+ IC is called the spectrum of the Frechet algebra (9M and is denoted by spec (9M. As might be expected, there is a natural one-to-one correspondence between spec (9M and the set of closed maximal ideals in (9M by associating to each T E spec (9M the ideal in (9M that is the kernel of the homomorphism T, but this result is not needed here and so will not be proved until later. For each point A EM, the evaluation mapping SA: (9M -+ IC defined by SA(f) = f(A) is clearly an element SA E spec (9M. The correspondence that associates to any point A E M the evaluation mapping SA E spec (9M is the natural mapping S: M -+ spec (9M. If the holomorphic functions on M separate points, the natural mapping S: M -+ spec (9M is clearly injective and so can be viewed as imbedding M in spec (9M. If E is a holomorphic extension of M, then the restriction mapping rE : (9E -+ (9M must actually be an isomorphism of Frechet algebras, for it is obviously a continuous mapping between Frechet spaces and is an algebraic isomorphism by definition of holomorphic extension and hence by the open mapping theorem must be an isomorphism of Frechet spaces. The inverse mapping ri 1 : (9M -+ (9E is thus also an isomorphism of Frechet algebras. For each point A E E the mapping S;: (9M -+ IC defined by S;(f) = (ri 1f)(A) is then clearly an element S; E spec (9M' and the correspondence that associates to any point A E E this composition S; E spec (9M is a well-defined mapping SE: E -+ spec (9M. If the holomorphic functions on E separate points, which can of course happen only if the holomorphic functions on M already separate points, the mapping SE: E -+ spec (9M is also injective, so can be viewed as imbedding E in spec (9M. In that case the imbedding S: M -+ spec (9M is merely the restriction S = SEIM. Next to any element f E (9M there can naturally be associated a function J on spec (9M by defining J(T) = T(f) for any T E spec (9M. In particular, to the coordinate functions PI' ... , Pn of the projection P: M -+ en there can be associated the functions PI' ... , Pn on spec (9M' and these functions can in turn be taken as the coordinate functions defining a mapping P: spec (9M -+ en. Note that as a consequence ofthese definitions the composition ofthe natural mapping S: M -+ spec (9M

H Envelopes of Holomorphy and Riemann Domains

77

en

with this mapping P: spec (!)M -+ is just the original projection P = po S. Note further that for any hoi om orphic extension E of M, the projection P* ofthe Riemann domain E must be given by P* = ri 1 (p). Hence, the composition of the mapping SE: E -+ spec (!)M with the mapping P: spec (!)M -+ is just the projection P* = poSE.

en

en

7. LEMMA. For any connected Riemann domain M with projection mapping P: M -+ the set spec (!)M can be given the structure of a Riemann domain for which the projection mapping is P: spec (!)M -+ en, and holomorphic functions on this Riemann domain separate points. Proof. Note first that for any fixed point T E spec (!)M there is a compact subset K of M such that IT(f)1 ~ IlfilK for all f E (!)M' Indeed if that were not so, then, when M = U. K. for some compact subsets K. £ M with K. contained in the interior of K.+ 1 , there would exist for each index v afunctionf. such that IT(f.) I > IIf.IIK,. After multiplying f. by a suitable constant and raising it to a sufficiently high power, it could even be assumed that T(f.) = 1 and Ilf.IIK, < T'. The series L.J. then converges in the topology of (!)M, but the series L. T(f.) diverges, contradicting the continuity of T. Now choose a positive number e such that e < t5M ,R(K) for the polyradius R = (1, ... , 1), and note that K. = UZEKL\(Z; eR) is a compact subset of M. It follows from the Cauchy inequalities, Theorem A5, that

for any function f E (!)M, any point A E K, and any multi-index I, where as usual differentiation is with respect to the natural coordinates imposed on M by the projection P. Consequently,

for any f

E (!)M'

The power series (1)

therefore converges for all points Z estimates,

E

A(P(T); eR) for any

f

E (!)M'

By the usual

so that for any fixed point Z E A(P(T); eR) the mappingf -+ LT(f, Z) is a continuous linear mapping LT(Z): (l)M -+ C. By Leibniz's rule for differentiation, the derivatives iJllIj/iJZ1 behave on a product function in the same way as the corresponding terms

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in the power series expansion of a product; since T is an algebra homomorphism, then LT(fg, Z) = LT(J, Z)LT(g, Z), and consequently LT E spec (!)M' The correspondence Z -+ Lr(Z) thus defines a mapping L T: .1.(P(T); eR) -+ spec (!)M' It is clear from (1) that LT(P(T» = T. On the other hand, since if I = (0, ... , 0) if I = (0, ... , 1, ... ,0) with 1 in the jth place otherwise

OIIlpi _ {Pi OZI 1

o

it also follows readily from (1) that

and hence that P(LT(Z» = Z whenever Z E .1.(P(T); eR). The desired topology can then be introduced by taking as the basic open neighborhoods in spec (!)M the sets L T(.1.(P(T»; eR) for all points TE spec (!)M and all sufficiently small positive numbers e. It is of course necessary to show that these sets satisfy the appropriate conditions to define a topology. In this case it is clearly sufficient merely to show that if T' = LT(Z') for some point Z' E .1.(P(T); eR), then LT(Z) = LT'(Z) whenever Z is sufficiently near Z'. For any function f E (!)M' note from (1) that

Hence the Taylor expansion of the function LT(J, Z) about Z' has the form

=

6)! :! T(:~:!)(Z

- P(T'»J(Z' - P(TW

since Z' = p(T'). On the other hand, T'(f) = LT(f, Z'), so from (1) again LT·(f, Z)

=

1 (OIJ'J) L.., T' ~J (Z J. uZ J

~

P(T'W

~ = ~ J!1 LT (OIJ'J) oZJ' Z' (Z - P(T'W

=

, h J!1 I!1 T (OII+J'J) azI+J (Z -

~

I

~,

P(T» (Z - P(T

» J

H Envelopes of Holomorphy and Riemann Domains

79

Thus, LT(f, Z) = LT'(f, Z) for all f E (!)M and all points Z sufficiently near Z', so = LT'(Z) for all points Z sufficiently near Z' as desired. With the topology thus introduced, it is evident that spec (!)M has the structure of a Riemann domain with projection F: spec (!)M -+ en; for LT is a homeomorphism from an open neighborhood of F(T) in en to an open neighborhood of T in spec (!)M and is the local inverse of the mapping F. It only remains to show that holomorphic functions separate points on this Riemann domain. For any f E (!)M the induced functionjis hoI om orphic on spec (!)M' since whenever T E spec (!)M' the compositionj(LT(Z)) = LT(f, Z) is a holomorphic function of Z near F(T); so by the definition of the spectrum of the Frechet algebra (!)M these functions j separate points on spec (!)M. That suffices to conclude the proof. LT(Z)

8. THEOREM.

In the category of Riemann domains for which holomorphic functions separate points, any such Riemann domain has a unique maximal holomorphic extension.

Proof. Let M be a Riemann domain with projection P: M -+ en, and assume that holomorphic functions separate points on M. It follows from Lemma 7 that spec (!)M has the structure of a Riemann domain with projection F: spec (!)M -+ en, and that holomorphic functions separate points on spec (!)M as well. It was noted before that the natural mapping S: M -+ spec (!)M is injective in this case. Since it was also noted that F 0 S = P, and Fand P are both nonsingular holomorphic mappings, it follows that S is a nonsingular holomorphic mapping as well. Thus, S is a biholomorphic mapping between M and its image S(M) ~ spec (!)M and can be viewed as imbedding M as an open subset of spec (!)M with the induced complex structure. Let E(M) be the union of those connected components of the complex manifold spec (!)M that meet the subset S(M) ~ spec (!)M; then S(M) is also an open subset of the complex manifold E(M) with the induced complex structure, and the natural restriction mapping (!)E(M) -+ (!)S(M) is injective. It was noted in the proof of Lemma 7 that for any function f E (!)M the induced functionjis holomorphic on spec (!)M. Sincejo S = f, it follows that the restriction mapping (!)E(M) -+ (!)S(M) is surjective as well and hence that E(M) is a holomorphic extension of S(M) = M. If E is any other holomorphic extension of M, where E is a Riemann domain with projection P*: E -+ en and holomorphic functions also separate points on E, then as noted before the mapping SE is also injective; so since F 0 SE = P*, it follows that SE is a biholomorphic mapping between E and its image SE(E) ~ spec (!)M. Since S = SEIM, it follows that S(M) ~ SE(E), and since each connected component of SE(E) must contain a connected component of M, it is furthermore clear that SE(E) ~ E(M) ~ spec (!)M. That shows that E(M) is a maximal holomorphic extension of M in this category, and since the uniqueness is obvious, that suffices to conclude the proof.

9. DEFINITION.

If M is a Riemann domain for which holomorphic functions separate points, the envelope of holomorphy E(M) of M is the maximal holomorphic extension of M among all those holomorphic extensions for which holomorphic functions separate points.

The existence of the envelope ofholomorphy and its uniqueness up to biholomorphic mappings are established in Theorem 8. That shows in particular that to

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every open subset D s en there corresponds a unique maximal holomorphic extension E(D) of D within the category of Riemann domains for which holomorphic functions separate points, but as shown by the example of Figure 7, E(D) need not be representable as an open subset of en containing D. The example of Figure 8 also indicates that there can be no analogous theorem without the assumption that holomorphic functions separate points. Actually the proof of Theorem 8 demonstrated somewhat more than what the statement of the theorem indicated, since the envelope of holomorphy E(M) of the Riemann domain M was constructed quite explicitly in terms ofthe spectrum of the Frechet algebra (!)M. This additional information was not emphasized because it is still incomplete in two quite interesting points. First, the envelope of holomorphy E(M) was constructed as a subset of spec (!)M' leaving open the question whether E(M) = spec (!)M. It was shown that the evaluation mapping at each point A E E(M) represents a point in the spectrum spec (!)M' but the question remaining is whether conversely every homomorphism in spec (!)M is just the point evaluation at some point of E(M). That is the case, as will be demonstrated in Theorem HU1 and consequently actually E(M) = spec (!)M. Second, spec (!)M was given the structure of a Riemann domain with projection P: spec (!)M -+ en by imposing an appropriate topology on the set spec (!)M' as described in Lemma 7; but that leaves open the . question of how this topology compares with any more natural or intrinsic topologies on the set spec (!)M. Now the construction of spec (!)M leads immediately to the functions] on spec (!)M associated to the functions f E (!)M. These functions] are all hoI om orphic in the complex structure imposed on spec (!)M' and indeed they are precisely the holomorphic functions on spec (!)M' since it is a hoi om orphic extension of M. The most natural topology to introduce on spec (!)M is the weakest topology in which all these functions]are continuous. The question is then whether this is a weaker topology than that imposed on spec (!)M in Lemma 7. It will be demonstrated in Theorem HIR5 that these topologies actually coincide-that is, that the topology of the complex manifold spec (!)M can be described as the weakest topology in which all global holomorphic functions are continuous. Thus, the eventual result will be that for any Riemann domain M for which holomorphic functions separate points, the envelope of holomorphy E(M) can be identified naturally with the spectrum of the Frechet algebra (!)M' where spec (!)M is given the weakest topology in which all the functions] for f E (!)M are continuous.

I Riemann Domains of Holomorphy

To any open subset D £::: en there is canonically associated its envelope of holomorphy E(D), the maximal holomorphic extension of D in the category of Riemann domains for which holomorphic functions separate points. If E(D) is also an open subset of en, then it is clear from Theorem G8 that E(D) is a domain ofholomorphy. The envelopes of holomorphy of subsets in en, or more generally the envelopes of holomorphy of Riemann domains for which holomorphic functions separate points, are thus natural candidates to be considered the analogues for Riemann domains of domains of holomorphy. From this point of view it is fairly natural to limit the consideration to those Riemann domains for which holomorphic functions are assumed to separate points, as will be done whenever convenient in this section. The extent to which such an assumption is necessary will be discussed in a later section. As in the case of open subsets of en, there are a number of equivalent characterizations of domains of holomorphy among Riemann domains; the characterization chosen here as the definition is the one closest to Definition G 1 but is slightly more complicated to state. Recall that if M is a Riemann domain with projection P: M -+ en, then for any point A E M and polyradius R there are open neighborhoods L\M(A; sR) of A in M which are mapped biholomorphically by P to open polydiscs L\(P(A); sR) in en whenever s < <>M,R(A). For any holomorphic function f E (!JM' the composition fA = f 0 (PIL\M(A; sRW l is a well-defined holomorphic function in an open neighborhood of P(A) in en whenever s is sufficiently small, and the power series expansion of this function about the point P(A) is of course independent of s. This power series converges at least in the polydisc L\(P(A); sR) where s = <>M,R(A), and may for some functions f and some points A converge in a properly larger polydisc.

1. DEFINITION. The radius of convergence of the holomorphic function f A E M in terms of the polyradius R is

E

(!JM at the point

PM.R(f; A) = sup{s: the power series expansion of £. about P(A) converges in L\(P(A); sR)} 81

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2. DEFINITION. A Riemann domain of hoiomorphy is a Riemann domain M for which there exists at least one function f E (9M such that PM.R(f; A) = c5M.R(A) for any polyradius R and any point A EM. Since it is always the case that PM.R(f; A) ~ c5M.R(A), the critical property in this definition is just that PM.R(f; A) ~ c5M. R(A); in a sense that amounts to the condition that f cannot be extended to a holomorphic function in any properly larger manifold. Obviously this definition reduces to Definition G 1 in the special case that M is an open subset of en. Any domain of holomorphy in en is thus an example of a Riemann domain of holomorphy. The following very simple observations are worth noting explicitly here.

3. LEMMA. A Riemann domain of holomorphy admits no properly larger holomorphic extension. Proof. Suppose to the contrary that M is a Riemann domain of holomorphy that does admit a properly larger holomorphic extension E. There must be some point A E Me E and polydisc ~E(A; BR) ~ E such that ~E(A; BR) is not contained in M and hence such that c5M.R(A) < B. Any f E (9M extends to a holomorphic function on E, so that evidently PM.R(f; A) ~ B > <>M.R(A). But since M is a Riemann domain of holomorphy, there exists at least one function f E (9M for which PM.R(f; A) = <>M.R(A), and that contradiction suffices to conclude the proof.

4. LEMMA. If for a Riemann domain M there exist a point A E M, a polyradius R, and a constant <> > <>M.R(A) such that PM,R(f; A) ~ <> for every f E (9M' then M admits a properly larger holomorphic extension. Proof.

Let E' be the disjoint union of the Riemann domain M and the polydisc <>R) ~ en. With the projection P of the Riemann domain M on the subset M ~ E' and the identity mapping on the polydisc ~(P(A); <>R) ~ E', the set E' has the structure of a Riemann domain. Any function f E (9M can be viewed as a holomorphic function on the subset M ~ E'. Since f 0 (PI~M(A; <>M.R(A)RW 1 is holomorphic in ~(P(A); <>M.R(A)R) and by hypothesis extends to a holomorphic function on ~(P(A); <>R), this provides an extension of f to a holomorphic function on all of E'. The class offunctions so constructed form a subalgebra (9;'. ofthe algebra (9E' of all holomorphic functions on E', and under the natural restriction mapping f E (9E' -+ flM E (9M' this subalgebra is isomorphic to (9M' Now introduce the equivalence relation defined by setting Z '" B where Z E ~(P(A); <>R) and B E M whenever f(Z) = f(B) for all f E (9;'.. It follows readily as in the proof of Theorem H4 that the space of equivalence classes E = E' / '" is a Riemann domain with (9E = (9;'., and it is evident that E is a properly larger extension of M. That suffices to conclude the proof. ~(P(A);

These two lemmas taken together suggest but do not quite demonstrate that Riemann domains of holomorphy are the same as Riemann domains admitting no properly larger holomorphic extension.s. For open subsets of en this equivalence

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83

was easily shown by using the auxiliary concept of holomorphic convexity; that notion is equally useful in the case of Riemann domains.

5. DEFINITION. A Riemann domain M is holomorphically convex if for any compact subset K S;;; M the set KM

=

{A EM: If(A)1 ~ IlfilK for allf E (l)M}

is also compact. The set KM is the holomorphically convex hull of Kin M.

The basic result that Riemann domains of holomorphy are the same as holomorphically convex Riemann domains is unfortunately considerably more difficult to prove than the corresponding result for open subsets of en. In one direction the proof is essentially the same, as follows.

6. THEOREM.

If M is a holomorphically convex Riemann domain, then M is a Riemann domain of holomorphy.

Proof. Suppose that M is a holomorphically convex Riemann domain, which can of course be supposed to be connected. Since M is second countable by Theorem H3, it is possible to choose a sequence of compact subsets K. s;;; M, such that K. S;;; interior K.+l and U. K. = M, and a countable dense sequence of points A. E M. The holomorphically convex hulls K. are compact, so if L\M(A.; R.) is a maximal polydisc in M centered at A., there must be a point Z. E L\M(A.; R.) such that Z. ¢ K•. Then, as in the first part of the proof of Theorem G5 it follows readily that there are functions f. E (l)M for which f.(Z.) = 1 and IIf.IIK. < v- 12-·. The infinite product f = 0.(1 - f.t is then a nontrivial holomorphic function on M and has total order at least v at the point Z •. In order to complete the proof, it is enough just to show that PM,R(f; A) ~ bM,R(A) for any polyradius R and any point A E M. Suppose to the contrary that PM,R(f; A) > bM,R(A) for some polyradius R and some point A EM. If Us;;; M is the connected component of P-l(L\(P(A); PM,R(f; A)R» containing L\M(A; bM,R(A)R), it is clear that L\{P(A); bM,R(A)R)

S;;;

P(U) c: L\(P(A); PM,R(f; A)R)

S;;;

en

with the second inclusion necessarily a proper inclusion, and that there exists a point BE L\(P(A); PM,R(f; A)R) such that BE oL\(P(A); bM,R(A)R) 11 oP(U). There must exist a subsequence {A •.J } of the countable dense sequence {A.} such that A •.J E U and P(A.) ~ B. Since B ¢ L\(P(A.); R •. ), the polyradii R •. must approach zero, so P(z.) ~ B and it can be assumed that L\~(A'j; R.) S;;; u. N~w the nontrivial function fA = f 0 (PI U)-l extends to a holomorphic function in all of L\(P(A); PM,R(f; A)R) and has total order ~ Vj at the points P(Z.) converging to B; but as in the proof of

Theorem G5 that is impossible, and with that contradiction the proof is concluded. The converse ofthe preceding theorem is considerably harder to demonstrate. The direct extension to Riemann domains of the second half of the proof of Theorem G5 really yields only the following weaker result.

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7. THEOREM. If M is a Riemann domain of holomorphy, then for any compact subset K eM and any polyradius R,

Proof. Since K ~ K M, it is of course trivial that <5M,R(K) ~ <5M,R(KM). Suppose that, to th~ contrary of the desired result, <5M,R(K) > <5M,R(KM). There is then a point A E KM for which <5M,R(A) < <5M,R(K) = <5. Now for any function f E (!)M the argument of Lemma G4 shows that the power series expansion about the point P(A) of the function fA = f 0 (PIAM(A; <5M,R(A)RW 1 converges in the polydsic A(P(A); <5R) and hence that PM,R(f; A) ~ <5 > <5M,R(A). But that contradicts the hypothesis that M is a Riemann domain of holomorphy, and with that contradiction the proof is completed.

For the case that M is an open subset of en the conclusion of the preceding theorem implies that KM is disjoint from an open neighborhood of 8M and is consequently compact; thus, any Riemann domain of holomorphy in en is holomorphically convex, However, a general Riemann domain may be infinitely sheeted over en, in which case the conclusion of the preceding theorem does not directly imply that KM is compact. In special cases the argument can be salvaged, though. It is convenient to say that a Riemann domain M with projection P: M _ en is locally finitely sheeted if the number of points in p- 1 (Z) is a locally bounded function of Z in en. 8. THEOREM. If M is a Riemann domain of holomorphy and is locally finitely sheeted, then M is holomorphically convex. Proof. If K is any compact subset of M, then the coordinate functions of the projection P: M - C" are bounded on KM by their bound on K, so that the image P(K M) is a bounded subset ofC". Therefore for any sequence {A.} of points of K M, after passing to a subsequence if necessary, it can be assumed that the projections P(A.) converge to some point Z E en. If 0 < e < <5M,R(K) = <5M,R(KM) for some polyradius R, then for all sufficiently large indices v it is clear that Z E P(AM(A.; eR»; hence, there must exist points B. E AM(A.; eR) such that P(B.) = Z. However, there are only finitely many distinct such points B., since M is locally finitely sheeted; so for one of these points B, there must be infinitely many points A. E AM(B; eR). Now these points A. must then converge to B, and B E KM since KM is a closed subset of M. That shows that KM is compact and concludes the proof.

It should be pointed out that the argument of the preceding theorem shows that if M is a locally finitely sheeted Riemann domain with projection P: M - C" and if L is any closed subset of M such that P(L) is bounded and bM,R(L) > 0 for some polyradius R, then L is necessarily compact. If M is not locally finitely sheeted, this is of course not true. It is still true, though, that a Riemann domain of holomorphy is holomorphically convex, but the proof is considerably more involved. One proof using properties of plurisubharmonic functions and the notion

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Riemann Domains of Holomorphy

85

of pseudoconvexity will be given in section P, after the introduction of the relevant additional machinery. To bring this part of the discussion to a reasonable conclusion for those readers who do not care to get involved in such machinery, an ingenious elementary proof due to E. Bishop will be included here. That proof requires the additional hypothesis that holomorphic functions separate points, a hypothesis that will not be needed in the later treatment of this topic. Those readers not inclined to go into the rather special argument needed for this proof can skip the remainder of this section and wait for the completion of this circle of ideas later. 9. LEMMA. Let M be a Riemann domain with projection P: M --+ en, and assume that holomorphic functions separate points on M and that (jM,R(K) = (jM,R(K M) for every compact subset K ~ M for some polyradius R = (r l , ... , rn). If M is not holomorphically convex, there exists a compact subset L ~ M such that for some e > 0 the set LM contains irifinitely many disjoint open polydiscs A. = AM(A.; R) with P(A.) = Zo for all indices v. Proof. If M is not holomorphically convex, there exists a compact subset K ~ M such that KM is not compact. Choose a number e such that 0 < 4e < (jM R(K) = (jM,R(K), and introduce the compact subset ' L =

U KM(A; 4eR) ~ M AEK

and the open subsets L* =

U

AM(A; 2eR) ~ M

AEKM

and L** =

U

AM(A; 4eR) ~ M

AEKM

The set L * is itself a Riemann domain with the projection PI L *, and KM is a closed subset of L* such that P(K M) is bounded and (jL*,R(K M ) = 2e > O. This Riemann domain L * cannot be locally finitely sheeted, for otherwise, as noted after the proof of Theorem 8, the set KM would be compact. Therefore, for any positive integer Il there is a point Z/l E en such that P-l(Z/l) n L* contains at least Il distinct points Since P(L*) is bounded, it can be assumed by passing to a subsequence if necessary that the points Z/l converge to some point Zo E en, and for all indices Il sufficiently large, Zo E A(Z/l; eR). For any such indices Il, the Il polydiscs AM(B~; 2eR) are well-defined disjoint open subsets of L**; for since Bi/l E L*, it follows that . ' . ~ B~ E AM(C~; 2aR) ~ AM(C~; 4aR) for some C~ E K M. There must furthermore be some point A~ E AM(B!; 2aR) such that P(A~) = Zoo The Il polydiscs AM(Ai; aR) are then well-defined disjoint open subsets of L **. Since this is the case for all s;fficiently large indices Il, there mu.st actually be infinitely many distinct polydiscs A. among all these polydiscs AM(A~; aR).

B;, ... ,B:.

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To complete the proof it is only necessary to show that L ** ~ LM • For any given point BE L**, there must be some point A E KM such that B E L\M(A; 4&R). Now any function f E (!)M has a power series expansion in L\M(A; 4&R) in terms of the coordinates induced on that polydisc by the projection P; and in terms of that expansion,

(CP1) az (A)(P(B) -

1 f(B) = ~ I!

Recalling that A

~

l

E

P(A))1

KM and using the Cauchy inequalities, Theorem AS, show that

IlflldI!)(4&R)-1

Consequently, If(B) I ~ L Ilflld4&R)-I(P(B) - P(AW 1

~

CllfllL

where C = LI(4&Rr l (p(B) - P(AW < 00, since P(B) E MP(A); 4&R). The same inequality also holds for the function for any positive integer v, so that necessarily

r

If(B) I ~ Cl/' IlfilL

So if v tends to infinity, it follows that If(B)1 ~ suffices to conclude the proof.

IlfilL and hence that BE LM. That

It will next be demonstrated that the conclusion of Lemma 9 leads to a contradiction ifit is assumed in addition that holomorphic functions separate points on M. With the notation as in the statement of Lemma 9, for any point Z E L\(Zo; &R), let A.(Z) be that point of the polydisc L\. = L\M(A.; &R) for which P(A.(Z)) = Z; thus, in particular A. = A.(Zo). 10. LEMMA.

If in addition to the hypotheses of Lemma 9 it is assumed that holomorphic functions separate points on M, then there is a holomorphic function f E (!)M such that {Z E L\(Zo; &R}: f(A,,(Z}) = f(A.(Z» for some J1. :F v} is a subset of L\(Zo; &R) of measure zero.

Proof. Since holomorphic functions separate points on M by hypothesis, it is clear that for any indices J1. :F v the set {J E (!)M :f(A,,) :F (A.}} is a dense open subset of the Frechet space (!)M' The intersection of all these sets for all indices J1. :F v is then nonempty by the Baire category theorem; hence, there exists a function f E (!)M such that f(A,,) :F f(A.} whenever J1. :F v. Now whenever J1. :F v, the set

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Riemann Domains of Holomorphy

87

{Z E A(Zo; eR) : f(AiZ)) = f(Av(Z))} is a proper subset of the polydisc A(Zo; eR) and hence by Corollary A9 must be a subset of measure zero in A(Zo; eR). The union of all these sets for all indices Jl. =F v is then also of measure zero, and the proof is thereby concluded.

In terms of the function f of the preceding lemma, introduce for each point = {f(A v (Z)) } consisting of the values of the function f at the various points Av(Z). The conclusion of Lemma 10 implies that Tz* is an infinite point set for all points Z E A(Zo; eR) outside a subset of measure zero, and the desired contradiction will be found by constructing a polynomial function vanishing on Tz* for all points Z E A(Zo; eR) in a subset of positive measure. To simplify the description of this construction it is convenient to introduce some temporary terminology. A polynomial Z

E

A(Zo; eR) the set Tz*

is of degree(a l ,

... ,

a") ifit is of degree aj in Xj' and is normalized ifmax vIFv, ... v",

=

1.

Let K be a compact subset of a Riemann domain M with projection P: M --+ en, and let f be a function in (!)M. There are a constant r < 1 and an integer Po > 0 such that for all a ~ P ~ Po there exists a normalized polynomial F E C [X 1, ... , X" +1 ] of degree (a, ... , a, P) for which IF(P(A), f(A))1 ~ raP'!" whenever A E K.

11. LEMMA.

Proof. Choose a constant e with 0 < e < dM(K), and choose finitely many points Aj E K, 1 ~ j ~ N, such that the balls BM(Aj; e) cover K. The set K. = E K BM(A; e) is also compact, so there is a constant c ~ 1 such that Ipj(A)1 ~ c and If(A)1 ~ c whenever 1 ~ j ~ n and A E K •. The set of all polynomials F E C[X 1 , .•• , Xm + l ] of degree (a, ... , a, P) form a complex vector space of dimension (a + I)"(P + 1). There exists a nonzero polynomial F in this space such that the function F = F(P, f) E (!)M satisfies

UA

whenever 1 ~ j ~ N and III < y, where y is the greatest integer strictly less than N-1/"(a + I)(P + WI" and as usual differentiation is with respect to the natural coordinates imposed on M by the projection P; for the number oflinear conditions this imposes on the space of polynomials is at most y"N, and y"N < (a + I)"(P + I). This polynomial F can even be taken to be normalized; it thus consists of at most (a + 1)"(P + 1) monomials Fv 1 ... v"+1 XVI'··· X"v~rl' with IFv 1 ... Vn +I:S: 1. From the lchoice of the constant c, note that IF(A)I ~ (a + I)"(P + l)c"a+ P whenever A E K •. lt then follows from Schwarz's lemma, Theorem A6, that

whenever A

E

BM(Aj; e). There is a constant (j in the interval 0 <

(j

< 1 such that the

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balls BM(Aj; &) also cover K. Hence,

whenever A E K. By the choice of the constant y, note that y + 1 ~ N- 1/ n(IX (P + 1)1/n ~ N- 1/nIXp 1/n. Hence,

whenever A

E

+ 1)

x

K, where

log r(IX, P) = N- 1/n log ()

+ IX- 1p- 1/n[n log (IX +

1)

+ 10g(P +

1)

+ (nIX + P) log c -log ()]

Clearly, log r(IX, P) ~ log r for some r < 1 whenever IX ~ P ~ Po for sufficiently large Po, since N- 1/ n log () < 0 and the remaining terms in the above expression for log r(IX, P) tend to zero as Po becomes large. That suffices to conclude the proof. 12. LEMMA. For any polydisc ~(O; R) s en with 0 < rj < 1, there are positive integers IXo, Y such that if FEe [X l ' ... , Xn] is any normalized polynomial of degree «(1., ••• , (1.) with (1. ~ IXo, and if St = {Z E ~(O; R): IF(Z)I < t"'} for some t in the interval 0 < t < 1, then the Lebesgue measure IStl of the set St satisfies IStl ~ y/log C 1. Proof.

Since F is normalized, there is a multi-index I such that oiliF OZI (0) = I!

For Sj

= (1

- rj )/3, note that

I~(O; S)IIIFII&(o;s) ~ f

IF(Z)I dv(Z)

&(O;S)

~

S

(2n)n oiliF ( - I I ~Zl (0)

(2n)"

n

Sj

Pj=O

~

•

j

(

U

np/

J+

J

s!.t+ 2 ) _.1_ Ij

+2

Hence there is a point Zo E A(O; S) such that

) 1 dp1··· dPn

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Riemann Domains of Holomorphy

89

Letting .1 = .1(Zo; (1 + 2rj)/3) and using the preceding inequality and Jensen's inequality, Theorem A8, show that

;;:: nnn ( -

1 + 2ro)2 log I 2 n __ (s 3 (a + l)(a + 2)

.. s )IZ/n

J

j

0

1

I

n

Now note also that .1 ~ .1(0; 1); so since F is a sum of at most (a + It monomials Fv,ooovnzl' ... z;n with IFv,oo.v", ~ 1, it follows thatl(a + l)-nF(Z)1 ~ 1 in.1, or equivalently that logl(a + 1)-nF(Z)1 ~ 0 in .1. Note further that .1 ;;2 .1(0; R) ;;2 S, and

f

logl(a

+ l)-nF(Z)1 dV(Z)

~

&

~

r logl(a + l)-nF(Z) I dV(Z)

J~

Js,r 10gIF(Z)1 dV(Z) < IS,la log t

Combining this and the preceding inequality, and recalling that log t < 0, imply that

1 (1 n + [1n

IS,I ~ -----1 nnn log t

- 32ro)2 -J -log(sl ... Sn)

j

0

1

-log(a

(X

(X

+ -log 2 -

1

+ 1) -

1

-log(a

+ 2)

]

(X

<-y= log C 1 for some constant y whenever (X is sufficiently large, since the expression in brackets converges to (lin) log(s1 ... sn) as n tends to infinity. That suffices to conclude the proof. 13. THEOREM, If M is a Riemann domain for which holomorphic functions separate points, then the following conditions are equivalent:

Mis holomorphically convex. M is a Riemann domain of holomorphy. M admits no properly larger holomorphic extension. M = E(M) For a~ polyradius R and any compact subset K ~ M, necessarily bM,R(K) = bM,R(KM)· (vi) There exists afixedpolyradius Rsuch that bM,R(K) = bM,R(KM)!orany compact subset K £; M.

(i) (ii) (iii) (iv) (v)

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Volume I Function Theory

Proof. That (i) implies (ii) follows from Theorem 6, and that (ii) implies (iii) follows from Lemma 3. It is fairly obvious that (iii) and (iv) are equivalent, for (iii) obviously implies (iv), while if M admits any properly larger holomorphic extension, the construction of Theorem H4 easily leads to a properly larger holomorphic extension for which holomorphic functions separate points. That (ii) implies (v) was proved in Theorem 7, and essentially the same argument shows that (iii) implies (v). Indeed, if there exist a polyradius R and a compact subset K £ M for which c5M,R(K) > c5M,R(K M), then there is a point A E KM for which c5M,R(A) < c5M,R(K) = c5, and as in the proof of Theorem 7, it follows rather readily that PM,R(f; A) ~ c5 > c5 M,R(A) for any function f E (1)M; but then Lemma 4 shows that M must admit a properly larger hoI om orphic extension. It is obvious that (v) implies (vi). Then to complete the proof, it is only necessary to show that (vi) implies (i), which is the only difficult part of the whole argument. To proceed by contradiction, suppose that (vi) holds but that M is not holomorphically convex. It follows from Lemmas 9 and 10 that there exist a compact subset L £ M, an open polydisc .1(Zo; eR) £ en for some e > 0, and a function f E (1)M' such that Tz = {J(A): A ELM and P(A) = Z} is a finite point set only for a subset of points Z E .1(Zo; eR) of measure zero. Here as usual P: M -+ Cn is the projection of the Riemann domain M. To simplify the notation it can be assumed that Zo = O. It then follows from Lemma 11 that there are a constant r < 1 and an integer Po > 0 such that for all (X ~ P ~ Po there exists a normalized polynomial Fap E C[X 1 , ... , X n+1] of degree «(X, ... , (X, P) for which lFap(P(A), f(A))1 ~ r ap "

n

(1)

whenever A E L, and the same inequality of course also holds whenever A ELM' If FaP(X) = fapv(X l' ... , X n)X:+ 1 , the polynomials fapv E C [X l' ... , Xn] are of degree «(X, ... , (X), and for at least one index Vo = vo«(X, P) the polynomial fapvo = fap is normalized. For these polynomials fap it follows from Lemma 12 that there are positive integers (xo and y such that whenever (X ~ (xo, the Lebesgue measure of the set

Ie=o

satisfies

2y I -1 ISap 1 < = p1/n og r

(2)

for e can be supposed chosen sufficiently small that erj < 1. Now fix an integer P ~ Po sufficiently large that ISapl < !1.1(0; eR)1 for all (X ~ (Xo; that is certainly always possible in view of the inequality (2). If Z E .1(0; eR) and if there are infinitely many values (X ~ max(p, (Xo) such that Z ¢ Sap, then for each such (x, max Ifapv(Z) I ~ I fap(Z) I ~ r(l/2)ap' /n v

Furthermore, for each such (x, the polynomial Ua(Xn+1 ) = Fap(Z, X n+ 1 )/max v Ifapv(Z)1

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Riemann Domains of Holomorphy

91

is a normalized polynomial of degree /3; and whenever a E Tz , from (1) it follows that g,,(a) 1 ~

1

r(1/2)"P"n

(3)

Now for some subsequence of these indices (x, the normalized polynomials g" converge to a normalized polynomial g, which is also of degree at most /3, and it follows from (3) that in the limit g(a) = O. Therefore, whenever Z E L1(O; eR) is such that Z ¢ S"p for infinitely many values of (x, it follows that the set Tz consists of at most /3 points. Now if S*p = lim infIX S"p = {Z E L1(O; eR) : Z E S"p for all but finitely many (X}, it is an elementary result from measure theory that IS*pl = llim inf"S"pl ~ lim sup" 1S"p I· Since /3 was chosen so that 1S"p 1 ~ tlL1(O; eR)I, it follows that IS*pl < 1L1(O; eR)I. Thus the complement L1(O; eR) - S*p is a set of positive measure with the property that Tz is finite whenever Z E L1(O; eR) - S'I'P' But that contradicts the result that Tz is finite only for a subset of L1(O; eR) of measure zero, and thereby concludes the proof. It should be mentioned here that there are still other conditions on a Riemann domain M that are equivalent to M being a Riemann domain of holomorphy, paralleling results already obtained for subsets of en. For example, the proof of Theorem G7 carries over immediately to the case of Riemann domains; thus, if M is a Riemann domain for which holomorphic functions separate points, then M is holomorphically convex if and only if for any discrete sequence of distinct points A. E M there exists a function f E (!)M such that lim suP. If(A.)1 = 00. Furthermore, the argument of Corollary G6 shows that condition (v) of Theorem 13 is equivalent to the same condition for the Euclidean distance function dM • Since these results will not be needed here, the details will be omitted. Finally, since the envelope of holomorphy E(M) of any Riemann domain M for which holomorphic functions separate points is clearly a Riemann domain of holomorphy, it follows from the preceding theorem that E(M) is holomorphically convex.

J Subharmonic Functions

Although subharmonic functions have played a rather minor role in the classical theory of holomorphic functions of one variable, the situation is quite different for the theory of holomorphic functions of several variables. A special class of subharmonic functions has implicitly played a significant role there from the time of the pioneering work of Hartogs early in the 20th century. For the sake of completeness a brief review of some of the basic properties of subharmonic functions of one complex variable will be included in this section to provide background for the discussion of the relevant special class of subharmonic functions of several complex variables in the next section. The functions to be considered here are not just real-valued functions defined on open subsets D £;; C but rather mappings from D into the half-open real line [-00, +(0); thus, these functions have as their values either real numbers or -00. Recall that such a mapping u is upper semicontinuous if {z E D: u(z) < r} is an open subset of D for every real number r, or equivalently if u(a)

~ lim sup u(z) = z-+a

lim ( .-+0

sup O
U(Z»)

(1)

for every point a E D. An upper semicontinuous mapping u on an open subset D £;; C is bounded from above on any compact subset K £;; D; for the sets {z E K: u(z) < r} for all real numbers r are relatively open in K and cover K, so since K is compact, finitely many of these sets serve to cover K. An upper semicontinuous mapping is Lebesgue measurable, and its integral over any compact set has a well-defined value, either a real number or - 00. 1. DEFINITION.

A mapping u: D -+ [-00, +(0) defined in an open subset D £;; C is a subharmonic function if: (i) u is upper semicontinuous in D, and (ii) each point a E D has an open neighborhood Ua

1 u(a) ~ 2n

f21t

Jo

u(a

whenever ~(a; r) !: Ua.

£;;

D such that

+ re i8 ) dO

(2)

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Subharmonic Functions

93

There a number of equivalent characterizations of subharmonic functions, which will be discussed in this and the next section. The particular characterization chosen as the definition here is one of several commonly used as the definition and is perhaps the most primitive one. It is familiar that harmonic functions satisfy (2)-indeed, with an equality in place of the inequality. That is one way in which subharmonic functions can be viewed as lying below harmonic functions, motivating the terminology; other ways will be discussed later in this section, especially in Theorem 7. Condition (i) of the preceding definition is really just a regularity condition imposed on the mapping u. Some sort of regularity is of course necessary in order that the integral in (2) can be defined, and upper semicontinuity is really the most natural form of regularity condition to use for subharmonic functions. Intuitively the integral inequality (2) implies that the value of a subharmonic function u at a point a is bounded from above in terms of the values of the function u near a, while the upper semicontinuity inequality (1) implies that the value of u at a point a is bounded from below in terms of the values of u near a. So conditions (i) and (ii) of Definition 1 are in some sense naturally complementary. The deeper reasons that upper semi continuity is the natural regularity condition to impose will be discussed in the next two sections. As a word of caution, it should be pointed out that the function that is identically equal to - 00 is subharmonic according to this definition but is not considered a subharmonic function in other definitions. A brief catalog of some of the elementary general properties of subharmonic functions is as follows. 2. THEOREM. [ -00,

Let D be an open subset of C and let u, u 1 , +(0).

U2

be mappings from D into

(a) A mapping u is subharmonic in D precisely when it is sub harmonic in an open neighborhood of each point of D. (b) If u is subharmonic in D and if u(a) = sUPz E D u(z) for some point a E D, then u is constant in the connected component of D containing a. (c) If u is subharmonic in D, then u(a) = lim SUPz-+a u(z) for each point a E D. (d) If u is subharmonic in D and if ¢J: IR --+ IR is a convex and monotonically increasing function, then the composition ¢J 0 u is also subharmonic in D, where ¢J( -(0) = lim x -+_ oo ¢J(x). (e) If u 1 , U2 are subharmonic in D and if c 1 , C 2 are positive real numbers, then the mapping u defined by u(z) = C 1 U 1 (Z) + C2U2(Z) is also subharmonic in D. (f) If u 1 , U2 are subharmonic in D, then the mapping u defined by u(z) = sup(u 1 (z), U 2 (z)) is also subharmonic in D. (g) Any harmonic function is also subharmonic, and if both u and - u are subharmonic, then they are actually harmonic.

Proof.

(a) This assertion is an immediate consequence of the definition, since upper semicontinuity is a local condition and the integral inequality (2) is only required. to hold locally. (b) Suppose that u is subharmonic in D and that u(a) = SUPzEDU(Z) for some point a E D. If u(a} = - 00, then of course u(z) = - 00 for all points ZED

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Function Theory

and the desired result is trivially true; so assume that u(a) "# - 00. The set E = {z ED: u(z) = u(a)} = {z ED: u(z) ~ u(a)} is a closed subset of D, since u is upper semicontinuous. On the other hand, if bEE and if Ub is an open neig~borhood of b in which the inequality (2) holds, then u(z) = u(b) = u(a) whenever z E A(b; r) ~ Ub; for if ~(b; r) ~ Ub and u(b + re i80 ) < u(b) for some Bo, then u(b + re i8 ) < u(b) for all values B sufficiently near Bo , since u is upper semicontinuous, but u(b + re i8 ) ~ u(b) for all values of B, so it follows from (2) that 1 u(b) ~ 2n

Jr

27t

0

u(b

+ re i8 ) dB <

u(b)

a contradiction. Therefore, E is also an open subset of D, and since a E E, it follows that E contains at least the connected component of D containing a, as desired. (c) Since u is upper semicontinuous, u(a) ~ lim SUPz_a u(z) for any point a E D. If u(a) > lim SUPz_a, then from (1) it follows that u(a) > sUPO O. But in that case u(a) = sUPzeA(a;E) u(z), so as a consequence of part (b) of this theorem u(z) = u(a) whenever z E A(a; Il), and that is impossible since u(z) < u(a) whenever 0 < Iz - al < Il. That shows that the desired equality holds. (d) Suppose that u is subharmonic in D and that rfo: ~ --+ ~ is a convex and monotonically increasing function. The composition rfo 0 u is upper semicontinuous, since rfo is continuous and monotonically increasing and u is upper semicontinuous. The convexity of rfo implies that for any point Xo E ~ there is a real constant c such that rfo(x) ~ rfo(xo) + c(x - xo) for all x E ~. In particular, for Xo = u(a) "# - 00 and x = u(a + re i8 ), this inequality becomes

+ re i8 )) ~ rfo(u(a)) + cu(a + re i8 )

rfo(u(a

-

cu(a)

If Ua is an open neighborhood of a in which the inequality (2) holds, then from that inequality it follows that 1 2n

Jr

27t

0

rfo(u(a

c

+ re i8 )) dB ~ rfo(u(a)) + 2n ~

rfo(u(a)

Jr

27t

0

+ cu(a) -

u(a

+ re i8 ) dB - cu(a)

cu(a)

= rfo(u(a))

whenever ~(a; r) ~ Ua, so that rfo 0 u is subharmonic as desired. (e) This assertion is an almost immediate consequence of the definition of subharmonic funtions. (f) Suppose that U 1 and U 2 are subharmonic in D and that u is defined by u(z) = sup(u 1 (z), u 2 (z)). The function u is upper semicontinuous in D, since {z ED: u(z) < r} = {z ED: u1(z) < r} n {z E D: u 2 (z) < r}. If a E D and if Ua is an open neighborhood of a in which the inequality (2) holds for both Ul and u 2 , then from that inequality and the observation that Uj(z) ~ u(z) it follows that

J

for j = 1 or 2 whenever L\(a; r)

S;;;

1 u(a) = sup(u 1 (a), u2(a» ~ 2n

Subharmonic Functions

95

Uu • Therefore,

J(2" u(a + re 0

i9 )

dO

whenever L\(a; r) S;;; Ua , so that u is subharmonic as desired. (g) Recall that the harmonic functions in an open subset D S;;; C can be characterized as those mappings u: D -+ ~ such that (i) u is continuous in D, and (ii) each point a E D has an open neighborhood Ua such that 1

u(a) = 2n

Je" u(a + re 0

i9 )

dO

(3)

whenever L\(a; r) s;;; Ua • It is then obvious that any harmonic function is subharmonic. On the other hand, if both u and - u are sub harmonic, then u is both upper and lower semicontinuous and so is necessarily continuous, while both u and - u satisfy the inequality (2) so u necessarily satisfies the equality (3). Therefore, u is harmonic, and - u is of course also harmonic. That suffices to conclude the proof of the entire theorem. The first part of the preceding theorem can be rephrased as the assertion that subharmonicity is a local property; that is a very useful observation and will be used repeatedly in the sequal. Part (b) ofthe theorem can be viewed as a maximum theorem for subharmonic functions, analogous to the maximum modulus theorem for holomorphic functions but slightly weaker. The maximum modulus theorem for holomorphic functions, Theorem A4, asserts that a holomorphic function that attains a local maximum modulus at a point in a connected set D must be constant in D, whereas the theorem above refers to a global maximum value for a subharmonic function. Of course, if a subharmonic function u attains a local maximum value at some point a, it follows that u must be constant in an open neighborhood of a, but there is no identity theorem for subharmonic functions, so such a function need not be constant everywhere. For example, if u is any subharmonic function in a connected open set D S;;; C, then it follows from part (f) of the theorem that the function u+ defined u+(z) = sup(u(z), 0) is also subharmonic. The function u+ is identically zero in the open subset {z ED: u(z) < O}, which may be a nonempty proper open subset of D, but u+ is not necessarily identically zero throughout D. Part (c) of the theorem shows that subharmonic functions necessarily satisfy a regularity condition somewhat stronger than upper semicontinuity; it follows naturally from the interaction of the two complementary conditions involved in the definition of subharmonicity. Part (e) of the theorem can be interpreted as the statement that the set of all subharmonic functions in an open subset D S;;; C form a convex cone. The subharmonic functions do not form a vector space over ~, though, since - u need not be a subharmonic function if u is a subharmonic function. Indeed, it is clear from part (g) of the theorem that the maximal linear subspace contained in the cone of subharmonic functions in D consists precisely of the harmonic functions inDo

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The interrelations between subharmonic and harmonic functions can be used to derive some of the deeper properties of the former from known properties of the latter. As a first instance of this it will be demonstrated that the basic inequality (2) for subharmonic functions holds not just locally but globally. As a convenient notation for this purpose, for any subharmonic function u in an open subset D s;;; C and any circle {z E C : Iz - al = r} = a~(a; r) contained in D introduce the circular mean 1

Mu(a; r) = 211:

f271 0

u(a

(4)

+ re i6 ) dO

This defines a function Mu of the variables a and r in the region {(a, r) E C x [0, (0): a~(a; r) s;;; D}, where the values of this function are either real numbers or - 00. Note particularly that for (4) to be defined, it is only necessary for the bounding circle a~(a; r) to be contained in D; it is not required that the full closed disc A(a; r) lie in D, not even that its center a lie in D.

3. THEOREM. If u is subharmonic in an open set D s;;; C, then u(a) ~ Mu(a; r) whenever A(a; r) S;;; D. Moreover, Mu(a; r) is a monotonically increasing function of the variable r in any interval [0, R) such that A(a; r) S;;; D whenever ~ r < R.

°

Proof. For any closed disc A(a; r) S;;; D the restriction ofu to the boundary a~(a; r) of that disc is an upper semicontinuous mapping on that boundary circle. As is well known, there is a monotonically decreasing sequence of continuous real-valued functions I. on aMa; r) such that fv(z) converges to u(z) at each point z E a~(a; r). Indeed, the functions fv defined by

fv(z) =

sup {ea&(a;r)

[u(,) -

v·" -

zl]

have the desired properties. As is also well known, each function fv on a~(a; r) can be extended to a function Fv that is continuous on the closed disc A(a; r) and harmonic in its interior ~(a; r). Indeed, the function Fv is given by the Poisson integral

Since u(z) ~ J.(z) = Fv(z) whenever z E a~(a; r), it follows from the maximum theorem for subharmonic functions, Theorem 2(b), that u(z) ~ Fv(z) for all points z E A(a; r). Indeed, u - Fv is upper semicontinuous on A(a; r) and so must attain its maximum value there at some point Zo E A(a; r), and if u(zo) - Fv(zo) > 0, then necessarily Zo E ~(a; r), so by Theorem 2(b) the function u - Fv must be constant on ~(a; r), contradicting the assumption that it is positive at Zo and zero on aMa; r). In particular, it follows from this observation that u(a) ~ Fv(a). Now since F. is harmonic in ~(a; r) and continuous on A(a; r), it follows from (3) that F.(a) = MF.(a; r) = MJ.(a; r), and lim. Mf.(a; r) = Mu(a; r) by a straightforward

J

Subharmonic Functions

97

application of the Lebesgue monotone convergence theorem. Consequently, u(a) ~ Mu(a; r). On the other hand, it is also the case that u(a + pe i9 ) ~ Fv(a + pe i9 ) for all p ~ r and all e, so Mu(a; p) ~ MFv(a; p); but again Mf.(a; p) = Fv(a) = Mf.(a; r) and lim vMfv(a; r) = Mu(a; r), so that Mu(a; p) ~ Mu(a; r). Thus, the function Mu(a; r) is monotonically increasing as a function of r, and the proof is thereby concluded. This establishes that the basic inequality (2) holds for all discs contained in the set in which u is subharmonic, not just for sufficiently small discs. It is then possible to consider infinite sequences of subharmonic functions in a fixed domain with the assurance that (2) will hold for all the functions simultaneously. That leads to the following additional entry in the catalog of elementary general properties of subharmonic functions. 4. THEOREM. If {uv } is a monotonically decreasing sequence of sub harmonic functions in an open subset D ~ C, then u = limv U v is also subharmonic in D. The function u is upper semicontinuous in D, since {z ED: u(z) < r} = Uv {z ED: uv(z) < r} for any real number r. For any closed disc L\(a; r) ~ D it follows from Theorem 3 that uv(a) ~ Muv(a; r); but u(a) ~ uv(a) and limv M uv(a; r) = Mu(a; r) by a suitable application of the Lebesgue monotone convergence theorem, so u(a) ~ Mu(a; r). That shows that u is subharmonic in D and concludes the proof. Proof.

Theorem 3 can be viewed as a form of regularity theorem for subharmonic functions, since it clearly implies that whenever u is subharmonic in an open set D ~ en and u(a) # - 00 at a point a E D, then u is integrable over the boundary circle oL\(a; r) of any closed disc L\(a; r) ~ D. There are some useful extensions of this regularity theorem, as follows.

5. THEOREM.

If u is subharmonic and not identically equal to -

00

in a connected open subset

~

C, then u is locally Lebesgue integrable in D, and hence {z ED: u(z) = - oo} is a subset of D of Lebesgue measure zero. D

Proof. As a preliminary observation, note that if u(a) # - 00 for some point a E D, then it follows from Theorem 3 that for any closed disc L\(a; r) ~ D,

-00

< u(a) = 2r- 2

J:

u(a)p dp

= 1L\(a; r)I- 1

r

u(z) dV(z)

JA(a;r)

where dV(z) is Lebesgue measure in C and 1L\(a; r)1 = JA(a;r) dV(z), the equality of the iterated and multiple integrals following from Fubini's theorem for positive func-

98

Volume I Function Theory

tions (Tonelli's theorem), since u is bounded from above on the compact set A(a; r). Therefore, u is integrable over A(a; r). Now let E be the subset of D consisting of those points a E D such that u is integrable over an open neighborhood of a. Clearly, E is an open subset of D, and since u is not identically equal to - 00 by hypothesis, it follows from the preliminary observation that E is nonempty. On the other hand, if bED - E and if A(b; 2r) s D, it also follows from the preliminary observation that u is identically equal to - 00 in A(b; 2r); for if u(a) > - 00 at some point a E A(b; r), then A(a; r) s A(b; r) s D and u is integrable over A(a; r), and since b E A(a; r), it follows that bEE, a contradiction. Thus, A(b; r) s D - E so that D - E is also open. Since D is connected necessarily E = D, and that suffices to conclude the proof. Recall that Theorem 3 described properties of the circular means Mu(a; r) of a function u subharmonic in an open subset D s C only under the assumption that the full disc A(a; r) lies in D. If it is merely assumed that just the boundary circle oA(a; r) lies in D, it is still possible to say something, as in the following theorem. Note, though, that this result is rather different from that of Theorem 3, not just a weaker result than the previous one; it really provides additional information even under the hypotheses of Theorem 3.

6. THEOREM.

If u is subharmonic and not identically equal to - 00 in a connected open subset D S C, then u is Lebesgue integrable over any circle contained in D. M oreover,for any fixed point a E C, the real-valued function Mu(a; r) is a convex function of log r in the open set {log r E R: oA(a; r) S D}. For any closed annulus A = {z E C: r1 ~ Iz - al ~ r 2 } contained in D with < r2 , the restriction of u to the boundary oA is an upper semicontinuous mapping

Proof. r1

there, so as before there is a monotonically decreasing sequence of continuous real-valued functions J. on the two circles forming the boundary of A such that f.(z) converges to u(z) at each point z E oA. Each function J. on oA can be extended to a function F. that is continuous on A and harmonic in the interior of A. The solution of the Dirichlet problem in this case requires more than just the use of the Poisson integral, but it is a standard result -indeed, one that can be proved following Perron by using known properties of harmonic functions and the just demonstrated properties of subharmonic functions. Since u(z) ~ J.(z) = F.(z) whenever z E oA, it follows from the maximum theorem for subharmonic functions as in the proof of Theorem 3 that u(z) ~ F.(z) for all points z E A. Consequently, Mu(a; r) ~ MF.(a; r) whenever r 1 ~ r ~ r2 • Note that from Theorem 5 and Fubini's theorem it follows immediately that Mu(a; r) > -00 for almost all values r in the interval [rl' r2 ]. Note also that since F is harmonic, then M F.(a; r) = 0(. log r + P. for some real constants 0(., P•. Indeed, in terms of polar coordinates (r, 9) centered at a, the Laplacian can be written

Hence,

J

Subharmonic Functions

99

which implies that MFv does have the form asserted. The constants a v and Pv are determined by the two conditions M Fv(a; r) = fv(a; rj ) = (Lv log rj + Pv for j = 1, 2, and Mu(a; r) ~ (Lv log r + Pv whenever r 1 ~ r ~ r2 • The constants (Lv and Pv are monotonically decreasing as v tends to 00 and are bounded from below, since Mu(a; r) > - 00 for almost all r E [rl' r2 ], and hence converge to some finite values ex, P for which Mu(a; r) ~ ex log r + P whenever r 1 ~ r ~ r2 • Moreover, it follows from an application of Lebesgue's monotone convergence theorem that these limiting values are determined by the two conditions (L log rj + P= lim v (Lv log rj + Pv = lim v M!.«(L; rj ) = Mu(a; rj ) for j = 1, 2. That implies first that Mu(a; r) > - 00 and hence shows that u is integrable over every circle in D, and implies further that Mu(a; r) is convex as a function of log r. This completes the proof. The only property of subharmonic functions used in proving Theorems 3 and 6 was that the maximum theorem holds for the difference u - h between a subharmonic function u and a harmonic function h; the remainder of the arguments rested on familiar properties of harmonic functions. That suggests an alternative characterization of subharmonic functions, a characterization that can be expressed in a variety of ways, some of which are sometimes taken as the definition. For the purposes of this characterization, a mapping u: D -+ [ - 00, + 00) in a connected open subset D !;;; C will be said to satisfy the maximum principle in D if u(a) = sUPz E D u(z) for a point a E D only when u is constant in D. For an upper semicontinuous mapping u: D -+ [ -00, +00) in an open subset D s; C the following conditions are equivalent:

7. THEOREM.

(i) u is subharmonic in D. (ii) For any harmonic function h in a connected open subset U !;;; D, the difference u - h satisfies the maximum principle in U. (iii) Whenever h is a harmonic function in an open neighborhood of a compact subset K S; D and u(z) ~ h(z) at each point z E oK, then u(z) ~ h(z) at each point zEK. (iv) Whenev~ p is a complex polynomial and u(z) ~ Re p(z) at each point z E ol\(a; r) where l\(a; r) S; D, then u(z) ~ Re p(z) at each point z E l\(a; r). (v) Whenev~ p is a complex polynomial and u(z) ~ Re p(z) at each point z E ol\(a; r) where l\(a; r) S; D, then u(a) ~ Re p(a). Proof. It follows from Theorem 2(b) that condition (i) implies condition (ii), while it is quite obvious that condition (ii) implies condition (iii), which implies condition

100

Volume I

Function Theory

(iv), which in turn implies condition (v). Indeed, (iv) is a special case of (iii), since the real part of a polynomial is a harmonic function, while (v) is evidently a special case of (iv). It only remains to show that condition (v) implies condition (i). If u satisfies condition (v) for a disc /i(a; r) s; D, it suffices to verify the inequality (2) for that disc. Choose a monotonically decreasing sequence of continuous functions Iv on oLl(a; r) such that limv-+oo Iv (z) = u(z) at each point z E oLl(a; r), as in the proof of Theorem 3. As is well known, each function!. can be approximated uniformly on oLl(a; r) by trigonometric polynomials, merely summing the Fourier series expansion of Iv by arithmetic means and applying Fejer's theorem. Therefore, for each index v there is a trigonometric polynomial q., a finite sum of the form qv(O) = Ll avleil8, such that Ifv(a + re i8 ) - qv(O)1 < l/v for all real O. The ordinary polynomial Pv(z) = avo + 2Ll>o aVlr-l(z - a)l then has the property that Re pv(a + re i8 ) = qv(O); hence, IIv(z) - Re Pv(z)1 < l/v whenever z E oLl(a; r). Now since u(z) ~ Iv(z) < Re(pv(z) + l/v) whenever z E oLl(a; r), it follows from condition (v) that u(a) < Re(pv(a) + l/v). Using these various results and noting that (3) holds for the harmonic function Re Pv show that

-1 2n

1 2

0

"

u(a

+ re l8. ) dO =

1 lim -2 v

n

1 2

"

I.(a

+ re l8. ) dO

0

1Joe" [Re pv(a + re .l8 ) - ;IJ dO

~ limv inf 2n

~ limv inf [ Re pv(a) - ~J ~ limv inf [u(a) - ~J =

u(a)

Thus u is subharmonic, and that concludes the proof. Unlike harmonic functions, subharmonic functions are not necessarily smooth. However, for those subharmonic functions that are differentiable, there are still other parallels with harmonic functions, such as the following result. A real-valued function u of class C 2 in an open subset D s; C is subharmonic in D precisely when Llu = 4 02U/ OZ OZ ~ 0 throughout D.

8. THEOREM.

Proof. It is a minor calculation to verify that Ll = 02/0X 2 + 02/oy2 = 4 02U/ OZ oz. As a preliminary observation, note that if Llu > 0 throughout D, then u is subharmonic in D. Indeed, if u is not subharmonic, then by Theorem 7 there must exist a harmonic function h in D and a point a E D such that u - h has a local maximum value at a; but then the Hessian matrix of the function u - a must be negative semidefinite at the point a, an impossibility since Ll(u - h) = Llu > 0 at a. Then for the proof of the theorem itself, first suppose that Llu ~ 0 throughout D. For any e > 0, note that the function u. defined by u.(z) = u(z) + elzl 2 satisfies Llu.(z) =

J

Subharmonic Functions

101

+ 48 > 0 at each point zED. Hence, by the preliminary observation, u. is sub harmonic in D. The functions u. are a monotonically decreasing family of functions converging to u as 8 tends to zero, so u is subharmonic by Theorem 4. Next suppose that u is subharmonic in D. If ~u < 0 at some point a E D, then by the preliminary observation the function - u is also subharmonic near a; but then ~u = 0 near a, which is a contradiction. Thus, ~u ~ 0 throughout D, and the proof is thereby concluded. ~u(z)

K Pluriharmonic and Plurisubharmonic Functions

From the point of view of complex analysis, harmonic functions in e are particularly interesting as being at least locally the real parts of holomorphic functions. Only a subclass of the harmonic functions in en are locally the real parts of holomorphic functions when n > 1, and it is clearly of some importance to investigate more closely this subclass. It is a consequence of Hartogs's theorem that hoi omorphic functions of several variables can be characterized as those functions that are holomorphic in each variable separately, but there is no corresponding characterization of the real parts of holomorphic functions. For example, the function u(z 1, z2) = Xl Y2 is the real part of a holomorphic function of each variable separately but is easily seen not to be the real part of a holomorphic function of two variables. However, it is possible to characterize the real parts of holomorphic functions of several variables at least locally as those continuous functions such that their restrictions to all complex lines, not just to complex lines parallel to the coordinate axes, are real parts of holomorphic functions. The complex line in en through a point A Een in the direction of a vector BEen is the one-dimensional complex submanifold of en described parametrically as {A + tB: t EC}. 1. DEFINITION. A real-valued function in an open subset D ~ en is a pluriharmonic function if it is continuous in D and its restriction to any complex line through any point of D is a harmonic function on that line in D. If f is a holomorphic function in an open subset D ~ en, then the restriction of f to any complex line through any point of D is a holomorphic function on that line; hence, the real part of f is a pluriharmonic function in D. On the other hand, the function U(Zl' Z2) = X 1 Y2 mentioned above is harmonic in e 2 but is not pluriharmonic, since u(t, it) = (Re t)2.

The pluriharmonic functions in an open subset D ~ en are harmonic and hence are of class coo in D. A real-valued function u of class C 2 in D is a pluriharmonic function precisely when aau = 0 throughout D.

2. THEOREM.

Proof. If u is continuous in D and dJ-l is the standard surface measure on the boundary aB(A; r) of any closed ball B(A; r) ~ D, then the spherical average

K PI uri harmonic and Plurisubharmonic Functions

103

loB(A; r)I- 1 faB(A;r) u(Z) dfJ.(Z) can be written as an integral average of the average values of u over the circles oB(A; r) n L for all complex lines L through A; it is not really necessary to know the explicit formula. If u is plurisubharmonic in D, then the restriction ulL is harmonic, so the average value of u over the circle oB(A; r) n L reduces to the value u(A), and hence the spherical average also reduces to the value u(A); as is well known, that ensures that u is harmonic in D. If u is any function of class C 2 in D, then for any complex line {A + tB: t E C} through a point A E D, the complex form of the chain rule for differentiation shows that the Laplacian of the restriction of u to this complex line can be written in the form (1)

If u is a pi uri harmonic function in D, then ~tu(A + tB) = 0 for all A, B, t, with A + tB E D; hence, 02U(Z)/OZj OZk = 0 throughout D. Conversely, if 02U(Z)/OZj OZk = 0 throughout D, then ~tu(A + tB) = 0 for all A, B, t with A + tB E D; hence, u is pluriharmonic in D. Since oau(Z) = Ljk(02U(Z)/OZj Ozk) dZj A k, that suffices to conclude the proof.

az

This shows that the second-order linear differential operator oa plays for pluriharmonic functions the role the operator aplays for holomorphic functions. The pluriharmonic functions are characterized by the system of partial differential equations oau = 0, a more complicated situation than the characterization of harmonic functions by the single partial differential equation ~u = O. When expressed in terms of the underlying real coordinates Xj' Yj in C" = R211 , the system of partial differential equations oau = 0 takes the form (2)

3. THEOREM. The pluriharmonic functions in a simply connected open subset D precisely the real parts of the holomorphic functions in D.

S;

en are

Proof. Suppose u is a pluriharmonic function in D, so that by the preceding theorem u is of class COO and satisfies oau = 0 throughout D. The differential form OU is then a closed hoi om orphic I-form in D, so since D is simply connected, the indefinite integral f(Z) = ou is a well-defined holomorphic function in D. Note that df = ou, so that d(f + 1 - u) = df + dj - du = OU + au - du = 0; but then f + 1 - u = c is a real constant, and therefore u = f + 1 - c = Re(2f - c) is the real part of a holomorphic function in D. The converse was already observed, and that concludes the proof.

n

Since only a subclass of harmonic functions in C" are of particular interest in studying holomorphic functions of several variables, it might be expected that correspondingly only a subclass of subharmonic funCtions in e" are of particular interest in this context. That is indeed the case, as has been made quite evident by

104

Volume I

Function Theory

the pioneering work of Oka and Lelong. The relevant concept is the following one. A mapping u: D -+ [ -00, +(0) defined in an open subset D £; en is a plurisubharmonic function if: (i) u is upper semicontinuous in D and (ii) the restriction of u to any any complex line through any point of D is a subharmonic function on that line in D.

4. DEFINITION.

The parallels between Definitions 1 and 4 are quite apparent. As in the case of subharmonic functions, upper semicontinuity is the natural regularity condition to require in this context, but upper semicontinuity on each complex line is evidently not enough to imply upper semicontinuity in en, so the latter condition must be required separately. The reasons for requiring this form of regularity will become apparent in the subsequent discussion. It should also be pointed out here that the function that is identically equal to - 00 is plurisubharmonic according to this definition, but is sometimes not considered to be plurisubharmonic in other definitions. Most of the general elementary properties of subharmonic functions in e discussed in the last section extend quite trivially to hold for plurisubharmonic functions in en. For convenience the following brief catalog of such properties is included here. Let D be an open subset of en and u, u l , u 2 , ... be mappings from D into +(0). (a) A mapping u is plurisubharmonic in D precisely when it is plurisubharmonic in an open neighborhood of each point of D. (b) If u is plurisubharmonic in D and ifu(A) = SUPZEDU(Z) for some point A e D, then u is constant in the connected component of D containing A. (c) If u is plurisubharmonic in D, then u(A) = lim SUPZ-+AU(Z) for each point AeD. (d) If u is plurisubharmonic in D and if rP: IR -+ IR is a convex and monotonically increasing function, then the composition rP 0 u is also plurisubharmonic in D, where rP( -(0) = lim x -+_ oo rP(x). (e) If u l , U 2 are plurisubharmonic in D and if c l , C2 are positive real numbers, then the mapping u defined by u(Z) = ClUl(Z) + C 2 U 2 (Z) is also plurisubharmonic inDo (f) If u l , U2 are plurisubharmonic in D, then the mapping u defined by u(Z) = sup(u l (Z), u 2 (Z)) is also plurisubharmonic in D. (g) If {u v} is a monotonically decreasing sequence of plurisubharmonic functions in D, then the mapping u defined by u(Z) = lim v uv(Z) is also plurisubharmonic in D. (h) Any pluriharmonic function is also plurisubharmonic, and if both u and - u are plurisubharmonic, they are actually pluriharmonic.

5. THEOREM.

[ -00,

Proof. All but parts (c) and (g) of this theorem follow almost immediately from the definitions and Theorem 12. Part (c) follows from part (b) as in the proof of the

K Pluriharmonic and Plurisubharmonic Functions

105

corresponding assertion in Theorem J2, while part (g) follows directly from Theorem J4. The first part of the preceding theorem can be rephrased as the assertion that plurisubharmonicity is a local property; as in the case of subharmonicity, that is a very useful observation and will be used repeatedly. Part (b) is a maximum theorem for plurisubharmonic functions; as noted in the discussion ofthe corresponding result

for subharmonic functions, this is analogous to but slightly weaker than the usual maximum modulus theorem for holomorphic functions. Parts (e) and (h) can be reinterpreted as the assertions that the set of plurisubharmonic functions in D form a convex cone, with the pluriharmonic functions in D forming the maximal linear subspace contained in that cone.

The integrability properties of subharmonic functions in e discussed in the preceding section also extend to hold for plurisubharmonic functions in en, but with some restrictions and a bit more effort. It follows immediately from Theorems J5 and J6 that if u is plurisubharmonic in an open subset D ~ en and if the restriction of u to some complex line L = {A + tB: t E C} is not identically equal to - 00 in a connected component U of D n L, then u is locally Lebesgue integrable in U and is integrable over every circle contained in U. The restriction of u to some line L may of course be identically equal to -00 even when u is not identically equal to -00 in D, so some care must be taken in applying the preceding observations. However, the restriction of u is not identically equal to - 00 on almost all lines in en, as a consequence of the following natural extension of Theorem J5.

6. THEOREM.

If u is plurisubharmonic and not identically equal to - 00 in a connected open subset D ~ en, then u is locally Lebesgue integrable in D. Hence, {Z ED: u(Z) = - 00 } is a subset of D of Lebesgue measure zero.

Proof. The proof is essentially a repetition of the proof of Theorem J5, the corresponding result for sub harmonic functions. If u is plurisubharmonic in D and if A(A; R) ~ D, then the restriction U(Zl' a2"'" an) is subharmonic in A(a 1 ; rd, so as in the proof of Theorem J5, u(A)

~ I~(al; rdl-

1

r

J.~(al;rl)

U(Zl' a2' ... , an) dV(zd

For a fixed Zl there is a similar inequality for U(Zl' Z2"'" zn) viewed as a subharmonic function of Z2 in A(a2; r2), and so on. Altogether, u(A)

~ I~(A; R)I- 1

f

~~

u(Z) dV(zd'" dV(zn) =

I~(A; R)I- 1

r

J~~

u(Z) dV(Z)

where dV(Z) is Lebesgue measure in en, the equality of the iterated and multiple integral following immediately from Fubini's theorem for positive functions

106

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Function Theory

(Tonelli's theorem), since u is bounded from above on the compact set A(A; R). This formula leads immediately to the useful preliminary observation that if u(A) #- - 00 and A(A; R) £: D, then u is Lebesgue integrable over A(A; R). Now let E be the subset of D consisting of those points A E D such that u is integrable over an open neighborhood of A. Clearly E is an open subset of D, and since u is not identically equal to - 00 by hypothesis, it follows from the preliminary observation that E is nonempty. On the other hand, if BED - E and if A(B; 2R) £: D, it also follows from the preliminary observation that u is identically equal to - 00 in A(B; R); for if u(A) > - 00 at some point A E A(B; R), then A(A; R) £: A(B; 2R) £: D and u is integrable over A(A; R), but since BE A(A; R), it follows that BEE, a contradiction. Thus, A(B; R) £: D - E, so that D - E is also open. Since D is connected, necessarily E = D, and that suffices to conclude the proof. Next for the special case of smooth plurisubharmonic functions, there is a natural extension of the characterization of smooth subharmonic functions given in Theorem J8, for which extension it is convenient to introduce some further notation.

7. DEFINITION.

If u is a function of class C2 in an open subset D £: a point ZED is the Hermitian form

for any vectors B, C E

en, the Levi form of u at

en. For simplicity of notation also set Lu(Z; B, B) = Lu(Z; B).

The Levi form is thus the Hermitian form naturally associated to the complex Hessian matrix a 2 u(Z)/az; Ozj of the function u, so it is the Hermitian version of the skew-Hermitian form also associated to that Hessian matrix. The usefulness of this form in studying holomorphic functions of several variables was made clear in the work of E. E. Levi.

aau

8. THEOREM.

A real-valued function of class C 2 in an open subset D £: en is plurisubharmonic in D precisely when its Levi form is positive semidefinite at each point of D.

Proof. If u is a function of class C 2 in D, then for any complex line {A + tB: t E C} through a point A EDit follows from the complex form of the chain rule for differentiation as in (1) that Atu(A + tB)lt=o = Lu(A; B). It is then an immediate consequence of Theorem J8 that u is plurisubharmonic in D precisely when Lu(A; B) ~ 0 for all points A E D and all vectors BEen-hence, precisely when the Levi form of u is positive semidefinite at each point of A. That suffices for the proof.

A convenient method for deriving some additional properties of plurisubharmonic functions is to show that sufficiently smooth plurisubharmonic functions

K Pluriharmonic and Plurisubharmonic Functions

107

have the desired properties and then to approximate general plurisubharmonic functions by smooth plurisubharmonic functions in such a way that the properties of interest also hold in the limit. One ofthe standard smoothing operators in analysis provides a useful approximation for such purposes. To introduce this operator choose once and for all a real-valued COO function a in the complex plane such that a(z) depends only on Izl, a(z) ~ 0 for all z E e, a(z) = 0 whenever Izl ~ 1, and Ie a(z) dV(z) = 1 where dV(z) is Lebesgue measure on e. Then set a(Z) = a(z 1)' .. a(zn) in en, and let dV(Z) be Lebesgue measure on en. For any locally Lebesgue integrable mapping u: D ~ [ -00, +00) defined in an open set D ~ en and any real constant e > 0, let

9. DEFINITION.

u£(Z)

= [

u(Z

+ eW)a(W) dV(W)

(3)

JWEC n

whenever Z is contained in the open subset D£

= {Z E D: t5D (Z) > e}

~ D

The notation t5D is that introduced in Definition G3.1f ZED., then Li(Z; e)~D, and since the integration in (3) can be restricted to the closed polydisc Li(O; 1) containing the support of a and Z + eW E Li(Z; e) ~ D whenever WE Li(O; 1), it is clear that the function u£ is well defined in D. By the obvious change of variables, the integral (3) can be rewritten u£(Z) = e- 2n

[

JWECn

u(W)a (W -

e

Z)

dV(W)

(4)

It is clear from the latter formula that u£ is actually a function of class Coo in D. That the functions u£ do generally approximate u is a well-known result, but for the sake of completeness a proof will be appended here.

10. LEMMA. If u: D ~ IR is continuous, the Coo functions u£ converge to u uniformly on any compact subset of D as e tends to zero. If u: D ~ [ - 00, + 00] is locally Lebesgue integrable, the Coo functions u£ converge to u in L1-norm on any compact subset of D as e tends to zero. If u: D ~ [ - 00, + 00] is locally square-integrable, then it is also locally integrable and the Coo functions u£ converge to u in L 2- norm on any compact subset of D as e tends to zero. Proof. If u is continuous on D, it is uniformly continuous on any compact subset of D. Given a compact ~ubset K ~ D and a constant '1 > 0, choose t5 > 0 sufficiently small that K' = UZEKA(Z; 0) ~ D and lu(Zd - u(Z2)1 < '1 whenever Zl and Z2 are any two points of the compact set K' such that IIZl - Z211 < 0. Then whenever o ~ e < 0 and Z E K, it follows that

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Volume I Function Theory

lu.(Z) - u(Z)1 =

r

(u(Z

r

'1U(W) dV(W)

I

+ eW) -

u(Z»t5(W) dV(W)1

JWe4(O;1)

~

= '1

JWe4(O;1)

as desired. Next suppose that u is locally Lebesgue integrable in D, that V is an open subset of D such that [j is a compact subset of D, and that '1 > 0 is any given positive constant. Choose an open neighborhood of V of [j such that V is also a compact subset of D, and choose a continuous function v of compact support in V such that IIu - vII 1. v < '1/3, where 11'11 1. V denotes the usual L 1- norm on V. That there exists such a function v is a well-known result from the theory of the Lebesgue integral. Whenever e > 0 is sufficiently small that [j ~ then the functions u. and v. are well defined in V and

v.,

I u. -

f ~f

v. I 1. U =

Iu.(Z) -

v.(Z)1 dV(Z)

ZeU

r

lu(Z

+ eW) -

v(Z

+ eW)lu(W) dV(W) dV(Z)

Ze U JWe4(O; 1)

The order of integration can be interchanged by Fubini's theorem, and since [j ~ V. and the last integral is increased by integrating over a larger set, it follows that

IIu.-v.II1.U~

r

IIu-v ll 1 • v u(W)dV(W)<}

JWe4(O;l)

Now it is clear from the first part of this lemma that IIv - v.II l,U < '1/3 whenever e is sufficiently small. Thus for all small enough values of e, IIu - u.II1.u ~ IIu - vll 1 ,u

+

IIv - v.II1.u

+ IIv. -

u.II1.u < '1

thus demonstrating the second assertion. Finally, suppose that u is locally square integrable in D, that V is an open subset of D such that [j is a compact subset of D, and that '1 > 0 is any given positive constant. Since locally square-integrable functions are locally integrable by Schwarz's inequality, the functions u. are well defined in D•. Choose an open neighborhood V of [j such that V is also a compact subset of D, and choose a continuous function v of compact support in V such that IIu - vll2,v < '1/3, where II' 112,vdenotes theusualL 2-norm on V; again, the existence of such a function v can be taken as known. Whenever e > 0 is sufficiently small that [j ~ then u. and v. are well defined in V and

v.,

K Pluriharmonic and Plurisubharmonic Functions

Ilu. -

f ~f

v.II~,u =

109

lu.(Z) - v.(ZW dV(Z)

ZeU

r

lu(Z

+ eW) -

v(Z

+ eWWa(W) dV(W) dV(Z)

ZeU JWe4(0;1)

since by Schwarz's inequality lu.(Z) - v.(Z) I

~

f

lu(Z

+ eW) -

v(Z

+ eW)la(W)1/2·a(W)1/2·dV(W)

4(0;1)

~ {f

lu(Z

+ eW) -

v(Z

+ eWWa(W) dV(W)}1/2

4(0;1)

x

{f

a(W) dV(W)}1/2

4(0;1)

The order of integration can be interchanged by Fubini's theorem, and since [J it follows that

lIu. -

v.lltu

~

r

JWe4(0;1)

lIu _

vll~,va(W)dV(W) <

!;;;

v.,

(j)2

Now again it is clear from the first part of this lemma that Ilv - v.112,V < '1/3 whenever e is sufficiently small. Thus, for small enough values of e,

thus demonstrating the last assertion and concluding the proof. For the special case in which the mapping u is plurisubharmonic, the preceding result can be considerably strengthened, as follows.

11. THEOREM.

If u is a plurisubharmonic function that is not identically equal to - 00 in a connected open set D !;;; en, the COO functions u. are plurisubharmonic in D. and form a monotonically decreasing family of functions converging to u at each point of D as e tends to zero.

Proof. First note as a consequence of Theorem 6 that u is locally Lebesgue integrable in D, so that the functions u. are well defined and of class COO in D•. To see that u is plurisubharmonic in D., note that whenever the disc {A + tB : It I ~ I} is contained in D., then since u is plurisubharmonic,

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Volume I

Function Theory

-1

2n

f2" u,(A + Be

i8 )

de

2n

f 12"

r

u(A

= -1

0

~

WeCn

u(A

+ eW + Be i8 ) de· a(W) dV(W)

8=0

+ eW)a(W) dV(W) = u,(A)

Jwecn

The interchange of the order of integration is justified as usual by Fubini's theorem, the domain of integration really being compact since the support of a is contained in the compact polydisc A(O; 1) and the integrand hence being bounded from above there. Thus, u, is subharmonic on any line {A + tB : t E IC} n D, and hence is plurisubharmonic in D,. Next introduce polar coordinates Wj = rj e i8j and note that the integral (3) can be written U , (Z)

=

f

oo

12"

,=0

u(z.J

i8j ) de ... de a(r ) ... a(r)r ... r dr ... dr + er.e J 1 n1 nl n1 n

(5)

8=0

Since u is plurisubharmonic in D, it follows that

~ u(Z)

r

Jc

a(W) dV(W) = u(Z) n

On the other hand, from Theorem 13 the circular means g" u(Zj + ert8j) dej are monotonically increasing functions of e, so it follows immediately that u,(Z) is also a monotonically increasing function of e. Thus, the functions u, form a monotonically decreasing family of functions as e tends to zero, and the limit function v satisfies v(Z) = lim,-+o u,(Z) ~ u(Z) at each point ZED. Furthermore, since the mapping u is upper semicontinuous u(Z) ~ lim sUP,-+o u(Z + W), and since the function u is bounded from above on any compact subset of D, it follows from Fatou's lemma in measure theory that u(Z)

=

f<1.(0; 1) u(Z)a(W) dV(W)

~

f

<1.(0; 1)

lim sup u(Z ,-+0

~ lim sup ,-+0 ~

f

<1.(0;1)

u(Z

+ eW)a(W) dV(W) + eW)a(W) dV(W)

lim sup u,(Z) = v(Z) ,-+0

K Pluriharmonic and Plurisubharmonic Functions

111

Thus the limit function v must coincide with u at each point of D, and the proof is thereby concluded. Some more properties of plurisubharmonic functions can now easily be demonstrated by applying the preceding theorem. If D !:; em and E !:; en are open subsets, F: D -+ E is a holomorphic mapping from D into E, and u is a plurisubharmonic function in E, then the composite mapping u 0 F is a plurisubharmonic function in D.

12. THEOREM.

Proof. If u is a plurisubharmonic function of class C 2 in E and if Zj are the coordinate functions in D and Wr are the coordinate functions in E, then by the complex form of the chain rule for differentiation,

where W = F(Z). It then follows immediately from Theorem 8 that u 0 F is plurisubharmonic in D. For a general plurisubharmonic function u, if u is identically equal to - 00, then so is u 0 F and the desired result is trivial; otherwise, it follows from Theorem 11 that u = lime-+o Ue where U e are a monotonically decreasing family of Coo plurisubharmonic functions in the open subsets Ee !:; E. The functions Ue 0 F are plurisubharmonic in F-l(Ee) !:; D by the first part of the proof, and since they obviously form a monotonically decreasing family of plurisubharmonic functions converging to u 0 F in Ue F-l(Ee) = D, it follows from Theorem 5(g) that u 0 F is plurisubharmonic in D. That concludes the proof. One corollary of this result is that the class of plurisubharmonic functions is preserved by biholomorphic mappings. It is thus possible to speak of plurisubhar-

monic functions not just in open subsets of en but also on arbitrary complex manifolds. It is evident as a further corollary that if u is plurisubharmonic in an open subset D !:; en and if M is a complex submanifold of D, then the restriction u/M is a plurisubharmonic function on the complex manifold M. If D !:; en is a tube domain with base B!:; ~n, if u: D -+ [ - 00, + (0) is not identically equal to - 00 on any connected component of D, and if u(Z) depends only on the real part of Z for all ZED, then u is plurisubharmonic in D precisely when it is convex when viewed as a function on B.

13. THEOREM.

If u is a function of class C 2 in D and if u(Z) depends only on the real part X of Z, then the Levi form of u reduces to the real form Proof.

Thus by Theorem 8 the function u is plurisubharmonic in D precisely when the real

112

Volume I Function Theory

Hessian matrix a2u(X)/aX/Jxk is positive semidefinite at each point X E B, and as is well known that is just the condition that u is a convex function B. That establishes the theorem for functions of class C 2 • Next for the general case of the theorem, if u is plurisubharmonic and not identically equal to - 00 on any connected component of D, iffollows from Theorem 11 that u = lim.-+o u. where u. are a monotonically decreasing family of Coo plurisubharmonic functions in D. as e tends to zero. If, moreover, u(Z) depends only on the real part of Z, so does u.(Z), since u.(Z

+ iR) =

r u(Z + eW + iR)u(W) dV(W) = J~r u(Z + eW)u(W) dV(W) = u(Z)

J~

for any R E ~n. It then follows from the first part of the proofthat u. is convex when viewed as a function on B•. But it is well known that the limit of a monotonically decreasing sequence of convex functions is convex; hence, u is convex when viewed as a function on B as desired. On the other hand, if u is a convex function on B, then it is necessarily continuous and can be viewed as a continuous function u on D such that u(Z) depends only on the real part of Z. The functions u. also depend only on the real part of Z as before and moreover are convex in B., for if A and A' are any two points of B. such that the line segment between them lies in B., then for any value t E [0, 1], u.(tA

+ (1

- t)A') =

=

~ ~

r

u(tA

r

u(t(A

+ eW) + (1

- t)(A'

r

[tu(A

+ eW) + (1

- t)u(A'

tu.(A)

+ (1

JweC n

JweC n

JweC n

+ (1

- t)A'

+ eW)u(W) dV(W) + eW))u(W) dV(W) + eW)]u(W) dV(W)

- t)u.(A')

It then follows from the first part of the proof that the functions u. are plurisub-

harmonic in D•. However, the functions u. converge uniformly to u on compact subsets of D as e tends to zero, by Lemma 10, and it is obvious that the limit function u must also be plurisubharmonic. That suffices to conclude the proof of the theorem. It is apparent from the definitions that subharmonicity and plurisubharmonicity are properties somewhat analogous to convexity, and the results so far established show that these properties are preserved by many of the operations that preserve convexity. The analogy is sometimes quite helpful in thinking about these properties, as will be more evident in the later discussion. The preceding theorem shows that in some cases the relation between these properties and ordinary convexity is quite close indeed; but that theorem is probably more interesting for providing other simple examples of plurisubharmonic functions than for exhibiting any general relationship between plurisubharmonicity and convexity.

K Pluriharmonic and Plurisubharmonic Functions

113

Another application of Theorem 11 leads to an extension of the plurisubharmonicity criterion of Theorem 8 applicable to more general functions than those of class C 2 , essentially by interpreting the Levi form in terms of distributions. For this purpose, though, it is not really necessary to introduce any of the general machinery of the theory of distributions, only to use the following simple auxiliary result. 14. LEMMA. If u and v are functions of class C 2 in an open subset D £; of them has compact support in D, then their Levi forms satisfy

t

u(Z)Lv(Z; A) dV(Z) =

for any vector A

E

t

en and if at least one

v(Z)Lu(Z; A)dV(Z)

en, where dV(Z) is the usual Lebesgue measure in en.

Proof. For n = 1 the Levi form is just Lu(z; a) = laI 2 a 2 u(z)/az az, so Lu(a; z) dV(z) = (i/2) lal 2aau(z) and the desired result is just that JD u(z)aav(z) = JD v(z)aau(z) whenever the functions u and v are of class C2 and at least one has

compact support. This is a simple consequence of Stokes's theorem, for since the differential forms u av and v au have compact support in D, then 0 = JaD u av = JD~JU av) = JDU aav + JDau A av and 0 = JaDv au = JDd(v au) = JDV aau + JD av Aau, and the desired result follows immediately from a comparison of the last two formulas. For n ~ 1 and A = (1,0, ... ,0), the Levi form is just Lu(Z; A) = a 2 u(Z)/az l az1 , and after replacing the multiple integral by an iterated integral using Fubini's theorem, the desired result follows immediately from what has already been established in the special case n = 1. Finally for n ~ 1 and an arbitrary A E en, choose a matrix ME GL(n, IC) such that M- 1 A = (1,0, ... ,0). It then follows from the complex form of the chain rule for differentiation that the change of variables Z = F(W) = MW transforms the Levi form according to the rule Lu 0 F(W; M- 1 A) = Lu(Z; A). Therefore, from the results already established, it follows that

r u(Z)· Lv(Z; A) dV(Z) JF-I(D) r u =

0

F(W)· Lv

0

F(W; M- 1 A)ldet MI2 dV(W)

0

F(W)· Lu

0

F(W; M- 1 A)ldet MI2 dV(W)

JD

=

r

JF-I(D)

=

Iv

v

v(Z)· Lu(Z; A) dV(Z)

and that suffices to conclude the proof. A mappingu: D -+ [ - 00, + 00) not identically equal to - 00 in any connected component of an open subset D £; en is a plurisubharmonic function in D if and only if: (i) u is locally Lebesgue integrable in D and for each point A E D,

15. THEOREM.

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Volume I Function Theory

.-0

u(A) = lim IL\(A; 8)1- 1

f

u(Z) dV(Z)

a(A;.)

where dV(Z) is the usual Lebesgue measure in en and IL\(A; 8)1 = fa(A;') dV(Z); and (ii) for every nonnegative Coo function v having compact support in D and every vector A Een,

t

u(Z)· Lv(Z; A) dV(Z)

~0

Proof. If u is a function of class Coo in D, then u automatically satisfies condition (i) of the theorem, and from Theorem 8 it follows that u is plurisubharmonic in D precisely when Lu(Z; A) ~ 0 for every point A E D and every vector A E C"; but that is clearly equivalent to the condition that fD v(Z)· Lu(Z; A) dV(Z) ~ 0 for every nonnegative Coo function v having compact support in D and every vector A E en, and by Lemma 14 that is in turn equivalent to condition (ii) of the theorem. Thus, the theorem has been established for the special case of Coo functions u. Then for the general case of the theorem, first suppose that u is plurisubharmonic and not identically equal to - 00 in any connected component of D. The function u is locally Lebesgue integrable in D by Theorem 6, and as in the proof of that theorem, u(A)

~ IL\(A; 8)1-

1

f

u(Z) dV(Z)

a(A;.)

for any point A 5(c),

E

D and any closed polydisc ~(A; 8) s; D. Moreover, by Theorem

u(A) = lim sup u(Z) Z-A

= lim

sup

u(Z)

.-0 Z E a(A;.).Z,.A

~ lim IL\(A; 8)1.-0

1

f

u(Z) dV(Z)

a(A;.)

and a comparison of the last two inequalities shows that u satisfies condition (i). It follows from Theorem 11 that the associated functions u. are Coo plurisubharmonic functions in D•. If v is a nonnegative Coo function with compact support in D, then the support of v lies in D. whenever 8 is sufficiently small, and for such values 8 it follows from the first part of the proof that JD, u.(Z)· Lv(Z; A) dV(Z) ~ 0 for all vectors A E en. The functions u. converge to u in L I-norm on the support of v as 8 tends to zero by Lemma 10, and consequently,

r u(Z)· Lv(Z; A) dV(Z) = lim r u.(Z)· Lv(Z; A) dV(Z) ~ 0

JD

for all vectors A

£-+0 JDa

E

en. so that u also satisfies condition (ii).

K Pluriharmonic and Plurisubharmonic Functions

115

Finally, suppose that u: D -+ [ - 00, + (0) is a mapping that satisfies conditions (i) and (ii) of the theorem, and consider the associated Coo functions u. in D•. For any Coo function v with compact support in D. and any vector A E en, note that

f

u.(Z)· Lv(Z; A) dV(Z)

=

ZED,

r

JWE4(O;1)

r

=

JWE4(O;1)

f f

u(Z

+ 8W)Lv(Z; A) dV(Z)' a(W) dV(W)

ZED,

u(Z)Lv(Z - 8W; A) dV(Z)'a(W) dV(W)

ZED,+.W

~o

since for any fixed point WE .::\(0; 1), the Coo function v'(Z) = v(Z - 8W) has compact support in D. + 8 W ~ D and the function u by assumption satisfies condition (ii). It then follows from the first part of the proof that u. is plurisubharmonic in D•. Now Theorem 11 can be applied to this plurisubharmonic function u.; so for any (j > 0, the function u"ij defined by u•. ij(A) =

=

r

Jc

u.(A

+ (jZ)a(Z) dV(Z)

n

r r

Jc Jc n

u(A

+ (jZ + 8W)a(W) dV(W)'a(Z) dV(Z)

n

is a Coo plurisubharmonic function in D.,ij c D., and these functions are a monotonically decreasing family converging to u. as (j tends to zero. Since it is clear from Fubini's theorem that u"ij = uij,., it follows that whenever 8 1 ~ 8 2 and Z E D'2 ~ D. I , then u' l (Z) = limij--+o u.I,ij(Z) = limij--+o Uij"1 (Z) ~ limij--+o Uij"2(Z) = limij--+o u. 2,ij(Z) = u. 2 (Z). Thus, the functions u. also form a monotonically decreasing family offunctions as 8 tends to zero, so by Theorem 5(g) the limit function uo(Z) = lime--+o u.(Z) is a plurisubharmonic function in D. Note that for any closed polydisc K(A; (j) ~ D, it follows from an application of the monotone convergence theorem that the functions u. converge to U o in L 1 -norm on K(A; (j) as 8 tends to zero, while by Lemma 10 the functions u. also converge to u in L 1-norm on K(A; (j) as 8 tends to zero. Consequently, u and U o are equal almost everywhere in D. However, u satisfies condition (i) by hypothesis, while U o being plurisubharmonic satisfies condition (i) by the preceding part ofthe proof. Hence, since these two functions are equal almost everywhere, they must actually coincide, so that u is plurisubharmonic in D as desired. That suffices to conclude the proof. For some purposes it is convenient to rephrase what was proved in Theorem 15 in the following form. A locally Lebesgue integrable mapping u: D -+ [ - 00, + (0) in an open set en is equal almost everywhere to a plurisubharmonic function in D if and only

16. COROLLARY. D

tf

~

116

Volume I

Function Theory

Lu(Z)·Lv(Z;A)dV(Z)~O

(6)

for every nonnegative Coo function v with compact support in D and every vector A E C n • In any set of functions satisfying (6) and equal almost everywhere in D, there is a unique plurisubharmonic function uo, defined in terms of any other function u of that set by uo(A) = lim IL\(A; e)rl £-+0

Jr

u(Z) dV(Z)

(7)

.:1(A;£)

Proof. A plurisubharmonic function u satisfies the inequality (6) by Theorem 15, and that inequality is unchanged if the function u is changed arbitrarily on a set of measure zero in D. On the other hand, the argument in the last part of the proof of Theorem 15 really showed that if u satisfies the inequality (6) for all v and A, then u is equal almost everywhere in D to the plurisubharmonic function U o defined by (7). Any plurisubharmonic function equal to U o almost everywhere in D necessarily coincides with Uo as another consequence of Theorem 15, and that suffices to conclude the proof. It is evident from Theorem 15, particularly when restated in the form of Corollary 16, that condition (i) of that theorem is an eminently natural regularity condition to require for a plurisubharmonic function that is not identically equal to - 00 in any connected component of its domain of definition. Any mapping satisfying conditions (i) and (ii) of Theorem 15 is plurisubharmonic as a consequence of that theorem and in particular is upper semicontinuous. But it is clear that neither upper semicontinuity nor the slightly stronger condition of Theorem 5(c) is an appropriate regularity condition in the context of Theorem 15, since both are sensitive to alterations in the values of a function even on a set of measure zero. On the other hand, upper semicontinuity in the context of Definition 4 does serve to imply all the other forms of regularity and is one of the weakest forms of regularity that will so suffice, so these observations may be almost convincing arguments for the naturality of the definition of a plurisubharmonic function. It is well known that in any open subset D ~ en the space oflocally Lebesgue integrable functions-or more precisely the space of equivalence classes of locally Lebesgue integrable functions, where two such functions are considered equivalent if they agree outside a subset of D of Lebesgue measure zero-has the natural topology of a Frechet space. If {Kv} are countably many compact subsets of D such that the interiors of these sets cover D, then the pseudonorms IIfllv = fKy If(ZW dV(Z) define this topology, the topology of local U-convergence. By Corollary 16 the mapping that assigns to each plurisubharmonic function that is not identically equal to - 00 in any connected component of D the equivalence class of that function in the Frechet space of locally Lebesgue integrable functions in D is an injective mapping. Thus, the set of plurisubharmonic functions that are not identically equal to - 00 in any connected component of D can naturally be

K Pluriharmonic and Plurisubharmonic Functions

117

identified with a subset ofthe Frechet space oflocally Lebesgue integrable functions in D. It is worth restating this simple consequence of Corollary 16 explicitly as follows. 17. COROLLARY. The set of plurisubharmonic functions that are not identically equal to - 00 in any connected component of an open subset D ~ en can naturally be identified with a closed convex cone in the Frechet space oflocally Lebesgue integrable functions in D.

Proof. That the set of such plurisubharmonic functions form a convex cone follows, of course, from Theorem 5. That this cone is closed follows from the plurisubharmonicity criterion of Corollary 16, since it is apparent from that corollary that if U v are locally Lebesgue integrable plurisubharmonic functions in D and converge to u in the topology of local L I-convergence, then u is necessarily plurisubharmonic in D. That suffices for the proof.

The preceding theorem also yields an extension of the characterization of plurisubharmonic functions in Theorem 2, as follows.

18. COROLLARY. D~

A locally Lebesgue integrable real-valued function u in an open subset

en is equal almost everywhere to a plurisubharmonic function in D if and only if

In u(Z)· Lv(Z; A) dV(Z) ~ 0

(8)

for every nonnegative Coo function v with compact support in D and every vector A E en. In any set of functions satiifying (8) and equal almost everywhere in D, there is a unique plurisubharmonic function u o, defined in terms of any other function u of that set by uo(A) = lim IL\(A; 8)1- 1 £-0

f

u(Z) dV(Z)

(9)

.1.(A;£)

Proof. Since u is pluriharmonic precisely when both u and - u are pi uri subharmonic, this result is an immediate consequence of Corollary 16.

L Special Classes of Plurisubharmonic Functions

Two special classes of plurisubharmonic functions are of particular usefulness and interest and will be discussed briefly here. The first and by far the most important is the subclass of strictly pluriharmonic functions. Recall from Theorem K8 that a real-valued function of class C 2 in an open subset D ~ en is plurisubharmonic in D precisely when its Levi form is positive semidefinite at each point of D. Among the plurisubharmonic functions of class C 2 in D one extreme case evidently consists of those functions with Levi forms identically zero; by Theorem K2 these are just the plurisubharmonic functions in D. The complementary extreme case consists of those functions with Levi forms strictly positive definite at each point of D; these are quite naturally called strictly plurisubharmonic functions and play a useful special role in the theory. Of course, it is not really necessary to restrict these considerations to functions of class C 2 if the Levi form is interpreted in the sense of distributions, as in Theorem K15 and its corollaries, and that is perhaps the most natural way in which to introduce the full class of strictly plurisubharmonic functions. But it is in some ways easier formally to introduce this class of functions in the following equivalent but more primitive form. 1. DEFINITION. A mapping u: D -+ [ - 00, + (0) not identically equal to - 00 in any connected component of an open subset D ~ en is strictly plurisubharmonic in D if in an open neighborhood UA of any point A E D the restriction of u can be written as a sum ul UA = u~ + u;';', where u~ is a function of class C 2 with Levi form strictly positive definite at each point of UA and u;';' is plurisubharmonic in UA • It is obvious from this definition that a strictly plurisubharmonic function is in particular a plurisubharmonic function, and that the condition that a mapping be strictly plurisubharmonic is a local condition. Moreover, it is easy to see that a function of class C 2 is strictly plurisubharmonic in D precisely when its Levi form is strictly positive definite at each point of D. Note also that by this definition a strictly plurisubharmonic function in D is not identically equal to - 00 in any open subset of D. In an alternative and frequently used terminology, a strictly plurisubharmonic function is called a strongly plurisubharmonic function. Some equivalent 118

L Special Classes of Plurisubharmonic Functions

119

characterizations of strictly or strongly plurisubharmonic functions are as follows. A mapping u: D - [ - 00, + 00) not identically equal to - 00 in any connected component of an open subset D ~ en is strictly plurisubharmonic in D if and only if for any coo function v with compact support in D, there is a constant ~v > 0 such that u + tv is plurisubharmonic in D whenever - ~v < t < ~v.

2. THEOREM.

Proof. First suppose that u: D - [ - 00, + 00) is a mapping satisfying the conditions of this theorem. For any point A ED, choose an open neighborhood UA of A such that its closure [JA is a compact subset of D, and choose a Coo function VA such that vA(Z) = IIZI1 2 whenever Z E UA and that VA has compact support in D. Since u was assumed to satisfy the conditions of the theorem, there is a constant t > 0 such that u - tVA is plurisubharmonic in D; but then u = tVA + (u - tVA)' where L(tvA)(Z; B) = t· LvA(Z; B) = tilBI12 is positive definite at each point Z E UA and u - tVA is plurisubharmonic in UA; hence, u is strictly plurisubharmonic in D as desired. Next suppose that u: D - [ - 00, + 00) is strictly plurisubharmonic in D and that V is a Coo function with compact support in D. For any point A in the support ofvchoose an open neighborhood UA of A such that ul UA = u~ + u~, where u~ is a function of class C 2 with Levi form strictly positive definite in UA and u~ is plurisubharmonic in UA • After shrinking the neighborhood UA if necessary, it can be assumed that [JA is a compact subset of D and that the minimum eigenvalue of the Levi form Lu~ is bounded away from zero on [JA' Since the eigenvalues of Lv are bounded on the compact set [JA' there is evidently a constant ~A > 0 such that u~ + tv is plurisubharmonic on UA whenever It I < ~A' and then of course u + tv = (u~ + tv) + u~ is also plurisubharmonic in Uk Finitely many such neighborhoods UA J_ will cover the compact support of v, so for ~v = infJ'~A_J > 0, it follows that u + tv is plurisubharmonic on the support of v and hence in all of D whenever It I > ~v. That suffices to conclude the proof. A mapping u: D - [ - 00, + 00) defined in an open subset D ~ plurisubharmonic in D if and only if:

3. THEOREM.

(i) u is locally Lebesgue integrable in D, and for each point A u(A)

1

= lim IA(A; 8)1- 1 .-+0

a(A;.)

E

en is strictly

D,

u(Z) dV(Z)

(ii) for any open subset U ~ D such that [J is a compact subset of D, there is a constant (ju > 0 such that

L

u(Z)· Lv(Z; A) dV(Z)

~ (ju1l A 1I2 So v(Z) dV(Z)

whenever v is a nonnegative Coo junction with support contained in U and A

E

en.

120

Volume I

Function Theory

Proof. First suppose that u: D --+ [ - 00, + 00) is a mapping satisfying the two conditions of this theorem. In order to show that u is strictly plurisubharmonic in D, it is enough by Theorem 2 to show that for any COO function v with compact support KeD there is a constant (jv > 0 such that u + tv is plurisubharmonic in D whenever - (jv < t < (jv, since u is not identically equal to - 00 in any connected component of D by condition (i). Choose an open neighborhood U of K such that [J is also a compact subset of D, let (ju > 0 be the constant associated to the set U so that condition (ii) holds, and let Av > 0 be any constant such that ILv(Z; A)I ~ Av II A 112 for all ZED and A E en. Observe that there obviously exists such a constant, since v is Coo with compact support in D. The constant (jv = A;l(jU then has the desired property; that is, u + tv is plurisubharmonic in D whenever - (jv < t < (jv' To verify this note first that conditions (i) and (ii) imply that u itself is plurisubharmonic in D as a consequence of Theorem K15. The function u + tv coincides with u in D - K and hence is plurisubharmonic there, so to show that it is plurisubharmonic in D, it is enough just to show that it is plurisubharmonic in U. For this purpose the plurisubharmonicity criterion of Theorem K15 will again be used. It is clear thatu + tv satisfies condition (i) of Theorem K15, since u does so by hypothesis and v is Coo in U. As for condition (ii) of Theorem K 15, if w is any nonnegative Coo function having compact support in U and A is any vector in C", then recalling Lemma K14 shows that

In

[u(Z)+tV(Z)]' Lw(Z; A) dV(Z) =

In

u(Z)· Lw(Z; A) dV(Z)+t

In

w(Z)· Lv(Z; A) dV(Z)

~ (ju11 A 112 In w(Z) dV(Z) -ltlAvll A l12 In w(Z) dV(Z) by condition (ii) of this theorem and the definition of the constant Av. The last expression above is ~O whenever It I < A;l(ju = (jv; hence, u + tv is plurisubharmonic in U whenever It I < (jv as desired. Next suppose that u is a strictly plurisubharmonic function in D. The function u is in particular plurisubharmonic in D and is not identically equal to - 00 in any connected complement of D, so it follows from Theorem K15 that u satisfies condition (i) of the present theorem. As for condition (ii), consider an open subset U ~ D such that [J is a compact subset of D. In an open neighborhood UA of any point A E [J, the restriction ul UA = u~ + u~, where u~ is a function of class C 2 with Levi form strictly positive definite at each point of UA and u~ is plurisubharmonic in Uk After shrinking the neighborhood UA if necessary, it can be assumed that the minimum eigenvalue of the Levi form of u~ is bounded away from zero on UA and hence that there is a constant (jA > 0 for which Lu~(A; B) ~ (jA IIBII2 whenever Z E UA and 1!. E C". Choose finitely many such neighborhoods UAJ covering the compact set U, and let (ju = in~ (jA > O. If v is a nonnegative Coo function with support contained in U, write v = Vj' where Vj is a nonnegative Coo function with , by using a Coo partition of unity for the covering {UA .} of support contained in U A _ J J U as usual. Then from Lemma K14 it follows that

L

l

=~ J

i

uAj

Special Classes of Plurisubharmonic Functions

[Vj(Z)·

121

Lu~/Z; B) + u~JZ)· LVj(Z; B)] dV(Z)

~ ~ b )B11 2LA. viZ) dV(Z) A

}

~ bu IIBI12

L

v(Z) dV(Z)

for SUA u~/Z)· LVj(Z; B) dV(Z) ~ 0 by Theorem K15, since u~/Z) is plurisubharmonic.} Thus, u satisfies condition (ii) also, and that suffices to conclude the proof of the theorem. The preceding theorem provides a characterization of strictly plurisubharmonic functions paralleling the characterization of ordinary plurisubharmonic functions given in Theorem K15. Condition (i) is one of the regularity conditions satisfied by all plurisubharmonic functions nowhere identically equal to - 00, while condition (ii) is just the condition that the Levi form interpreted in the sense of distributions is strictly positive definite everywhere. In some ways this is the most natural characterization of general strictly plurisubharmonic functions, while the characterization given in Theorem 2 is perhaps the most functionally useful. The special role played by strictly plurisubharmonic functions will become more evident in the course of the subsequent discussion, so nothing will be said about that here. However, it is useful to have available a catalog of some of the operations on plurisubharmonic functions that preserve the subclass of strictly plurisubharmonic functions. It is evident that no limiting processes can be expected to preserve this subclass, and that in particular the strictly plurisubharmonic functions do not form a closed subset of the space of plurisubharmonic functions in the topology of local U-convergence; but most purely algebraic operations do generally preserve this subclass.

4. THEOREM.

If u, v are strictly plurisubharmonic functions in an open subset D s; a, b are nonnegative real constants, then (a) au

en and

+ bv is strictly plurisubharmonic in D; and

(b) sup(u, v) is strictly plurisubharmonic in D. Proof. (a) It is obvious that the first assertion is true for strictly plurisubharmonic functions of class C 2 • In general, in an open neighborhood UA of any point A ED, the functions u, v can be written ul UA = u~ + u~, vi UA = v~ + v~, where u~, v~ are strictly plurisubharmonic functions of class C 2 and u,;" v';' are plurisubharmonic functions; but then (au + bv)IUA = (au~ + bv~) + (au';' + bv~), where au~ + bv~ is

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Function Theory

strictly plurisubharmonic of class C 2 while au~ + bv~ is plurisubharmonic; hence, au + bv is strictly plurisubharmonic as desired. (b) It follows from Theorem 2 that for any COO function w with compact support in D, there is a constant t5w > 0 such that u + tw, v + tw are plurisubharmonic in D whenever - t5w < t < t5w ; but then by Theorem K5(f) the function sup(u, v) + tw = sup(u + tw, v + tw) is also plurisubharmonic in D, so from Theorem 2 again it follows that sup(u, v) is strictly plurisubharmonic as desired. That suffices to conclude the proof. If D £; em and E £; en are connected open subsets, F: D --+ E is a nonsingular holomorphic mapping from D into E, and u is a strictly plurisubharmonic function in E, then the composite mapping u 0 F is either identically equal to - 00 or a strictly plurisubharmonic function in D.

5. THEOREM.

Proof. Suppose that u 0 F is not identically equal to -00 in D. For any point A E D, it follows from Definition 1 that in an open neighborhood U of the point F(A) E E, the function u can be written as the sum ul U = u' -\- u", where u' is a strictly plurisubharmonic function of class C 2 in U and u" is plurisubharmonic in U. It follows directly from the complex form of the chain rule for differentiation that the Levi form of the composite function u' 0 F is given by L(u' 0 F)(Z; B) = Lu'(F(Z); JF(Z)· B) for any point Z E F-1(U), and since Lu' is positive definite and the Jacobian matrix JF(Z) is nonsingular by hypothesis, it follows that L(u' 0 F) is also positive definite and hence that u' 0 F is strictly plurisubharmonic in F- 1 (U). On the other hand, u" 0 F is plurisubharmonic in F-1(U) by Theorem K12; hence, u 0 F = u' 0 F + u" 0 F is also strictly plurisubharmonic in F-1(U), and the proof

is thereby concluded. It is clear that the non singularity of the mapping F is a necessary hypothesis in the preceding theorem, since it is even necessary in the special case of C2 strictly plurisubharmonic functions. However, it still follows just as for ordinary plurisubharmonic functions that the class of strictly plurisubharmonic functions is preserved by biholomorphic mappings. It is thus possible to speak of strictly plurisubharmonic functions on an arbitrary complex manifold. Furthermore, if u is a strictly plurisubharmonic function in an open subset D £; en and M is a connected complex

submanifold of D, then the restriction ulM is either identically equal to strictly plurisubharmonic function on M.

-00

or a

The other special class of functions to be considered here is a properly larger class of functions than the plurisubharmonic functions, consisting of functions that are equal almost everywhere to a plurisubharmonic function but that may not be plurisubharmonic because they fail to be upper semicontinuous. Such functions do arise in practice, and an examination of them here may serve to clarify the role that upper semicontinuity plays in the theory of plurisubharmonic functions. It is convenient to begin the discussion by recalling some more general properties of upper semicontinuous functions. For any mapping u: D --+ [ - 00, -\- 00] defined in an open set D s;;; en, the upper envelope of u is the mapping u*: D --+ [ - 00, -\- 00] defined by

L Special Classes of Plurisubharmonic Functions

123

u*(A) = sup (U(A), lim sup U(Z)) z~A

= lim ( sup U(Z)) e~O

(1)

Z E B(A;e)

The upper envelope has the following properties, familiar from elementary analysis.

6. LEMMA.

If u*, v* are the upper envelopes of mappings u, v: D -. [ - 00, + 00]: (a) u(A) ~ u*(A) at each point A E D. (b) If u(A) ~ v(A) at each point A ED, then also u*(A) ~ v*(A). (c) If u is locally bounded from above, then u* is upper semicontinuous. (d) If u is upper semicontinuous, then u = u*.

Proof. (a), (b) These assertions are trivial consequences of the definition (1) of the upper envelope. (c) If u is locally bounded from above, it follows immediately from (1) that u*(A) < 00 at each point A E D and hence that u* is actually a mapping u*: D -. [ - 00, + (0). If u*(A) < r at some point A E D, it further follows from (1) that there is a value t: > 0 for which SUPZEB(A;e)U(Z) < r, and then it is clear from (1) again that u*(Z) < r at each point Z E B(A; t:). Thus, {Z ED: u*(Z) < r} is an open subset of D for any real number r, so that u* is upper semicontinuous as desired. (d) If u is upper semicontinuous, then u(A) ~ lim SUPZ~A u(Z), and it then follows from (1) that u*(A) = u(A) at each point A E D. That suffices to conclude the proof.

Note that if u: D -. [ - 00, + (0) is a mapping that is locally bounded from above, if v: D -. [-00, +(0) is an upper semicontinuous mapping, and if u(Z) ~ v(Z) at each point ZED, then it follows immediately from parts (a), (b), and (d) of the preceding lemma that u(Z) ~ u*(Z) ~ v*(Z) = v(Z) at each point ZED, while u* is itself upper semicontinuous as a consequence of part (c) of that lemma. Therefore, the upper envelope of a mapping u: D -. [ - 00, + (0) that is locally bounded from above can be characterized as the least upper semicontinuous mapping u*: D -. [ - 00, + (0) such that u ~ u*. In terms of this construction then, introduce the following special class of functions.

A mapping u: D -. [-00, +(0) in an open subset D S; en is called a nearly if its upper envelope u* is a plurisubharmonic function in D and u(Z) = u*(Z) for all points ZED outside a subset of D of Lebesgue measure zero.

7. DEFINITION.

plurisubharmonic function

It is clear that the condition that a mapping be a nearly plurisubharmonic function is a local condition. Any plurisubharmonic function is obviously also a nearly plurisubharmonic function, so that the class of nearly plurisubharmonic functions is an extension of the class of plurisubharmonic functions. Note that a nearly plurisubharmonic function can be viewed as a mapping derived from a

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Function Theory

plurisubharmonic function by decreasing the values of the latter function on a set of measure zero. Indeed, any such operation necessarily leads to a nearly plurisubharmonic function as a consequence of the following observation. If v is a plurisubharmonic function in an open set D ~ en, and if u: D --+ [-00, +(0) is a mapping such that u(Z) ~ v(Z) for all points ZED while u(Z) = v(Z) for almost all points ZED in the sense of Lebesgue measure, then u is a nearly plurisubharmonic function in D.

8. THEOREM.

Proof. It suffices to show that the hypotheses of this theorem imply that u* = v; for v is plurisubharmonic, and it follows from Lemma 6 that u(Z) ~ u*(Z) ~ v*(Z) = v(Z) for all points ZED and hence that u(Z) = u*(Z) almost everywhere in D. On the one hand, u*(Z) ~ v(Z) for all points ZED as just observed. On the other hand, if u*(A) < v(A) at some point A ED, then it follows immediately from the definition (I) of the upper envelope that there are constants 8 > 0, () > 0 such that supz e B(A;d) u(Z) = v(A) - 8. But since v is plurisubharmonic and v = u almost everywhere in D, it then follows from Theorem KI5 that

r-+O

J.~(A;r)

r

v(Z) dV(Z)

lim IA(A; r)I- 1 r-+O

Jr

u(Z) dV(Z)

v(A) = lim IA(A; r)I- 1

=

~

v(A) -

L1(A;r)

8

a contradiction. Therefore, u* = v as desired, and the proof is thereby concluded. Let u 1 , u 2, ... be nearly plurisubharmonic functions in an open subset D ~ en. (a) If ai' a2 are nonnegative real numbers, then al Ul + a2 U2 is also nearly plurisubharmonic in D. (b) If the functions U v form a monotonically decreasing sequence, then the limit function u = limv U v is also nearly plurisubharmonic in D. (c) If the functions U v are uniformly bounded from above on any compact subset of D, then the functions u, v defined by u(Z) = supv uv(Z) and v(Z) = lim sUPv-+oo uv(Z) are also nearly plurisubharmonic in D.

9. THEOREM.

Proof. (a) If u = a 1 u 1 + a2u2, it is clearfrom the definition (1) of the upper envelope that u(Z) ~ u*(Z) ~ a 1 uT(z) + a 2u!(Z) for all points ZED. But a l uT + a2u! is plurisubharmonic in D while u(Z) = a 1 uT(Z) + a 2u!(Z) for almost all points ZED since the functions Ut> U 2 are nearly plurisubharmonic, so it follows immediately from Theorem 8 that u is nearly plurisubharmonic in D. (b) It is obvious from Lemma 6 that the functions u~ also form a monotonically decreasing sequence, and since each ofthese functions is plurisubharmonic, the limit w = limv u~ is plurisubharmonic by Theorem K5(g). But clearly u(Z) ~

L Special Classes of Plurisubharmonic Functions

125

w(Z) for all points ZED, while u(Z) = w(Z) for almost all points ZED, so it follows immediately from Theorem 8 that u is nearly plurisubharmonic in D. (c) It can obviously be assumed without loss of generality that none of the functions u. is identically equal to - OCJ on any connected component of the set D. The same is also true of the pI uri sub harmonic functions u~, which are then locally integrable in D by Theorem K6. For the auxiliary function w defined by w(Z) = suP. u~(Z), it is clear that u(Z) ~ w(Z) for all points ZED, while u(Z) = w(Z) for almost all points ZED. The functions u~ are evidently locally uniformly bounded from above, so their supremum w is locally bounded from above, and since w ~ ui, the function w is necessarily locally integrable as well. The sequence of plurisubharmonic functions w. defined by w.(Z) = sup(ui(Z), ... , u~(Z» converges pointwise to w, and since ui ~ w. ~ w, it follows readily from the Lebesgue dominated convergence theorem that the sequence of functions w. also converges to w in the topology of local L l-convergence. Therefore, from Corollary K 17 the function w is equal almost everywhere in D to a plurisubharmonic function w. Since u~(Z) ~ w(Z) = w(Z) for almost all points ZED while both u~ and ware plurisubharmonic, it follows from condition (i) of Theorem K15 that actually u~(Z) ~ w(Z) for all points ZED. Consequently, w(Z) = suP. u.(Z) ~ w(Z) for all points ZED, while w(Z) = w(Z) for almost all points ZED. Combining this with the relation between the functions u and w observed above shows that u(Z) ~ W(Z) for all points ZED while u(Z) = W(Z) for almost all points ZED. It then follows immediately from Theorem 8 that u is nearly plurisubharmonic in D. Finally, since v(Z) = lim sup .....oo u.(Z) = lim .....oo v.(Z) where v.(Z) = sup(u.(Z), U.+ l (Z), ... ), the functions v.(Z) are nearly plurisubharmonic by what has just been proved, and since they form a monotonically decreasing sequence, it follows from part (b) of this theorem that their limit v is also nearly plurisubharmonic in D as desired. That suffices to conclude the proof.

The first assertion of the preceding theorem shows that the nearly plurisubharmonic functions form a convex cone, while the second assertion shows that the set of nearly plurisubharmonic functions is closed under one of the basic limiting processes known to preserve plurisubharmonic functions. The third assertion is really the most interesting part of the theorem and is essentially the reason for introducing this class of functions. It shows that the nearly plurisubharmonic functions are closed under more general operations than those that preserve the ordinary plurisubharmonic functions. Whereas the supremum of any finite collection of plurisubharmonic functions is necessarily plurisubharmonic, that is not necessarily the case for the supremum of a countably infinite collection of plurisubharmonic functions, even when they are locally uniformly bounded from above; the latter supremum may be only nearly plurisubharmonic. Actually for the case of nearly plurisubharmonic functions, it is not even necessary to restrict to a countably infinite collection of functions for the preceding theorem to hold. A special case of a general observation of Choquet leads immediately to the following result. 10. THEOREM. If rUt} is any collection of nearly plurisubharmonic functions in an open subset D £; en, and if these functions are uniformly bounded from above on any

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Volume I Function Theory

compact subset of D, then the function u defined by u(Z) = SUPt ut(Z) is nearly plurisubharmonic in D. Proof. Let {Up} be a countable basis for the open subsets of D such that each set Up is a compact subset of D, and for each integer v ~ 1, let up. be one of the functions {u t } for which

sup up.(Z) ZeU.

~

1 1 sup ut(Z) - - = sup u(Z) - V

t,ZeU.

(2)

V

ZeU.

The function v defined by v(Z) = sUPP , ' up.(Z) is nearly plurisubharmonic in D by Theorem 9(c), and obviously v(Z) ~ u(Z) and hence v*(Z) ~ u*(Z) for all points ZED. In order to complete the proof, it is only necessary to show that v*(Z) = u*(Z) for all points ZED; for v* is plurisubharmonic in D, and since v(Z) ~ u(Z) ~ u*(Z) = v*(Z) at all points ZED, while v(Z) = v*(Z) at almost all points in D, then u(Z) = u*(Z) at almost all points in D. If, in contradiction to the desired result, v*(A) = u*(A) - 2e < u*(A) for some point A ED and some e > 0, then it follows directly from the definition (1) ofthe upper envelope v* that sUPZeU v(Z) < u*(A)-e for one of the basic open sets Up containing A and hence that ·suPzeu. up.(Z) < u*(A) - e for every integer v. But then using (2) and recalling the definition (1) of the upper envelope u* show that u*(A) - e > sup up.(Z) ZeU.

~

1 1 sup u(Z) - - ~ u*(A) - ZeU.

V

V

for every integer v, and that is impossible whenever v is so large that l/v < e. Therefore, v*(Z) = u*(Z) for all points ZED, and the proof is thereby concluded. It is perhaps worth pointing out explicitly that if, for example, {u t } is a collection of nearly plurisubharmonic functions in an open set D £; en indexed by the points t in an open set T £; em, then for any point to E T, the function Vto defined by vto(z) = lim SUPt-+to ut(Z) is also nearly plurisubharmonic in D; for Vto = lim.-+o vto" where vto.(Z) = SUPteB(to;') ut(Z), the function Vto' is nearly plurisubharmonic in D by Theorem 10, and since the functions Vto' are monotonically decreasing as e tends to zero, the limit function Vto is nearly plurisubharmonic in D by Theorem 9(b).

M Pseudoconvex Subsets of

en

Plurisubharmonic functions have proved quite useful in the further investigation of domains of holomorphy and related topics, following the pioneering work of K. Oka. Recall from section G that an open subset D s;;; en is a domain of holomorphy precisely when it is holomorphically convex, and that the latter condition is expressed in terms offunctions ofthe form If I where f is holomorphic in D. Now it is a familiar and obvious consequence of the Cauchy integral formula that the absolute value u = If I of a holomorphic function f of a single variable satisfies the integral inequality J(2) and hence is a subharmonic function. And since the restriction of a holomorphic function of several variables to any complex line is a holomorphic function of one variable, it is evident that the absolute value If I of a holomorphic function f of several variables is a plurisubharmonic function. That suggests looking at corresponding notions of convexity of domains defined in terms of plurisubharmonic functions, as a fairly natural generalization of holomorphic convexity. This suggestion turns out to be very fruitful indeed, as will become apparent from the subsequent discussion. Perhaps the easiest place to begin such an investigation is with the following. An open subset D S;;; en is pseudoconvex if for any compact subset K S;;; D the set KD = {A ED: u(A) ~ SUPZeKU(Z) whenever u is a continuous plurisubharmonic function in D} is also compact. The set KD is the plurisubharmonically convex hull of KinD.

1. DEFINITION.

This definition is parallel to Definition G2, the definition of a holomorphically convex set and the holomorphically convex hull of a compact subset, with continuous plurisubharmonic functions in place of the moduli of holomorphic functions. Note that for any open subset D s;;; en and any compact subset K s;;; D, the plurisubharmonically convex hull KD is a bounded subset of en, since the plurisubharmonic functions IZjl, where Zj are the coordinate functions in en, are bounded on KD by their maximal values on K. Note further that the subset KD is always a relatively closed subset of D, since the functions u considered in the definition of the subset KD are assumed to be continuous. That is a convenience, simplifying the discussion somewhat, and it is for that reason that continuity was required in Definition 1; continuity is not really essential, though, as will later be seen. In view

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of the preceding obser~ations, the condition that KD be compact reallylust amounts to the condition that KD be a closed subset ofcn, or equivalently, that KD be disjoint from an open neighborhood of the boundary oD of D. Since the absolute values of holomorphic functions are always plurisubharmonic, it is apparent that the plurisubharmonically convex hull KD and the holomorphically convex hull KD of a compact subset K of any open set D ~ C" are related by KD ~ K D. If Dis holomorphically convex, then KD is a compact subset of D, and since K Dis a relatively closed subset of K D, necessarily K Dis also compact; thus, a holomorphically convex set is pseudoconvex. The importance of the notion of pseudoconvexity in studying holomorphic functions arises firstly from the fact that pseudoconvexity is actually equivalent to holomorphic convexity, and secondly from the observation that pseudoconvexity is really a much easier property to handle than holomorphic convexity because of the greater abundance and flexibility of plurisubharmonic than of holomorphic functions. The theorem that a pseudoconvex subset ofcn is holomorphically convex is quite nontrivial and will be proved later, after the introduction of some more machinery. The present section will be devoted to a discussion of pseudoconvex sets for their own sake, examining the properties of these sets that follow rather directly from the results about plurisubharmonic functions derived in the preceding sections. These properties then carry over immediately to domains ofholomorphy, after it is proved that a pseudoconvex set is holomorphically convex. There are a variety of useful alternative characterizations of pseudocdnvex sets, exhibiting a corresponding variety of other properties of such sets. The two characterizations to be considered next are of a rather geometric form, and among other things indicate the motivation for the terminology. One of them involves for an open set D ~ cn the distance functions d Dand bD,R of Definition G3. A standard condition that D be a convex subset of C" = 1R211 in the ordinary sense is that -log dD be a convex function of D; what will be demonstrated is that D is pseudoconvex precisely when -log dD or -log bD,R is plurisubharmonic. The other characterization refers to the continuity theorem, Theorem D6. That theorem asserts that a function holomorphic in an open neighborhood of an annulus (A(A'; R') - ~(A'; S')) x A" ~ .C"· X en, where A(A'; S') ~ ~(A'; R'), and also holomorphic in open neighborhoods of polydiscs A(A'; R') x A~, where A;' -+ A", must extend to a function holomorphic in an open neighborhood of the limit polydisc A(A'; R') x A". So if D ~ C" is a domain of holomorphy, then it must be impossible to fit such a family of polydiscs into C" in such a way as to lead to holomorphic extensions beyond D. Perhaps the easiest way to use the continuity theorem in this reverse direction is in the following terms. 2. DEFINITION. An open subset D ~ C" is pseudoconvex in the sense of Hartogs if whenever !:: [0, 1] x A(O; 1) -+ C" is a continuous mapping from the compact subset [0, 1] x ~(O; 1) ~ IR x C into en such that F is holomorphic in A(O; 1) for each fixed point of [0, 1] and that F(t, z) E D if either 0 ~ t < 1, Z E A(O; 1) or t = 1, Z E o~(O; 1), then F([O, 1] x A(O; 1)) ~ D.

In other words, in a suggestive but less precise reformulation, D is pseudoconvex in the sense of Hartogs if for any continuous family of holomorphic discs

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en for 0 ~ t ~ 1, whenever .1, ~ D for 0 ~ t < 1 and O~1 ~ D, then .11 ~ D. Pseudoconvexity in the sense of Hartogs is then equivalent to ordinary pseudoconvexity, as follows. ~, ~

3. THEOREM. (i) (ii) (iii) (iv) (v)

For any open subset D

~

en the following conditions are equivalent:

D is pseudoconvex. D is pseudoconvex in the sense of H artogs.

The function -log dD is plurisubharmonic in D. For any polyradius R, the function -log t5D ,R is plurisubharmonic in D. There is a polyradius R such that the function -log t5D ,R is plurisubharmonic in D.

Proof. The first step is to show that condition (i) implies condition (ii). Suppose therefore that Dis pseudoconvex and that F: [0, 1) x .1(0; 1] --+ en is a continuous mapping such that F is holomorphic in ~(O; 1) for each fixed point of [0, 1] and that F(t, z) E D if either t E [0, 1), z E .1(0; 1) or t = 1, z E o~(O; 1); what must be shown is that F([O, t] x .1(0; 1)) ~ D. Now the image F([O, t] x O~(O; t)) = K is a compact subset of D, so its plurisubharmonically convex hull KD must also be a compact subset of D. For any fixed t E [0, 1) and any continuous plurisubharmonic function u in D, it follows from Theorem K12 that the composition u(F(t, z)) is subharmonic in z E ~(O; 1). Since this composition is also continuous in z E .1(0; 1), it then follows easily from the maximum theorem for subharmonic functions, Theorem J2(b), that u(F(t, a)) ~ sUPzeo.1(O; 1) u(F(t, z)) for any point a E .1(0; 1). Thus, u(F(t, a)) ~ supz eK u(Z), and since this holds for all u, necessarily F(t, a) E KD' But that means that F([O, 1) x .1(0; 1)) ~ KD, and since KD is compact and F is continuous, it then follows that F([O, 1] x .1(0; 1)) ~ KD ~ D as desired. The next step is to show that condition (ii) implies conditions (iii) and (iv); essentially the same proof gives both implications. Suppose therefore that D is pseudoconvex in the sense of Hartogs, and as a convenient abbreviation let d stand for either dD or t5D ,R in this portion ofthe proof. Since d is easily seen to be continuous in D, in order to show that d is plurisubharmonic it is sufficient just to verify that its restriction to any complex line through any point of Dis subharmonic. For that purpose it is convenient to use the subharmonicity criterion of Theorem J7(iv). Thus, consider a closed disc, described by parameter values z E .1(0; 1), lying in the intersection of D with a complex line described parametrically by {A + Hz: z E q, and suppose that p is a complex polynomial in one variable such that -log d(A + Hz) ~ Re p(z) whenever z E o~(O; 1); what must be shown is that -log d(A + Hz) ~ Re p(z) whenever z E ~(O; 1). Now for any fixed value of z the inequality -log d(A + Hz) ~ Re p(z) can obviously be rewritten as d(A + Hz) ~ le-P(Z)I, and in view of Definition G3 that amounts to the condition that H(A + Hz; le-P(Z)!) ~ D if d = dD or to the condition that ~(A + Hz; le-p(z)IR) ~ D if d = bD,R' These two conditions in turn can quite evidently be rewritten in the common form A + Hz + We-p(z) E Dforall vectors Win an open subset E ~ en, where E = H(O; 1) if d = dD or E = ~(O; R) if d = bD,R' Therefore, if A + Hz + We-p(z) E D for all z E o~(O; 1), WEE, what must be shown is that A + Hz + We-p(z) E D for all z E ~(O; 1), WEE. For a fixed WEE, introduce the mapping F: [0, 1] x K(O; 1) --+ en defined by F(t, z) = A + Bz + Wte-P(:), noting that F is continuous and that F

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is holomorphic in L\(O; 1) for each fixed point in [0, 1]. The assumption above implies that F(t, z) E D_ if t E [0, 1], Z E aL\(O, 1), since clearly tW E E whenever t E [0, 1], while F(O x L\(O; 1)) £; D, since that is just the closed disc in D initially considered. If 1= {t E [0; 1]: F(t x L\(O; 1)) £; D}, it is evident from the continuity of F that I is an open subset of the unit interval, while it follows from the assumption that D is pseudoconvex in the sense of Hartogs that I is a closed subset of the unit interval. On the other hand, I is nonempty since 0 E I as observed above; hence the connectedness of the unit interval implies that 1= [0; 1]. That, of course, means that F([O; 1] x L\(O; 1)) £; D as desired. That condition (iv) implies condition (v) is quite obvious. It therefore remains only to show that condition (iii) or condition (v) implies condition (i), and again the same proof gives both implications. If -log dD is pseudoconvex and if K is a compact subset of D, then for any point A E KD necessarily -log dD(A) ~ C, where C = SUPZeK( -log dD(Z)), or equivalently dD(A) ~ e- c. But that means that KD is disjoint from the open neighborhood {Z ED: dD(A) < e- C } of the boundary aD in D, and as noted before that is all that is required to show that D is pseudoconvex. The same argument works for the distance function ~D.R for any fixed polyradius R, and the proof is thereby concluded. It is perhaps worth pointing out that the proof of the preceding theorem actually yields a slightly stronger result than stated, for the functions dD and ~D.R can be replaced by any function d of the form d(Z) = sup {t E ~ : Z + t WED for all vectors WEE}, where E is any open neighborhood of the origin in en with the property that tE £; E whenever t E [0, 1]. The particular shape of the set E, whether ball, polydisc, or something more general, really played no role in the proof. In practice, these more general distance functions d seem not as yet to have been very much used. Note that the condition that -log d is plurisubharmonic implies that lid = exp( -log d) is plurisubharmonic and the function lid also tends to + 00 upon approaching the boundary. So the theorem can be expressed in terms of properties of lid rather than of -log d if desired. It is also worth observing as a consequence of the preceding theorem that the condition that an open subset D £; en be pseudoconvex can be demonstrated by exhibiting a single plurisubharmonic function having suitable properties, such as the function -log dD , which tends to + 00 at each point of D. A variety of other functions will also suffice for this purpose. To be more precise, it is convenient to introduce the following useful auxiliary notion.

4. DEFINITION. A continuous function u: D -+ ~ in an open subset D £; en is an exhaustion function for D if for any real number r the set {Z E D: u(Z) ~ r} is a compact subset of D. In this definition the requirement that u be a continuous function is not unnatural, since the condition that the set {Z E D: u(Z) ~ r} be compact for any r implies that u is lower semicontinuous, while the functions of greatest interest here are also upper semicontinuous. It is, of course, possible to alter the preceding definition by requiring only that the closure of the set {Z E D: u(Z) ~ r} in C be

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a compact subset of D, or even just a subset of D. The latter condition merely amounts to requiring that the set {Z ED: u(Z) ~ r} be disjoint from an open neighborhood of iJD, and in both cases there is no longer any reason to require that the function u be continuous. The choice of definition is mostly a matter of taste, the essential results being more or less equivalent in any case. The form of the definition used here was chosen to parallel more closely the definitions of pseudoconvexity and holomorphic convexity. The relevance of this notion lies in the following simple consequence of the preceding theorem. An open subset D s;;; en is psuedoconvex if and only if it admits a continuous plurisubharmonic exhaustion function.

5. COROLLARY.

Proof. If D admits a continuous plurisubharmonic exhaustion function, then it is obvious from the definition that D is pseudoconvex. Conversely, if D is pseudoconvex, then by Theorem 3 the function -log dD is a continuous plurisubharmonic function in D. This function tends to + 00 at each point Z E iJD and so is itself a continuous exhaustion function if D is a bounded subset of en, but not necessarily otherwise. In the general case, though, note that the function v defined by v(Z) = IZll + ... + IZnl is a continuous plurisubharmonic function in D and hence that the function u = sup( -log dD , v) is also a continuous plurisubharmonic function in D. Now

{Z ED: u(Z)

~

r} = {Z ED: -log dD(Z)

~

r} n {Z ED: v(Z)

~

r}

and since the first set on the right-hand side of this equality is a closed subset of en contained in D while the second set is a closed bounded subset of en, their intersection is a compact subset of D. Therefore u is a continuous plurisubharmonic exhaustion function for D, and the proof is thereby concluded. It is perhaps worth remarking that the preceding corollary nicely exhibits the greater flexibility of plurisubharmonic than of holomorphic functions; a holomorphically convex subset D s;;; en does not admit an exhaustion function of the form If I where f is holomorphic in D. That is a well-known observation in the case n = 1, and is equally easy to see in the case n > 1. Indeed, if If I is an exhaustion function for a connected open subset D s;;; en where f E l!JD , n > 1, then f is nonzero outside a compact subset K s;;; D and thus Iffis holomorphic in D - K. It then follows from Hartogs's extension theorem, Theorem E6, that Iff is holomorphic in D. Since If I is an exhaustion function, Iff must attain its maximum modulus at some point of D and so must be constant by the maximum modulus theorem, and that is clearly an impossibility. Other properties of pseudoconvex domains, easily derived from Theorem 3, provide yet more examples illustrating the relative simplicity of pseudo convexity.

6. THEOREM.

If D. are pseudoconvex open subsets of en with D. s;;; D.+1' then their union

D= U.D. is also pseudoconvex.

Proof. If ZED,., so that the values dD,(Z) are well defined whenever v ~ J1., it is obvious that dD,(Z) ~ dD,+l (Z) and that dD(Z) = lim. dD,(Z). Consequently the functions -log dD,(Z) form a monotonically decreasing sequence of functions converg-

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ing to -log dD(Z). Since the sets Dv are assumed to be pseudoconvex, it follows from Theorem 3 that the functions -log dD,(Z) are plurisubharmonic, and by Theorem K5(g) the limit function -log dD(Z) is also plurisubharmonic. Indeed, -log dD(Z) is plurisubharmonic in any of the sets D" and hence in their union D, plurisubharmonicity being a local condition. Another application of Theorem 3 shows that D is then pseudoconvex, and that concludes the proof.

7. THEOREM.

The intersection of any two pseudoconvex sets in

en is also pseudoconvex.

Proof. If Dl and D2 are pseudoconvex sets in en and K £; Dl () D2 is a compact subset, then KD,nD2 £; lSD, () KD2 , since there are more plurisubharmonic functions in a smaller set. Thus, K D,nD2 is contained in a compact subset of Dl () D2 and so must itself be compact; that suffices to prove that Dl () D2 is pseudoconvex as desired.

8. THEOREM. If D is an open subset of en and if each point A E aD has an open neighborhood UA such that UA () D is pseudoconvex, then D is necessarily pseudoconvex. Proof. First suppose that D is a bounded open set in en satisfying the hypotheses of the theorem. It follows from Theorem 3 that for each point A E aD the function -log dUAnD is plurisubharmonic in the set UA () D. However, whenever Z E UA () D is sufficiently near A, it is obvious that -log dUAnD = -log dD(Z), and consequently -log dD is plurisubharmonic in D near A. But this is true for every point A E aD, and therefore -log dD is plurisubharmonic in an open neighborhood of aD in D. Since D is bounded, it follows that there is a compact subset K £; D such that -log dD is plurisubharmonic in D - K. Now choose a point A ¢ K and introduce the continuous plurisubharmonic function v in en defined by v(Z) = rLi IZi - ail, where r > 0 is sufficiently large that v(Z) ~ -log dD(Z) for all points Z in an open neighborhood of K. There obviously exists such a constant r, since v(Z) # 0 whenever Z E K and K is compact. The function u = sup (v, -log dD) is then plurisubharmonic in D - K, since both v and -log dD are, and coincides with v in an open neighborhood of K so is also plurisubharmonic there. Thus, u is a continuous plurisubharmonic function in D. On the other hand, u tends to + 00 upon approaching any point of aD, and since D is bounded, that implies that u is an exhaustion function for D. It then follows from Corollary 5 that D is pseudo convex as desired. Next if D is an arbitrary open set in en satisfying the hypotheses of the theorem, it is apparent that the intersection D () B(O; v) for any integer v ~ 0 is a bounded open set in en also satisfying the hypotheses of the theorem, in view of Theorem 7 and the observation that B(O; v) is a domain of holomorphy and hence is pseudoconvex. It then follows from the first part of the proof of this theorem that D () B(O; v) is pseudoconvex; but since D () B(O; v) £; D () B(O; v + 1) and D = UvD () B(O; v), it further follows from Theorem 6 that D is pseudoconvex, and that suffices to conclude the proof.

The preceding theorem shows that the pseudoconvexity of an open subset

D

£;

en is a local property of aD. If D is pseudoconvex, then Theorem 7 shows that

each point A

E

aD has arbitrarily small open neighborhoods UA such that UA

()

D

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is pseudo convex, when UA are arbitrarily small balls centered at A, for example, while Theorem 8 shows conversely that this local pseudoconvexity of D implies that Dis pseudoconvex. It is interesting to note, as an immediate corollary of Theorems 3 and 8, that if -log d Dis plurisubharmonic in an open neighborhood of aD in D, then it is necessarily plurisubharmonic throughout D. The same assertion of course holds for the function -log ~D.R for any polyradius R. As already observed, any holomorphically convex set in en is pseudoconvex. After the inverse implication is established, and it is thus demonstrated that holomorphic convexity and pseudoconvexity are equivalent notions for open sets in en, the preceding results extend immediately to hold for domains ofholomorphy. These results are very useful and not at all easy to demonstrate directly. The analogue for domains of holomorphy of Theorem 6, the assertion that an increasing union of domains ofholomorphy in en is again a domain ofholomorphy, was first established by Behnke and Stein. The problem of demonstrating that holomorphic convexity is a local property-that is, of demonstrating the analogue of Theorem 8 for domains ofholomorphy-is usually referred to as the Levi problem. This problem was solved in e 2 by Oka, and his solution was extended to en by Bremermann and Norguet. In one special case this result can be obtained quite simply, by comparing Theorems D11 and GlO, and the following. 9. THEOREM. A tube domain in en with base B is pseudoconvex precisely when B is a convex subset of ~n in the usual sense. Proof. For a tube domain D = B + mn £; en, it is evident that dD(Z) depends only on the real part of Z and hence can be viewed as a function in B, and it then follows from Theorem K13 that -log dD(Z) is a plurisubharmonic function of ZED precisely when -log dD(X) is a convex function of X E B. So from Theorem 3 and one of the known characterizations of convex sets, D is pseudoconvex precisely when B is convex. That suffices to conclude the proof.

With some of the basic elementary properties of pseudoconvex sets having been demonstrated, it is perhaps now worth returning to the definition of pseudoconvexity in order to examine other notions of convexity arising from the choice of families of plurisubharmonic functions other than the family of continuous plurisubharmonic functions. Actually all of these notions coincide, as will next be seen. First, to begin with the family of all plurisubharmonic functions, note the following result. 10. THEOREM. If D is a pseudoconvex open subset of en and K is a compact subset of D with plurisubharmonically convex hull K D, then for any plurisubharmonic function u in D and any point A E K D, necessarily u(A) ~ sUPZeKu(Z). Proof. The conclusion, of course, holds for any continuous plurisubharmonic function u by the very definition of pseudoconvexity; the point is to show that it holds for an arbitrary not necessarily continuous plurisubharmonic function u. Suppose, to the contrary, that there exist a plurisubharmonic function u and a point A E KD such that u(A) > sUPZeK u(Z). and choose a constant m with u(A) > m >

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suPZ EK u(Z). Since U is upper semicontinuous, the set U = {Z ED: u(Z) < m} is an open neighborhood of K in D. Choose a continuous plurisubharmonic exhaustion function v for the set D, noting that the existence of such a function is guaranteed by Corollary 5, since D is pseudoconvex by hypothesis. By adding a suitable real constant to v, it can evidently be arranged that v(Z) ~ 0 whenever Z E AuK. Then for any fixed value b > 0, the ~et V = {Z ED: v(Z) < b} is an open neighborhood of AuK in D, and its closure V ~ {Z ED: v(Z) ~ b} is a compact subset of D. Now consider the function u, associated to U as in Definition K9, where e is chosen sufficiently small that A(Z; e) ~ U ~ D whenever Z E K and that A(Z; e) ~ D whenever Z E V. Recall from Theorem K 11 that u, is a COO plurisubharmonic function in the open subset D, = {Z ED: bD(Z) > e}, and that u,(Z) ~ u(Z) whenever ZED,; here e has been chosen sufficiently small that AuK ~ V ~ D,. Note that whenever Z E K and WE A(O; 1), then Z + eW E A(Z; e) ~ U so that u(Z + eW) < m, and hence from equation K(3) it follows that u,(Z) = fWEd(O; 1) u(Z +eW)O'(W) dV(W) < fWEMO;l)mO'(W) dV(W) = m. Therefore, altogether, u,(A) ~ u(A) > m > u,(Z) ~ u(Z)

whenever

Z

E

K

(1)

Finally, choose a real constant r sufficiently large that u,(Z) < rb

+ m whenever Z E oV

(2)

as is evidently possible since oV is a compact subset of D" and introduce the functions v+ and w in D defined by v+(Z) = sup(v(Z), 0) for any ZED and w(Z)

= {SUP(U,(Z), rv+(Z) + m) rv+(Z)

+m

if Z E V if ZED-V

(3)

Note that v+ is a continuous plurisubharmonic function in D, while w is a continuous plurisubharmonic function in the two open subsets V and D - V of D. Moreover, if Z E 0 V, then it follows from the definition ofthe set V that v(Z) = b > 0, and hence from (2) that u,(Z) < rb + m = rv+(Z) + m. But then in view of the definition (3) of the function w, it is evident that w coincides with rv+ + m in an open neighborhood of oV in D, and therefore w is actually a continuous plurisubharmonic function in D. Now if Z E K ~ V, then from (1) it follows that u,(Z) < m ~ rv+(Z) + m and therefore w(Z) = rv+(Z) + m = m, since by construction v(Z) ~ 0 whenever Z E K. But on the other hand, for the point A E V it is also the case that v(A) ~ 0 by construction, so it follows from (1) that u,(A) > m = rv+(A) + m and consequently w(A) = u,(A) > m = SUPZEK w(Z). That contradicts the assumption that A E KD and thereby concludes the proof. This theorem shows that in a pseudoconvex open set D ~ en the plurisubharmonicially convex hull of any compact subset K ~ D can be defined either by using only continuous plurisubharmonic functions as in Definition 1 or by using arbitrary

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plurisubharmonic functions. Consequently, with the latter definition the plurisubharmonically convex hull is still necessarily a closed subset of D-indeed, a compact subset of D-even though the separate sets {A ED: u(A) ~ sUPzeKu(Z)} need not be closed when u is not continuous. Although this was established only for pseudoconvex open sets in en, it is true without the explicit assumption of pseudoconvexity in the following sense.

11. THEOREM.

For an open subset D

~

en the following conditions are equivalent:

(i) D is pseudoconvex. (ii) Whenever K ~ D is compact, then the subset { A ED: u(A)

~ sup u(Z) for all plurisubharmonic functions u in D} ZeK

is also compact.

(iii) Whenever K { A ED: u(A)

~

D is compact, then the subset

~ sup u(Z) for all plurisubharmonic functions u in D} ZeK

is disjoint from an open neighborhood of oD.

Proof. The subset of D described in the statement of condition (ii) is of course contained in KD , and if Dis pseudoconvex, it follows from Theorem 10 that this subset actually coincides with KD , which is compact; thus, condition (i) implies condition (ii). It is trivial that condition (ii) implies condition (iii). It remains only to show that condition (iii) implies condition (i), and for that purpose it is convenient to recall from Theorem 3 that D is pseudoconvex precisely when D is pseudoconvex in the sense of Hartogs. Suppose therefore that D satisfies condition (iii), and consider a continuous mapping F: [0, 1] x ;;\(0; 1) -+ en such that F is holomorphic in .::\(0; 1) for each fixed point of [0, 1] and that F([O, 1) x ;;\(0, 1» u F(1 x 0.::\(0; 1» ~ D. The image F([O, 1] x 0.::\(0; 1» = K is thus a compact subset of D, so by condition (iii) the set L = {A E D: u(A) ~ sUPZeKu(Z) for all plurisubharmonic functions u in D} is disjoint from an open neighborhood of oD, or equivalently its closure is a subset L ~ D. For any fixed point t E [0, 1), any constant e > 0, and any plurisubharmonic function u in D, note that since F(t x 0.::\(0; 1» ~ K and u is upper semicontinuous in D, then u(A) < SUPZeK u(Z) + e for all points A in an open neighborhood of F(t x 0.::\(0; 1» in D, and consequently u(F(t, z» < SUPZeK u(Z) + e for all points z in an open neighborhood of 0.::\(0; 1) in .::\(0; 1). But u(F(t, z» is actually a subharmonic function of z E .::\(0; 1), since F(t, z) is a holomorphic function of z, so it follows from the maximum theorem for subharmonic functions that u(F(t, z» < sUPZeKU(Z) + e for all points z E .::\(0; 1). This is true for any value e > 0, and therefore u(F(t, z» ~ sUPZeK u(Z) for all points z E .::\(0; 1). But the latter inequality is true for any plurisubharmonic function u, and therefore

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F(t, z) E L for all points Z E A(O; 1). Thus, altogether F([O; 1) x ~(O; 1» £ L, and since F is continuous while L £ D, it follows that F([O, 1] x ~(O; 1» £ L £ D. Therefore, D is pseudoconvex in the sense of Hartogs, and the proof is thereby completed.

Then for the consideration of the corresponding notions of convexity involving various naturally occurring subfamilies of the family of continuous plurisubharmonic functions, what is needed to settle matters is just the following result.

An open subset D £ en is pseudoconvex if and only if it admits a c n strictly plurisubharmonic exhaustion function.

12. THEOREM.

Proof. If D admits such an exhaustion function, then D is pseudoconvex as a consequence of Corollary 5. For the other implication, which is of course the main point of the theorem, suppose that D is pseudoconvex and choose a continuous plurisubharmonic exhaustion function u for D, the existence of such a function being guaranteed by Corollary 5. For any integer v, the set Dv = {Z ED: u(Z) < v} is then an open subset of D, and its closure Dv is a compact subset of D. Now consider the functions u. associated to u as in Definition K9, recalling from Theorem Kll that u. is a COO plurisubharmonic function in the open subset D. = {Z ED: t5D(Z) > e} and that u.(Z) ~ u(Z) whenever ZED•. Since u is continuous, the functions u. converge uniformly to u on any compact subset of D as e tends to zero by Lemma KI0. Thus, if ev is chosen sufficiently small, then Dv £ D., and u(Z) ~ u.,(Z) < u(Z) + 1 whenever Z E Dv. The expression IIZII 2 is easily seen to be a COO strictly plurisubharmonic function of Z E en, since a trivial calculation shows that its Levi form is LII·1I 2(Z; A) = IIAII2. Therefore, the function U v defined by uv(Z) = u. (Z) + t5v11Z112 is clearly a Coo strictly plurisubharmonic function in an open neighborhood of Dv for any t5v > 0, and if t5v is chosen sufficiently small, it can be arranged that u(Z) < uv(Z) < u(Z) + 1 whenever Z E Dv. Next choose a Coo monotonically increasing function f/J: IR -+ IR such that f/J(t) = 0 whenever t < -!, f/J(t) = 1 whenever t > 2, and f/J'(t) > 0, f/J"(t) > 0 whenever -! < t < !. In terms of this auxiliary function f/J and the previously constructed functions u v , introduce the further set of functions Vv defined by

vv(Z) =

{i(Uv(Z) + 2 -

v)

if Z E Dv if ZED - Dv

(4)

Note that u is a Coo function in an open neighborhood of Dv , and that if Z E 8Dv , then uv(Z) + 2 - v > u(Z) + 2 - v = 2; thus, uv(Z) + 2 - v > 2 for all points Z in an open neighborhood of the compact set 8Dv, so that f/J(uv(Z) + 2 - v) = 1 for all such points Z. It follows from this that Vv is actually a Coo function in the entire set D. Next note that whenever Z E Dv- s , then uv(Z) + 2 - v < u(Z) + 3 - v ~ -2 and f/J(uv(Z) + 2 - v) = 0; thus,

vv(Z)

=0

if Z

E

Dv - s

Further note that whenever Z

(5) E

Dv- 2 , then uv(Z) + 2 -

v < u(Z)

+3-

v ~ 1, and

M Pseudoconvex Subsets of C'

137

therefore uv(Z) + 2 - v < ~ in an open neighborhood of Dv- 2 • Since ¢ is a monotonically increasing convex function in ( - 00, ~), it follows from Theorem K5(d) that Vv is a plurisubharmonic function in an open neighborhood of Dv - 2 ' Finally note that whenever Z E Dv- 2 - Dv- 3 , then 1 ~ uv(Z) + 2 - v > u(Z) + 2 - v ~ -1, and therefore ~ > uv(Z) + 2 - v > -~ in an open neighborhood of the compact set D'-2 - D.- 3 • Since fil(t) > 0, ¢"(t) > 0 whenever -~ < t < ~, and the Levi form of the function v. can be written in D. as

it follows that v. is actually a strictly plurisubharmonic function in an open neighborhood of D.-2 - D.- 3 . From these properties it is easy to see that it is possible to choose real numbers c. ~ 1 sufficiently large that w. = C 1 V 1 + ... + c.v., which is a COO function in D, is plurisubharmonic in D'-2 for all integers v ~ 1. Indeed, to proceed inductively on v, set C 1 = 1 and note that W1 = V 1 is plurisubharmonic in an open neighborhood of D-1 as pointed out above. If Wv is plurisubharmonic in an open neighborhood of Dv - 2 for some integer v ~ 1, note that Vv+1 is plurisubharmonic in an open neighborhood of DV - 1 and indeed is strictly plurisubharmonic in an open neighborhood of the compact set DV - 1 - Dv - 2 ' Since Wv + 1 = Wv + Cv +1 Vv+1' it is then possible as in Theorem L2 to choose C v+1 ~ 1 sufficiently large that Wv+ 1 is plurisubharmonic in an open neighborhood of the compact set Dv - 1 - Dv - 2 , and then W.+1 will be plurisubharmonic in an open neighborhood of D'-1 as desired. Note as a result of (5) that V.+ 1 vanishes identically in D). whenever v ~ 2 + 4, and consequently W.+ 1 = W. + C.+ 1 V = w. identically in D). whenever v ~ 2 + 4. The limit w = lim .....oo w. is therefore a well-defined COO plurisubharmonic function in D, since the sequence w. is eventually stationary on any subset D). s; D. Moreover, note as a result of the definition (4) that whenever Z ¢ D)., then v.(Z) = 1 for all values v ~ 2, and therefore w(Z) ~ w).(Z) ~ C 1 + ... + c). ~ 2. The function w is therefore a COO plurisubharmonic exhaustion function for D. To obtain a strictly plurisubharmonic such exhaustion function, it is only necessary to replace w by the function associating to each point ZED the value w(Z) + bllZI1 2 for any b > 1; that suffices to conclude the proof. Example. That holomorphically convex sets are the same as domains of holomorphy might suggest that pseudoconvex sets are possibly the same as the natural domains of existence of plurisubharmonic functions. It follows from Corollary 5, for instance, that in every pseudoconvex open subset D s; en there is a plurisubharmonic function that tends to + 00 upon approaching aD and hence that cannot be extended to a plurisubharmonic function in any properly larger set. On the other hand, it is not the case that any plurisubharmonic function in an open subset D s; en necessarily extends to a plurisubharmonic function in the smallest pseudoconvex set containing D. The following simple example of that is due to H. Bremermann. Consider the tube domain D S; e 2 with base B = B1 U B 2, where BI = {(Xl' X2) E ~2: 2 < SUP(IXII, Ix 2 1) < 4} and B2 = {(Xl' X2) E ~2: 0 < Xl ~ 2, IX21 < I}, as sketched in Figure 9. The base B is not convex, so by Theorem 9 the

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Figure 9

set D is not pseudoconvex. If every plurisubharmonic function in D extends to a pseudoconvex set E ::::l D, then since D is preserved by arbitrary translations in the imaginary direction, it is evident that E must have the same property, so that E must also be a tube domain. Indeed, clearly E must be the tube domain with base B the convex hull of B, the set B = {(Xl' X 2 ) E ~2 : sup(lxd, Ix 2 1) < 4}. Now the function u in B defined by if if

(Xl' X 2 )

E Bl

(X l ,X2)EB2

is of class C 2 and is convex, since ipujaxi = 6(2 - xd ~ 0 in Bl and a2 ujaxi = 0 if Xl = 2. Thus by Theorem K13, when viewed as a function in D depending only on the variables (Xl' x 2 ), the function u is plurisubharmonic in D. It is clear that this function u cannot be extended to a plurisubharmonic function in E, for since u is zero in the tube over Bl but takes positive values at points in the tube over B2 , the extended function would attain a positive maximum value at an interior point of E, contradicting the maximum theorem for plurisubharmonic functions.

N Pseudoconvex Riemann Domains

The notion of pseudoconvexity can easily be extended to Riemann domains, and the results about pseudoconvex open subsets of en obtained in the preceding section can be shown also generally to hold for Riemann domains. 1. DEFINITION. A Riemann domain M is pseudoconvex if for any compact subset K !;; M the set KM = {A E M: u(A) ~ sUPzeKu(Z) whenever u is a continuous plurisubharmonic function on M} is also compact. The set KM is the plurisubbarmonically convex bull of K inM.

The distance functions dM and ~M.R can be introduced for Riemann domains as in Definition H2, paralleling the corresponding definitions for open subsets of en, and can be used to characterize pseudoconvex Riemann domains in the same way that they were used to characterize pseudoconvex open subsets of en. The definition of an exhaustion function can be extended without change to Riemann domains, and can be shown to play the same role for Riemann domains as for open subsets of en in the discussion of pseudoconvexity. On the other hand, the notion of pseudoconvexity in the sense of Hartogs requires some slight modification in its extension to Riemann domains, for the condition that an open subset D !;; en be pseudoconvex in the sense of Hartogs as in Definition M2 involved properties of mappings into the canonical pseudoconvex set en containing D, while Riemann domains cannot always be viewed as subsets of some canonical larger pseudoconvex Riemann domains. Indeed, it is perhaps worth mentioning here, without going into the details at present, though, that there exist noncom pact Riemann domains that are maximal in the sense that they cannot be viewed as proper subsets of any other Riemann domains. 2. DEFINITION. A Riemann domain M with projection P: M -+ en is pseudoconvex in tbe sense of Hartogs if whenever-.!: [0, 1] x [\(0; 1) -+ en is a continuous mapping from the compact subset [0, 1] x A(O; 1) !;; ~ x e into en such that F is holomorphic in A(O; 1) for each fixed point of [0, 1] and G: ([0, 1) x [\(0; 1» u (1 x oA(O; 1» -+ M

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is a continuous mapping such that P mapping G: [0, 1] x [\(0; 1) -+ M.

0

G = F, then G can be extended to a continuous

It should be noted that the continuity of the extension Gimplies that po G= F and that Gis uniquely determined; the mapping Gis of course also holomorphic in .1(0; 1) for each fixed point of [0, 1]. If M is an open subset of en and P: M -+ en is the inclusion mapping, then the preceding definition reduces immediately to Definition M2. In general, the preceding definition modifies Definition M2 merely by replacing the condition that the values of F lie in D by the condition that F lifts to a mapping into M.

3. THEOREM.

For a Riemann domain M, the following conditions are equivalent:

M is pseudoconvex. M is pseudoconvex in the sense of H artogs. The function -log d M is plurisubharmonic on M. For any polyradius R the function -log (jM,R is plurisubharmonic on M. There is a polyradius R such that the function -log (jM,R is plurisubharmonic on M. (vi) M admits a continuous plurisubharmonic exhaustion function. (vii) M admits a C''' strictly plurisubharmonic exhaustion function. (i) (ii) (iii) (iv) (v)

Proof. The demonstration that condition (i) implies condition (ii) is essentially the same as the demonstration of the corresponding assertion in the proof of Theorem M3. If M is a pseudo convex Riemann domain with projection P: M -+ en and if F: [0, 1] x [\(0; 1) -+ en and

G: ([0, 1) x [\(0; 1)) u (1 x aA(O; 1)) -+ M are mappings as in Definition 2 with po G = F, then the image K = G([O, 1] x aL\(O; 1)) is a compact subset of M and hence has a compact plurisubharmonically convex hull KM . For any fixed point Z E [\(0; 1) the set of points G(t, z) for t E [0,1) must therefore have at least one limit point G(I, z) E M as t approaches one; but since P is then a homeomorphism from an open neighborhood of G(I, z) in M onto an open neighborhood of P(G(I, z)) = F(I, z) in en, it follows immediately that G(t, z) extended to the value t = 1 by G(I, z) is actually a lifting of the full path {F(t, z): 0 ;£ t ;£ I}. That provides the desired extension of the mapping G and shows that M is pseudoconvex in the sense of Hartogs as desired. The demonstration that condition (ii) implies conditions (iii) and (iv) is also very much like the demonstration of the corresponding assertion in the proof of Theorem M3. Suppose that M is a Riemann domain that is assumed to be pseudoconvex in the sense of Hartogs, let P: M -+ en be its projection, and as in the proof of Theorem M3, let d denote either dM or (jM,R' Since it is easily seen that d is continuous, in order to show that -log d is plurisubharmonic, it is sufficient to show that its restriction to any complex line through any point of Mis subharmonic, the lines being those defined in terms of the coordinates induced on M by the projection P: M -+ ICn; again it is convenient t9 use the subharmonicity criterion of

N Pseudoconvex Riemann Domains

141

Theorem J7(iv) for this purpose. Thus, suppose that H is a hoi om orphic mapping from an open neighborhood of the closed unit disc L\(O; 1) ~ e into M such that P(H(z» = A + Bz for some A, BEen, and suppose that p is a complex polynomial in one variable such that -log d(H(z» ~ Re p(z) whenever z E oA(O; I)-that is, such that d(H(z» ~ Ie-p(Z) I whenever z E oA(O; 1). What must be shown is that -log d(H(z» ~ Re p(z) whenever z E A(O; I)-that is, that d(H(z» ~ le-P(Z)I whenever z E A(O; 1). Recall from Definition H2 that d(H(z» is the largest real number for which there exists an open neighborhood ofthe point H(z) in M that is mapped homeomorphically by the projection P to the open neighborhood P(H(z» + rE of the point P(H(z» in en, where E = B(O; 1) if d = dM or E = A(O; R) if d = c5M ,R' For any point WEE, introduce the mapping Fw: [0, 1] x L\(O; 1) -+ en defined by Fw(t, z) = P(H(z» + te-p(z)w. Note that Fw(O, z) = P(H(z», and therefore that the mapping Gw: 0 x L\(O; 1) -+ M defined by Gw(O, z) = H(z) has the property that P(Gw(O, z» = Fw(O, z). Now for any fixed point z E L\(O; 1), it is clearly possible to extend Gw to a continuous mapping Gw: [0, c5z ) x z -+ M such that P(Gw(t, z» = Fw(t, z). Since L\(O; 1) is compact, there is therefore an extension of Gw to a mapping Gw: [0, (j) x L\(O; 1) -+ M for some value (j > 0, such that P(Gw(t, z» = Fw(t, z); in particular, let c5 be the largest value in [0, 1] for which there exists such an extension, and note that necessarily c5 = 1. Indeed, since d(H(z» ~ le-P(Z)I whenever z E oA(O; 1) and since Fw(t, z) E P(H(z» + le-p(z)IE whenever t E [0, 1], it follows that the mapping Gw can be extended to a continuous mapping

Gw : ([0, (5) x L\(O; 1» u ([0, 1] x oA(O; 1» -+ M such that P(Gw(t, z» = Fw(t; z). But since M is assumed to be pseudoconvex in the sense of Hartogs, it follows that Gw can be extended to a continuous mapping

Gw: ([0, c5] x L\(O; 1» u ([0, 1] x oA(O; 1» -+ M such that P(Gw(t, z» = Fw(t, z), and if (j < 1 it is easy to see using continuity alone that there is a further extension of Gw to a continuous mapping Gw : [0, (j') x L\(O; 1) -+ M covering Fw for some value (j' > (j, contradicting the maximality of (j. Thus, there must exist a continuous mapping Gw: [0, 1] x L\(O; 1) -+ M such that Gw(O; z) = H(z) and P(Gw(t, z» = P(H(z» + te-p(z)W for any element WEE; but that is evidently enough to show that d(H(z» ~ le-P(Z)I whenever z E A(O; 1), as desired. That condition (iv) implies condition (v) is obvious. The proof that condition (iii) or (v) implies condition (vi) is necessarily somewhat more difficult than the proof of the corresponding implication for open subsets of en in Corollary M5, because it is necessary to take into account the possibility that the Riemann domain is not finitely sheeted. The proof is nonetheless considerably easier that the proof of the analogous results for holomorphic rather than for plurisubharmonic functions, as will be evident upon comparing the argument to follow with that found in section I; that reflects yet again the greater abundance and flexibility of plurisubharmonic functions. Let M be a connected Riemann domain with projection P: M -+ en, and suppose that -log d is plurisubharmonic

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on M where d stands for either dM or bM,R' The first step in the proof is the construction of an auxiliary continuous exhaustion function for M. Choose a fixed but arbitrary base point B E M, and note that since M is connected, any other point Z E M can be joined to M by a continuously differentiable path. The length of that path can be defined by projecting it locally to en by the projection P: M -+ en and calculating the length piecewise by the usual differential geometric formulas in en = 1R2n. Let w(Z) denote the infimum ofthe lengths of all such paths from B to Z; it is clearly a continuous exhaustion function w for M. The function w need not be differentiable, for instance, at those points Z that can be joined to B by several distinct paths of minimal length. It is nonetheless obvious, though, that whenever Zl and Z2 are points of M that are near enough to lie in an open subset of M mapped homeomorphically into en by the projection P, then (1)

The next step consists in modifying this function w to secure a set of COO strictly plurisubharmonic functions that are in some sense approximate exhaustion functions. For any positive integer v, introduce the subset M.= {Z EM: -log d(Z) < v} = {Z EM: d(Z) > l/v}, and note that there is a constant e > 0 such that each point Z E M. has an open neighborhood that is mapped homeomorphically onto an open neighborhood of the closed polydisc l\(P(Z); e) by the projection P: M -+ en. Indeed, since d(Z) > l/v, there exists an open neighborhood of Z mapped homeomorphically onto B(P(Z); l/v) if d = dM or onto A(P(Z); R/v) if d = bM,R' and it is only necessary to choose e sufficiently small that A(O; e) c: A(O; l/v) or A(O; e) c: A(O; R/v). Use the projection mapping P to identify an open neighborhood of Z E M with A(P(Z); e); it is then possible to introduce the function w. associated to w as in Definition K9, in terms of the coordinates in en. The function w. is thus a welldefined COO function on M., and it is easy to see that it is an approximate exhaustion function for M in the sense that for any real number c the closure of the set {Z EM: w.(Z) < c} is a compact subset of M. The function w. is of course not really an exhaustion for M, since it is not defined everywhere on M, nor is it an exhaustion function for M., since it is only asserted that the closure of the set {Z E M. : w.(Z) < c} is a compact subset of M rather than of M •. To see this, note that for any real number c the set X = {Z EM: w(Z) < c} has a compact closure in M, and therefore the set X. = Uzex l\(Z; e) is a compact subset of M, where l\(Z; e) is the closure of the open neighborhood of Z in M mapped homeomorphically to A(P(Z); e) under the projection P. If Z ¢ X., then the function w is at least equal to c throughout the open neighborhood A(Z; e), and it follows from the defining formula K(3) that w.(Z) ~ c; thus, {Z E M.: w.(Z) < c} must lie in the compact subset X. ~ M and hence must have compact closure. Next to calculate the Levi form of this function w. in terms of the natural coordinates on M induced by the projection P: M -+ en, identifying an open neighborhood ofthe point Z E M with the polydisc A(Z; e) ~ en and recalling the defining equation K(3) in which u(Z) = u(z 1)" . u(zn) imply that

N Pseudoconvex Riemann Domains

=

~

=

_!~

W(S)U(Sl - Zl) ... U(S" - Z")B- 2 " dV(S)

aZj aZk JSeA(Z;.) B aZj

a

1 -a = -B

=

r

Zj

_! lim B 6-+0

B

r

W(S)U(Sl -

B

JseA(Z;.)

f f

143

B

Zl)"'Ur(Sk - Zk) ... u(s" - Z")B- 2 " dV(S) B B

w(Z

+ BT)u(td'"

w(Z

+ ~Ej + BT) -

Ur(t k)··· u(t,,) dV(T)

TeA(O;l)

~

TeA(O;l)

w(Z

+ BT) u(t 1)' .. U,(t k)· .. U(t,,) dV(T)

where Ur = au/at and Ej = (0, ... , 1, ... ,0) is the unit vector with entry 1 in the jth place. From (1), note also that

and consequently that la 2 w.(Z)/azj Ozkl ~ CB- 1 , where C = IJeuf(t) dV(T)I. Altogether, therefore,

~ CB- 1

L Iajil I ~ CB- n211A112 1

k

j,k

for any point Z on Mv by

E Mv

and any vector A

E

en. Now introduce the function Vv defined

noting that ILvv(Z; A)I = ILw.(Z; A) + 2CB- 1 n211A11 21~ CB- 1 n211A1I2 and hence that Vv is a COO strictly plurisubharmonic function on M. Clearly this function too has the property that for any real number c, the closure of the set {Z E M: vv(Z) < c} is a compact subset of M. The final step consists in combining the plurisubharmonic functions -log d and Vv appropriately so as to obtain a continuous plurisubharmonic exhaustion function for M. For any real constants Cv the sets Kv = {Z EM: -log d(Z) < v and Vv+2(Z) < cv} and Lv = {Z E ~: -log d(Z) < v + t and V.+2(Z) < c. + I} have compact closures in M and Kv s; Lv; and beginning with C 1 = 1 and choosing

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sufficiently large constants Cy inductively on v yield sets for which Ly ~ K + 1 and Uy Ky = Uy Ly = M. For such constants Cyit is then possible to show by induction on v that there exists a sequence of functions Uy such that Uy is a continuous plurisubharmonic function on M y+1 , that uy(Z) > Jl. whenever Z E Ly - L" for Jl. < v, and that uy(Z) = U y- 1 (Z) whenever Z E K y- 1 . The function U defined by u(Z) = limy uy(Z) is then clearly a continuous plurisubharmonic exhaustion function on M as desired. The induction can be begun by setting u 1 = V2; then if the functions U 1 , ••• , U Y - 1 have been constructed for some index v ~ 2, it is only necessary to construct a continuous plurisubharmonic function U y in M Y+ 1 such that uy(Z) > v - I whenever Z E Ly - L y- 1 , that uy(Z) > v - 2 whenever Z E Ly - L y - 2 , and that uJZ) = U y- 1 (Z) whenever Z E Ky- 1 • For this purpose, note that the function uy defined on M Y+ 1 by Y

Uy(Z) = sup( -log d(Z) - v

+ 1, vy+ 1 (Z) -

cy-d

is a continuous plurisubharmonic function on M y+ 1 , and that K y- 1 = {Z EM: uy(Z) < O}. The compact subsets L y - L y- 1 and oL y_1 are disjoint from Ky- 1 ; hence, it is possible to choose a real constant by sufficiently large that byuy(Z) > v - I whenever Z E Ly - L y- 1 and that byuy(Z) > U y- 1 (Z) whenever Z is in an open neighborhood of oL y - 1 • The function U y defined in M Y +1 by

U

y

(Z) = {SUP(U Y _ 1 (Z), byuy(Z» byuy(Z)

if Z if Z

E

L y-

E

MY +1

1

-

L y-

1

is then easily seen to have the desired properties, thus concluding this part of the proof of the theorem. The proof that condition (vi) implies condition (vii) is exactly the same as the proof of the corresponding implication for open subsets of e" in Theorem MI2. From a continuous plurisubharmonic exhaustion function u on M, the COO functions U ycan be introduced on M as the proof of the preceding implication just above, and the remainder of the argument proceeds formally the same on a Riemann domain or in an open subset of e". Finally it is quite obvious that condition (vi) or (vii) implies condition (i), and that suffices to conclude the proof of the entire theorem. The preceding theorem provides a very direct extension of some properties of pseudoconvex open subsets of e" to the corresponding properties of pseudoconvex Riemann domains. The statements are the same in both cases, although some of the proofs are rather more complicated in the case of Riemann domains. On the other hand, there are difficulties even in extending the statements of some other properties of pseudoconvex open subsets of e" to the case of Riemann domains; for instance, it is not always possible to make sense of the intersection of two Riemann domains, nor is it always possible to speak of the boundary of a Riemann domain, since as mentioned earlier there are maximal noncompact Riemann domains. These difficulties can be avoided quite naturally by limiting the consideration to subsets of a fixed Riemann domain, reflecting the fact that the open subsets of e" are after all

N Pseudoconvex Riemann Domains

145

subsets of the same fixed domain en. Any open subset of a Riemann domain inherits the natural structure of a Riemann domain by restricting the projection mapping, so it is possible to speak of any subset of a Riemann domain as being a pseudoconvex set. From these observations, Theorems M6, M7, and M8 extend to Riemann domains as follows. 4. THEOREM. If a Riemann domain M can be written as the union of open subsets M. £;; M where M. £;; M.+1and each subset M. is pseudoconvex, then M is itselfpseudoconvex.

Proof. As in the proof of Theorem M6, the function -log dM is the limit of the monotonically decreasing sequence offunctions -log dM ,. If M. are pseudoconvex, then the functions -log dM , are plurisubharmonic by Theorem 3, the limit function -log dM is then also plurisubharmonic by Theorem K5 (g) and another application of Theorem 3 shows that Mis pseudoconvex as desired, thus concluding the proof. 5. THEOREM. If MI and M2 are pseudoconvex open subsets of a Riemann domain M, then MIn M 2 is also pseudoconvex.

Proof. As in the proof of Theorem M7, for any compact subset K £;; MI n M 2 , clearly KMlnM2 £;; KMI n K M2 , from which the desired result is an immediate consequence. 6. THEOREM.

If M is an open subset of a pseudoconvex Riemann domain N and if each point

A E aM has an open neighborhood UA in N such that UA n Mis pseudoconvex, then M is itself pseudoconvex.

Proof. First suppose that the point set closure of M in N is a compact subset M £;; N. As in the proof of Theorem M8, for each point A E aM it is clear that -log dM(Z) = -log duAnM(Z) whenever Z E JtA n M for a sufficiently small open subneighborhood JtA £;; UA of A. Since UA n M is pseudoconvex by hypothesis, it follows from Theorem 3 that -log dUAnM is plurisubharmonic inYA n M, and consequently -log dM is plurisubharmonic in UA (JtA n M). Since M is compact, it is evident that K = M - UA (VA n M) is also compact. Note that the function v on N defined by v(Z) = 1 + IPI (z)1 + ... + IpiZ) I is plurisubharmonic on N and bounded away from zero on the compact subset KeN, when P(Z) = (PI (Z), ... , Pn(Z)), For a positive constant r sufficiently large that rv(Z) > -log dM(Z) whenever Z E K, it follows that the function u = sup( -log dM , rv) is a continuous plurisubharmonic function on M. Since u tends to + 00 upon approaching the boundary of M, the function u is a continuous plurisubharmonic exhaustion function for M, and consequently M is pseudoconvex. Next for the general case ofthe theorem, since N is pseudoconvex by assumption, it follows from Theorem 3 that N admits a continuous plurisubharmonic exhaustion function u. If M is an arbitrary open subset of N satisfying the hypotheses of the present theorem, it is apparent that for any integer v the subset M. = {Z EM: u(Z) < v} also satisfies the hypothesis of this theorem and has a compact closure in M. It then follows from the first part of the present proof that M. is

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pseudoconvex, and since M. s;;; M.+l and U.M. = M, it follows from Theorem 4 that M is pseudoconvex. That suffices to conclude the proof. As in the proofs of Theorems MIO and Mll, it follows that for Riemann domains just as for open subsets of en, pseudoconvexity can be defined equivalently in terms of arbitrary rather than of continuous plurisubharmonic functions. It was already demonstrated in Theorem 3 that pseudoconvexity can also be defined equivalently in terms of COO strictly plurisubharmonic functions. Finally, it is useful to note the following special result.

7. THEOREM.

Anyone-dimensional Riemann domain is pseudoconvex.

Proof. Suppose M is a connected one-dimensional Riemann domain with projection P: M -+ e, and write M = U.M. where M. are open subsets of M such that M. is compact and M. s;;; M.+ 1 • For any point a E aM., some open neighborhood Ua of a in M is mapped biholomorphically by P to an open subset P(Ua ) S;;; Co The image P(Ua n M) S;;; e is a domain of holomorphy, as is any open subset of e, and hence is pseudoconvex; so by Theorem M3 the function -log dUanM• is plurisubharmonic in Ua n M •. Since it is clear that -log dM,(z) = -log dUanM,(z) whenever Z E Ua n M. is sufficiently near a, it follows that -log dM , is plurisubharmonic in U where U is the intersection with M. of some open neighborhood of the compact subset aM. of M. The function 1 + IP(z) I is plurisubharmonic in all of M, and since it is strictly positive, there is a constant c > 0 sufficiently large that c(l + IP(z)l) > -log dM,(z) whenever z E M. - U•. The function u. defined by u.(z) = sup( -log dM,(z), c(l

+ IP(z)l»

is then evidently a continuous plurisubharmonic exhaustion function for M., so that M. is pseudoconvex by Theorem 3. Incidentally, once having demonstrated that M. is pseudoconvex, it follows from Theorem 3 that -log dM • is itself a plurisubharmonic exhaustion function for M. An application of Theorem 4 shows then that M is pseudoconvex, to conclude the proof.

o Pseudoconvexity and Dolbeault Cohomology

en in section M that pseudoconvexity is actually equivalent to holomorphic convexity, and hence that the properties of pseudoconvexity established in that section automatically hold for holomorphic convexity as well; the same is true in the case of Riemann domains. The proof involves a rather detailed analysis of some formal properties of the aoperator and an application of some fairly standard techniques from the theory of linear partial differential equations to show that the Dolbeault cohomology groups Jep,q(D) of a pseudoconvex open set D £ en vanish whenever q > O. It then follows from Theorem G14 that D is a domain ofholomorphy as desired. The same argument with minimal modifications shows that these Dolbeault cohomology groups of a pseudoconvex Riemann domain also vanish, leading to the result that a pseudoconvex Riemann domain is a Riemann domain of holomorphy. Rather than repeating the argument or referring back to it in the later discussion of Riemann domains, the proof will be presented in that generality from the beginning; those readers interested only in this result for open subsets of en should not find the minor further generality at all burdensome. Suppose then that M is a Riemann domain with projection P: M -+ en. The image points Z = P(Z) E en provide natural local coordinates at all points Z E M, and differential forms on the complex manifold M will be written solely in terms ofthese coordinates in this section. Thus, a Coo differential form

-

1

_


1\'"

1\

dz,o p 1\ dzJ. '

1\'"

1\

dzJo•

(1)

where
00

(2)

147

148

Volume I

Function Theory

where (j l'

IJ,J denotes the summation over all multi-indices I = (i1' ... , ip) and J = jq) with 1 ~ i ip, j jq ~ nand It,J denotes the summation over

... ,

1, ... ,

1 , ••• ,

just those multi-indices I and J for which 1 ~ i1 < ... < ip ~ nand 1 ~ j1 < ... < jq ~ n. As a further useful notational convention, if I and J are each p-tuples of distinct integers and J is a permutation of I, let sign(IJ) denote the sign of this permutation. If there are repeated integers in either lor J or if J is not a permutation of I, set sign(IJ) = 0, while if I = (1, 2, ... , n), then set sign(IJ) = sign(J) for short. To simplify the discussion of some of the formal properties of differential forms on M, introduce the complex linear mapping

that associates to any differential form

- = jn(p+q+1) *
I*

I,J,K,L

- sign(KJ) sign(LI) dZI
1\

dZ-J

E

(3)

This duality operator has the following basic properties. 1, LEMMA.

For any differential forms
(a) If t/I = *
where dV = dX 1 1\ dY1

1\ ••• 1\

dX n 1\ dYn and C;,q is the complex constant

C;,q = 2n( _1)(1/2)n(n-1)i n(P-q) Proof. (a) If

- = ( _1)Pqi n(P+q+1) I*
t/I

(4)

I,J,K,J

since dZJ 1\ dZI = ( _1)(n- p)(n- q) dZI 1\ dZJ. On the other hand, i = ( -1)PQ It.J iJ.I dZI 1\ dZJ> so it follows from (3) that *i is also given by (4) as desired. (b) If

=

I*

I,J,K,L,M,N


1\

-J dZ

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Pseudoconvexity and Dolbeault Cohomology

149

Note that

L* sign(ML) sign(LI) =

(_1)P(n- p)

L

L* sign(ML) sign(lL) L

= (_l)p(n- p) sign(l M) and that there is a corresponding formula for the summation over K. It hence follows that

L*

*t/I = (_l)(n+l)(p+q)


sign(l M) sign(J N) dZI

1\

dZJ

1\

dZN

I,J,M,N

= ( _l)(n+l)(p+q)

1\

E

cfffjq, then rewriting *~ as in (4) yields

- = (_1)Pqi n(P+q+l).

L*

*
-

t/lM,N
sign(KJ) sign(LI) dZM

1\ dZ 1\ dZ I

J

I,J,K,L,M,N

It is evident that the nonzero terms in this summation can only be those for which

M = Land N = K, since both M and L must be complementary to I and both N and K must be complementary to J. Noting that

L* sign(KJ) sign(LI) dZ

L 1\

1\ dZ 1\ dZ

dZK

I

J

I,J

= (_l)q(n- p)

L* sign(KJ) sign(LI) dZL 1\ dZI 1\ dZ- K 1\ dZ-

J

I,J

while

and combining all these observations easily leads to the desired result and thereby concludes the proof. If u is any continuous real-valued function on M and dV denotes the lift to M by means of the projection mapping P: M --+ en of the usual Lebesgue measure on en, then e- U dV is a well-defined measure on M, and it is possible to introduce the spaces ~:.q of those differential forms (2) of bidegree (p, q) on M such that the

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Volume I Function Theory

coefficients ¢JI,I are square-integrable functions on M with respect to the measure e- u dV. Thus, 'p:,q = {¢J =

t ¢JI,J

dZI

1\

dZ/ : ¢JI,J are measurable functions on M and

1,1

(5)

It is clear from familiar properties of measure theory that 'p:,q are Hilbert spaces

with the inner products (6)

and by Lemma 1 these inner products can also be written as (7)

The associated norm is as usual defined by II¢JII~ = (¢J, ¢J)u' It is easy to see that the subspace 8!'ii c 8fiq consisting of those differential forms in 8fiq having compact support on M is a dense linear subspace of 'p:,q for any choice of u. Indeed, it is an immediate consequence of standard results of measure theory that the subspace of 'p:,q consisting of those differential forms with coefficients that are measurable functions of compact support in M is dense in 'p:,q. On the other hand, if ¢J E 'p:,q has compact support in M, then, if ¢J. is the COO differential form associated to ¢J by applying the construction of Definition K9 to each component of ¢J in terms of the natural local coordinates on M induced by the projection P: M --+ en, it is evident from that construction that ¢J. is a Coo differential form of compact support on M whenever e is sufficiently small, and it follows from Lemma KlO that the forms ¢J. converge to ¢J in the topology of the Hilbert space 'p:,q as e tends to zero. For any pair of continuous functions u and v on M the differential operator 8: 8!,i,l--+ 8!'i.tq+1 can be viewed as a densely defined linear mapping from the Hilbert space 'p:,q into the Hilbert space 'p:,q+l, and the adjoint of this linear mapping can be identified with another linear differential operator as follows.

2. LEMMA.

If u is continuous and v is continuously differentiable on M, then the linear differential operator l)uv: 8fi q+1 --+ 8fiq defined by

has the property that

o

Pseudoconvexity and Dolbeault Cohomologv

151

whenever r/J E tCf;q, I/J E tCf;q+l, and at least one of these two differential forms has compact support. Proof. If r/J E tCf;q, I/J E 8f;q+l, and one of these forms has compact support, then r/J /\ *iii E 8~i"-1 has compact support, and by (7),

= -( -1)P+q

fM r/J /\ 8*e- viii

since JM 8(e- vr/J /\ *iii) = JM d(e-vr/J /\ *iii) = 0 by Stokes's theorem. On the other hand, by Lemma 1 and (7) again

and therefore,

as desired, thereby concluding the proof. A more explicit expression for the differential operator 3)yV of the preceding lemma can be obtained by a straightforward calculation. As a preliminary, note first that whenever r/J E tCf;q has the form (2), then

=

and

* L ar/J L aK'1 sign(kK 1) dZI I,I,K k Zk

/\

dZI

(8)

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Volume I

Function Theory

~ "O¢JI K • = ( - 1)P L... L... ---;t- .slgn(kK

Now whenever 1/1

J) dZI

1\

I.l,K k

ZK

E rfffjq+1,

it follows from (3) that

L*

*0*1/1 = i n(2n+1-p-q)

-

(9)

dZJ

(O*l/1kL sign(KJ) sign(LI) dZI

1\

-

dZJ

I.l,K,L

while by (8),

and from (3) again

(*I/1)M,L = in(p+q+2)

L*

R,S

I/1R,S sign(RL) sign(SM)

Noting that

L* sign(RL) sign(LI) =

(_1)P(n- p )

L

L* sign(RL) sign(IL) L

= (_1)P(n- p ) sign(IR) and similarly that

L*

sign(SM) sign(mM K) sign(KJ) =

L* sign(SM) sign(mM

J)

M

~M

= (-lr sign(mJ S) and combining all these observations show that

*0*1/1 =

(-It(p+q)+Pi n

f

I,J,R.S

L OI/1R,S sign(IR) sign(mJ m

oZm

S) dZI

1\

dtJ

Applying this formula to the differential form e- v l/1 and recalling the defining

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Pseudoconvexity and Dolbeault Cohomology

153

equation for the operator !:luv as given in the statement of Lemma 2 lead immediately to the result that (10) It is useful also to note in passing a common special case of the preceding lemma. Iff and g are Coo functions on M, at least one of which has compact support, then applying Lemma 2 and (10) to the differential forms ,p E tf;;n-1 and 1/1 E tf;;n defined by 'I'"'" = f dz 1 A'" A dz n A dz 1 A'" dz·)-1 A dz·)+1 A' •• A dzn for a fixed index j and 1/1 = g dZ 1 A .•. A dZn A dZ 1 A ..• A dZn leads by a straightforward calculation to the result that

Thus, the differential operators o/OZj and v

v0

ov

-v

<>/, where the latter is defined by

of

<>·1= -e - e 1= - f - ) OZj OZj OZj

(11 )

are adjoint in the sense that (12)

The commutator of these two operators has the form (13)

Now consider three continuously differentiable functions u, v, w on M and the corresponding Hilbert spaces 'p.:',q, 'p:,q+1, .P$,q-1 of differential forms on M. As in Lemma 2 there is a linear differential operator !:luv: tffi q+1 --+ tffiq such that (a,p, I/I)v = (,p, !:luvl/l)u whenever ,p E tffiq, 1/1 E tffiq+1, and at least one of these forms has compact support. Correspondingly there is a linear differential operator !:lwu: tffiq --+ tffi q- 1 such that (a,p, I/I)u = (,p, !:lwul/l)w whenever ,p E tffi q- 1, 1/1 E tffiq, and at least one of these forms has compact support. These operators can be viewed as comprising sequences of linear mappings

-

,f.p,q-1 ('\ .P,p,q-1 ~ ,f.p,q ('\ .P,p,q eM

w

~wu

eM

u

a

..,.....--+ f~.. v

tf. Pi.l+ 1 ('\ .p,p,q+1 c

v

and a basic result is the following inequality due to Hormander.

(14)

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Volume I

3. LEMMA.

Function Theory

If u - w = v - u and ,p E 8{'ii for 0 < q ~ n, then

2 ~ Ilo,pllv

Proof.

~ + 2111)w",pllw2 + 2 L...

1,1

i

M

e- vi L o(v0 - u) ,pI,KI 12 dV Z"

II;

If ,p E 8{'ii for 0 < q < n, then 8,p E 8{'i.r 1 has the form (9) and hence

L* 1(8,p)1,J1 2 = L* L ~:;,!:.K o~:;, I,L sign(kK

1,1

I,I,K,L ",I uZ"

=

J) sign(tL J)

uZI

0"" oJ: L* L ~ ~ sign(kK I,K,L ",I m" OZI

tL)

For fixed indices I, K, k, where it must be supposed that k ¢ K, the summation over the indices L, t can be broken into two cases: the first that for which t = k and L = K, the second that for which t = ky for some index v in the range 1 ~ v ~ q and L is a permutation of (kKy) where Ky = (kl' ... , ky- 1 , ky+1 , ••. , k q ). Thus with due attention to signs, the whole sum above is

and the last summation can also be extended to indices k

E

K, since clearly

After a change of notation, it follows readily that

This equality also holds if,p E 8{'ii for q = n, since the left-hand side is zero in that case, while for N" = (1, ... , k - 1, k + 1, ... , n) it is evident that

hence that the right-hand side is also zero. Upon recalling (6) and using (12), this equality implies that

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Pseudoconvexity and Dolbeault Cohomology

155

(15) Next from the explicit form (10) for the differential operator that whenever tP E tf!;,l for 0 < q ~ n, then

l)wu,

it follows

In terms of the differential operator ~: defined as in (11), this can be rewritten

Hence by Cauchy's inequality,

Recalling that -v inequality that

+ 2(u -

w) = -wand using (12) show in view ofthe preceding

Adding (15) to (16) and using the commutator relation (13) lead immediately to the desired inequality and thereby conclude the proof. To apply the preceding lemma to differential forms not necessarily having compact support, it is necessary to examine more closely the approximation of arbitrary smooth differential forms by those of compact support. It was already observed that tftii is a dense linear subspace of the Hilbert space !l',J',q, so it is only necessary to consider the effects of the differential operators a and l)wu on this appr~imation. For this e..urpose, choose a sequence of open subsets U. £;; M such that U. is compact, that U. c: U.+ 1 , and that U. U. = M; and for each v choose a CX) function P. ofzompact support on M such that 0 ~ P.(Z) ~ 1 for all Z E M and P.(Z) = 1 for all Z E U•. With the notation introduced in the preceding lemma, let !=u-w=v-u.

156

Volume I

Function Theory

If rP E c9'fjq n !t',;,q, then the forms rP. = P.· rP E C!i.l converge to rP in the topology of !t',;,q. Moreover, if (}rP E !t'!',q+l and i)wurP E !t'$,q-l and if

4. LEMMA.

(17) for all indices v and all points Z E M, then the forms (}rP. E C!i.tq+1 converge to (}rP in the topology of !t'!',q+l and the forms i)wurP. E c9'!i.tq- 1 converge to i)wurP in the topology of !t'$,q-l. Proof.

From (6) and the properties of the functions P., note that

Since the functions e-ul rPI,J 12 are integrable over M as a consequence of the assumption that rP E !t',;,q, the last expression above tends to zero as v tends to infinity, thus demonstrating the first assertion of the lemma. Next it follows from a simple calculation using the explicit form (9) of the operator (} that

The first term on the right-hand side of this last inequality is bounded from above by

and so tends to zero as v tends to infinity as a consequence of the assumption that (}rP E !t'!',q+l. The second term is bounded from above by

and by (17) and the other properties of the functions P., this last expression is in turn bounded from above by

which also tends to zero as v tends to infinity as a consequence of the assumption that rP E !t',;,q. Similarly with the explicit form (10) of the operator t>wu it follows from a simple calculation that

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Pseudoconvexity and Dolbeault Cohomology

157

The first term on the right-hand side of this inequality is bounded from above by

and so tends to zero as v tends to infinity as a consequence of the assumption that 1:Jwu t/J E It'$,q-l, while the second term is bounded from above by

and so tends to zero as v tends to infinity as a consequence of the assumption that t/J E It',;,q. The remaining assertion of the lemma follows immediately from these observations, thus concluding the proof. It now follows rather easily from the preceding observations that on a pseudoconvex Riemann domain M every a-closed differential form in 8li q for q > 0 is a-exact at least as a distribution, in the following sense.

5. THEOREM. If M is a pseudoconvex Riemann domain and t/J E 8l;q for q > 0 is a Coo differential form on M with at/J = 0, then there exist Coo functions u, w on M and a differential form () of bidegree (p, q - 1) with locally square-integrable coefficients on M such that (i) (1:J wu O", ()w = (0", t/J)u whenever 0" E 8!'ii, and (ii) (a., ()w = 0 whenever • E

8!'i.tq -

2•

Proof. Write M = U. V. where V. are open subsets of M such that U. is compact and U. c V.+ 1 , and for each v choose a Coo function P. of compact support on M such that 0 ~ P.(Z) ~ 1 for all Z E M and P.(Z) = 1 for all Z E V•. Then choose a COO functionf on M such that Lk IOP.(Z)/OZkI 2 ~ ef(Z) for all indices v and all points Z E M. It is clear that there exists such a function f, for on each set VI' only the finitely many conditions corresponding to indices v > f1 are nontrivial. Since M is pseudoconvex, it follows from Theorem M12 or Theorem N3 that M admits a COO strictly plurisubharmonic exhaustion function g. The Levi form of 9 is positive definite at each point of M, so there is a continuous real-valued function r on M such that Lg(Z; A) ~ r(Z)IIAII 2 and r(Z) > 0 for each point Z E M and vector A E en. For any Coo convex and monotonically increasing function p: IR -+ IR, it follows from Theorem K5(d) that the composite function v = p 0 9 is also a Coo plurisubharmonic exhaustion function on M. A simple calculation shows that v is also strictly plurisubharmonic, indeed that _

Lv(Z; A)

_

_

_

= p'(g(Z»Lg(Z; A) + p"(g(Z))

12 L a og(Z) OZj

1

j --

j

~ p'(g(Z»r(Z) IIAII2

(18)

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Volume I

Function Theory

for each point Z EM and vector A E en. Now it is possible to choose the function p in such a manner that the given differential form rP E tf[;q lies in the Hilbert space 2,,"..'.j, that 'X:Jv- 2/,v-/rP E 2,,"..'.~fl, and that (19) for each point Z E M and vector A E en. To see this, introduce for any real number t the open subset M t £; M defined by M t = {Z EM: g(2) < t}, where M t is compact since 9 is an exhaustion function for M. Then for any positive integer v,

with

Thus, in order to have rP E 2,,"..'.j it is sufficient to choose a function p increasing sufficiently rapidly that the series Lye-P(Y)Cy converges. Similarly, in order to have 'X:Jv- 2/,v-/rP E 2,,"..'.~fl it is sufficient to choose a function p increasing sufficiently rapidly that the corresponding series with coefficients involving the differential form 'X:Jv- 2/,v-/rP also converges. Next, since

Then in view of(18), in order to have (19) hold it is enough to require that p'(t) ~ c(t) for all t, where c(t) is the monotonically increasing continuous function defined by

That too can be achieved by choosing a function p that increases sufficiently rapidly. In terms of these functions j and v, let u = v - j and w = v - 2j, so that u - w = v - u = j, consider the sequences of linear mappings (14) discussed in Lemma 3. It follows from (19) that for any differential form 1/1

E

$!'ii,

t JMr e-v{La_za2avz Iiil,kJI/II.tJ-2ILaaj I/Il'kJI2}dv~tL r e- v+/II/II,kJI 2 dV= 111/111; k,t k t k Zk k JM

I,J

1J

But then by Lemma 3,

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Pseudoconvexity and Dolbeault Cohomology

159

since Lt,J LdMe-vI8tPl,J/8zkI2 dV ~ 0 and hence this term can be dropped from the inequality. On the other hand, an application of Lemma 4, since f = v - u was constructed so as to satisfy (17), shows immediately that the inequality (20) also holds for any differential form t/! E tfffjq ( l 'p,:,q for which at/! E 'p:,q+l and !:>wut/! E .P$,q-l. The remainder of the argument is a straightforward application of the techniques of the theory of Hilbert spaces, requiring only the inequality (20) for this class of differential forms. To carry out the rest of the proof, introduce the auxiliary linear subspace ;Fp,q = {.I. If'

E

tffp,q ( l !l'.p,q , a· l• = 0 and l:l .1. M u · 'I' wu'l'

E

!l'.P,q-l} w

Note that the inequality (20) holds for any differential form t/! E ;Fp,q and that by construction the given differential form tP is contained in ;Fp,q. The closure in the Hilbert space 'p,:,q ofthe linear subspace;Fp,q £::: 'p,:,q will be denoted by [;Fp,q], and similarly the closure in the Hilbert space .P$,q-l of the linear subspace l:lwu;Fp,q = {l:lwut/!: t/! E ;Fp,q} will be denoted by [!:>;Fp,q]. Now for any element PE [!:>;Fp,q] there is a sequence of elements 0(. E ;Fp,q such that l:lwuO(. -+ P in the topology of .P$,q-l. Since {!:>wuO(.} is a Cauchy sequence in the metric in .P$,q-l and aO(. = 0, it follows from (20) that {O(.} is a Cauchy sequence in the metric in 'p,:,q, so that 0(. -+ 0( for some element 0( E [;Fp,q]. The mapping that associates to each element PE [!:>;Fp,q] the element 0( E [;Fp,q] as just constructed is evidently a well-defined linear mapping S: [l:l;Fp,q] -+ [;Fp,q]. It follows readily from the inequality (20) that S is actually a continuous linear mapping, since IISPilu = lim. 110(. IIu ~ lim V II !:>wu 0(. II w ~ II PII w; it is also in a sense the in verse of l:lwu, at least to the extent that S(!:>wuO() = 0( whenever 0( E ;Fp,q. In terms of this mapping S and the given differential form tP, introduce the continuous linear functional T;: [l:l;Fp,q] -+ C defined by T~(P) = (SP, tP)u, noting that the continuity of T~ follows from the continuity of S. As a consequence of well-known general properties of Hilbert space, there must exist an element () in the Hilbert space [l:l;Fp,q] such that T~(P) = (P, (})w' Of course, since the elements !:>wuO( are dense in [!:>;Fp,q] as 0( ranges over ;Fp,q, the characteristic property of the element () E [!:>;Fp,q] can be restated equivalently as the property that (!:>wuO(, (})w = T~(S!:>wuO() = (0(, tP)u for all 0( E ;Fp,q. The element () E .P$,q-l can be interpreted as a differential form of bidegree (p, q - 1) on M with locally square-integrable coefficients. It is then a simple matter to show that this differential form has the desired properEes. First, whenever 't' E tff/'Mq- 2 and 0( E ;Fp,q, then (a't', !:>wuO()w=(aa't', O()u=O. Thus, 8't'1. l:l;Fp,q, and since () E [l:l;Fp,q], it follows that (a't', (})w = O. Next any element U E tff/'Mq £::: 'p,:,q can be written as a sum U = 0( + 0(' where 0( E [;Fp,q] and 0(' ..1 [;Fp,q], and 0( = lim. 0(. for some differential forms 0(. E ;Fp,q. Since tP E ;Fp,q, it follows of course that(u, tP)u = (0(, tP)u = lim. (0(., tP)u = lim. (l:lwuO(., (})u, in view of the characteristic property of the element (). On the other hand, whenever y E tI/'Mq- 1 , then (!:>wuu, y)w = lim. [(!:>wuO(., y)w + CDwuu - !:>wuO(., Y)w] = lim. (l:lwuO(., y), for lim. (!:>wuu - l:lwuO(., y)w= lim. (u - 0(., ay)u=(O(', ay)u=o since ay E /Fp,q while 0(' 1. [:Fp,q]. Since tI/'Mq- 1 is dense in .P$,q-l, it follows actually that (l:lwuu, Y)w =

.J2

.J2

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limy (l)wucx., y) for all y E Sf!,q-1, in particular that (l)wuO", e)w = lim v (l)wucx v ' e); therefore, (0", (P}u = (l)wuO", e)w, and that concludes the proof, If e E Cl/q satisfies condition (i) of the preceding theorem, then (0", ae)u = (l)wuO", e)w = (0", rP)u whenever 0" E C!'ii; hence, ae = rP, For q > 1, there are a great many differential forms ewith locally square-integrable but nondifferentiable coefficients satisfying condition (i), and these forms are only solutions of the partial differential equation ae = rP in the weak sense specified in the statement of the preceding theorem. But it will be shown that any differential form e having locally square-integrable coefficients and satisfying both conditions (i) and (ii) is essentially a Coo differential form e E Cl/q and hence does satisfy ae = rP in the straightforward sense. The remainder of this section will be devoted principally to the demonstration ofthis regularity theorem. For this purpose it is convenient to use unweighted inner products on the spaces of differential forms on M, inner products (rP, 1/1) defined as in (6) but with weighting factor u = 0, so that (rP, I/I)u = (e-urP, 1/1) = (rP, e-ul/i). It then follows from Lemma 2 that there is a linear differential operator l): cfl/ q+1 -+ cfl/ q such that (arP, 1/1) = (rP, l)I/I) whenever rP E Cl/q, 1/1 E Cl/q+1, and at least one of these two differential forms has compact support. The operator l) has the explicit form (10) with u = v = 0 and hence is a first-order homogeneous linear partial d~fferential operator with constant coefficients. In terms of this operator the differential operator l)wu of Theorem 5 has the form l)wuO" = eWl)(e-uO"). Note that (l)wuO", e)w = (l)(e-uO"), e) and (0", rP)u = (e-uO", rP), so the conclusion of Theorem 5 can be rephrased as the assertion that the differential form satisfies

e

(i)

(l)cx, e) = (cx, rP) whenever cx

(ii)

(ap, e-We) = 0 whenever p E

E

C!'ii (21)

C!'i.r 2

The case q = 1 merits separate treatment, since in that case condition (ii) is vacuous and the desired regularity can readily be deduced from the following result, which is interesting in its own right and is in turn a simple consequence of previously demonstrated results. 6, THEOREM (weak form of the Cauchy-Riemann criterion).

A locally Lebesgue integrable function u in an open subset U ~ en is equal almost everywhere in U to a holomorphic function precisely when Ju u(Z) av(Z)/azj dV(Z) = 0 for I ~ j ~ n and for any Coo function v with compact support in U.

Proof. Whenever oc = and consequently

(u, l)cx) = -

~ J

Lj CXj dZj E Cc'b 1, it follows from (10) that l)cx = - Lj acxj/azj,

r u(Z) aa.a:i~Z) dV(Z)

Ju

Zj

If u is equal almost everywhere in U to a holomorphic function u', then (u, l)oc) = (u', l)oc) = (au', oc) = 0; hence, u evidently satisfies the conditions of the theorem.

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Pseudoconvexity and Dolbeault Cohomology

161

Conversely, if u satisfies the conditions of the theorem, then (u, nO() = 0 whenever 0( E Iffc~ 1; hence, in particular for any C'" function v with compact support in U and any vector A E en it follows upon setting O(j = Lk aiik iJvjiJzk that

= -

Iv

u(Z)· Lv(Z; A) dV(Z)

If v is a real-valued function, this last equality holds separately for the real and imaginary parts of u, and it then follows from Corollary K18 that the real and imaginary parts of u are equal almost everywhere in U to pluriharmonic functions. Thus, after modifying u on a set of measure zero, it can be supposed that u is a C'X! function in U; but then since 0 = (u, nO() = (au, O() whenever 0( E Iffc~ 1, it follows that au = 0 and hence that u is holomorphic in U as desired. That suffices to conclude the proof. 7. COROLLARY. If M is a pseudoconvex Riemann domain and ifJ there is a differential form 0 E Ifffio such that ao = ifJ.

E

Ifffi 1 satisfies aifJ = 0, then

Proof It follows from Theorem 5 that there exists a differential form 0 of bidegree (p,O) with locally square-integrable coefficients on M such that (nO(, 0) = (0(, ifJ) whenever 0( E Iff!'i}, from (21), and as already noted it is sufficient to show that 0 is equal almost everywhere to a C'X! differential form on M. If A is any point of M and U is an open neighborhood of A such that an open neighborhood of 0 is mapped biholomorphically into en by the projection mapping P: M --+ en and P(U) is a polydisc centered at P(A), then by Dolbeault's lemma, Theorem E3, there is a COO differential form 1/1 of bidegree (p, 0) in U such that ifJl U = al/l. Now (nO(, 0 - 1/1) = (nO(, 0) - (nO(, 1/1) = (0(, ifJ) - (0(, al/l) = 0 whenever 0( E Iff!'i}. Then from (10) it is easy to see that

so it follows from the preceding theorem that each coefficient 01 - 1/11 is equal almost everywhere in U to a holomorphic and hence COO function. Then 01 is also equal almost everywhere in U to a COO function, and the proof is thereby concluded. The general case of the desired regularity theorem cannot so readily be reduced to known results about holomorphic or harmonic functions, since condition (ii) of equation (21) involves the rather arbitrary auxiliary function w, but nonetheless can be demonstrated fairly handily using standard methods from the theory of elliptic partial differential equations. To introduce these methods, consider first an arbitrary linear partial differential operator D: Iff£jq --+ Iff£jq having constant coefficients when expressed in terms of the standard local coordinates on M. For instance, Dt/J may consist in applying the operator jJkjaz~ to one particular coefficient

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of rfo and multiplying all others by zero. It is easy to see that to any such operator D there corresponds an adjoint operator D*: cfl;q -+ cfl;q, another linear partial differential operator with constant coefficients and of the same order as D such that (DIX, fJ) = (IX, D*fJ) whenever IX E cfl;q, fJ E cfl;q, and at least one of these forms has compact support in M. Indeed, a differential form in the standard coordinates can be viewed merely as a collection of CCXl functions, with the inner product (6) in which u = 0, and any differential operator D can be expressed as some combination of the derivatives o/OZj and O/OZk acting on these coefficients. Since the operators OjOZj and -%~ are adjoint as in equation (12), it is a simple matter to calculate the desired operator D*, but the explicit forms will not be needed here, so further details will be omitted. Now let ~p,q be the space of all differential forms of bidegree (p, q) having Lebesgue measurable and locally square-integrable coefficients on M, and more generally for any integer v ~ 0, let 1f".p,q consist of all those forms rfo E ~p,q such that to each linear partial differential operator D: cfl;q -+ cfl;q with constant coefficients and of order ~ v there corresponds another form rfoD E ~p,q for which (rfo, D*IX) = (rfoD, IX) whenever IX E cf!'ii. Thus, "If;,p,q can be viewed as the subspace of ~p,q consisting of those forms for which each coefficient has square-integrable partial derivatives of order ~ v in the sense of distributions, and rfoD = Drfo also in the sense of distributions. The precise meaning of these statements consists of the definitions just given, though. It is a straightforward matter to verify that frfo E "If;,p,q whenever f E cf~o and rfo E "If;,p,q. Next note that to each differential form rfo E "If/rJ',q it is possible to associate the smoothed differential forms rfo. defined for any e > 0 by applying the construction of Definition K9 to each component of rfo when expressed in terms of the natural coordinates on M. Thus, rfo. is a well-defined CCXl differential form of bidegree (p, q) on the open subset M. = {Z EM: "M(Z) > e}, with the notation introduced in Definition H2, and it follows from Lemma K 10 that as e tends to zero, the differential forms rfo. converge to rfo in the norms (22) for any compact subsets K c M. If rfo E ~p,q, I/J E forms has compact support contained in M, then

=

t f- r

I,J

=

ZeM

Jwed

~p,q,

and at least one of these

rfol,J(Z + eW)O'(W)ifll,AZ) dV(W) dV(Z)

t f- JW'ed r_ rfol,AZ')O'(W')ifll,AZ' + eW') dV(W') dV(Z')

I,J

Z'eM

with the obvious change of coordinates

Z'= Z + eW, W' = -

W. If rfo E Gf;q, it is

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Pseudoconvexity and Dolbeault Cohomology

163

quite obvious from the formula of Definition K9 that D(r/J.) = (Dr/J). for any linear partial differential operator D with constant coefficients, so fOT a COO differential form r/J, the notation Dr/J. is unambiguous. On the other hand, if r/J E "If"vp,q and if D is a linear partial differential of order ~ v with constant coefficients, then (D(r/J.), IX) = (r/J.. D*IX) = (r/J, D*IX.) = (r/JD, IX.) = «r/JD).. IX) whenever IX E 8!'ii has support contained in M •. Consequently, D(r/J.) = (r/JD). at least in the open subset M. c M. These constructions and observations can be applied to the problem at hand by using the following elementary version of Sobolev's lemma.

8. LEMMA. If r/J E n~o "If"vp,q, then r/J is equal almost everywhere on M to a COO differential form in 8l;q. Proof. For any point A EM choose an open neighborhood .1 of A in M such that .1 is mapped biholomorphically by the projection P to an open polydisc P(.1) centered at A = P(A) in en. When considering just the neighborhood .1, it can be identified with the polydisc P(.1) without leading to any confusion. Also choose a COO function p on M such that p is identically equal to one in an open neighborhood of A and the support of p is a compact subset of .1. Then pr/J E "If':,p,q and pr/J has compact support in .1. The smoothed differential forms 1/1. = (pr/J). are COO differential forms with compact supports in .1 whenever 6 is sufficiently small, and 111/1. - Pr/JII;1 tends to zero as 6 tends to zero where the norm is that defined as in (22). It will be demonstrated that for any linear partial differential operator D: 8l;q -+ 8l;q with constant coefficients, the differential forms DI/I. converge uniformly on .1 as 6 tends to zero. That evidently implies that the differential forms 1/1. converge uniformly on .1 to a COO differential form 1/1 E 8f.,q as 6 tends to zero, and since 111/1 - Pr/JII;1 = lim._ o 111/1. - Pr/JII;1 = 0, then pr/J is equal to 1/1 almost everywhere in .1. Thus, r/J is equal to a COO differential form almost everywhere in a neighborhood of A, and that gives the desired result. To conclude the proof in this manner, consider in addition to D the linear mapping Do: 8l;q -+ 8l;q defined by applying the differential operator iP"/OX1 ... OX"OY1 ... oY" to each coefficient of a form in 8l;q, where as usual Zj = Xj + iYj. Since DoD(I/Io) = (pr/J)1joD in .1 whenever 6 is sufficiently small, it is also the case that IIDoD(I/I.) - (pr/J)DDo 11;1 converges to zero as 6 tends to zero and hence that the differential forms DoD(I/I.) form a Cauchy sequence in this norm. However, for any points B E P(.1) and C ¢ P(.1) in e" and for any COO function f of compact support in P(.1), note that

nv

~

I

IDof(Z)1 dV(Z)

from which it follows that

~ 1.111/2 II DoJII;1

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D"'. D"'.

The coefficients of the differential forms are therefore also a Cauchy sequence in the supremum norm over ~; hence, converges uniformly on ~ as e tends to zero. As already noted, that suffices to conclude the proof. 9. THEOREM. If M is a pseudoconvex Riemann domain and t/J E 8fjq for q> 0 satisfies at/J = 0, then there is a differential form e E tfffjq-l such that ae = t/J. Proof. Since the special case of the theorem in which q = 1 was already established in Corollary 7, it can be assumed for the remainder of this proof that q > l.1t follows from Theorem 5 that there exists a differential form E 1fQp,q-l satisfying conditions (21). The proof will proceed by showing by induction on v that this differential form e actually belongs to 1f:,P,q-l for all indices v ~ O. It then follows from Lemma 8 that is equal almost everywhere on M to a COO form E 8fjq-l, and as already noted this form e' has the desired property that ae' = t/J. Suppose therefore that eE 1f:,P,Q-l for some index v ~ O. Condition (i) of equation (21) asserts that rDa, e) = (a, t/J) whenever a E tff!ii, while condition (ii) asserts that (api, e-We) = 0 whenever pi E tff!Mq- 2 for some real-valued Coo function w on M. To rewrite condition (ii), set p = e-Wp', and note then that 0 = (e-Wa(eWp), e) = (ap, e) + (aw 1\ p, e). But since

e

e

e'

(ow

1\

p, e) = =

where",

E 1f:,p,q-2

-

L* L

I,I,K k

i

M

(-l)P ow .:l- PI,I sign(kKJ)()l,J dV(Z) uZk

(P, -"') is the contracted differential form

'" = owJe = (_l)P+l

ow L* Lk ~ eI,kl dZI 1\ dZI uZk

(23)

1,1

it follows that (ap, e) = (P, "') whenever p E 8!Mq- 2 • Now let D: 8fjq-l -+ 8fjq-l be an arbitrary linear partial differential operator of order ~ v with constant coefficients on M. For any fixed a E tff!Mq and all sufficiently small e, it follows from condition (i) that (a, aDe.) = (D*!)a, e.) = (tW*a., e) = (D*a., t/J) = (a, Dt/J.); hence, aDe. = Dt/J. on any compact subset of M for all sufficiently small e. Similarly, for any fixed p E tS'!MQ- 2 and all sufficiently small e, it follows from condition (ii) and the observation", E 1f:,P,Q-2 that (p, !)De.) = (D*ap, e.) = (aD*P., e) = (D*P.. "') = (P., ",D) = (P, (",D).); hence, !)DO. = (",D). on any compact subset of M for all sufficiently small e. It results therefore upon recalling Lemma KlO that as e tends to zero the differential forms aDO. converge to Dt/J and the differential forms !)DO. converge to ",D in the norms (22). Next for any compact subset K £; M, choose a Coo function p such that p is identically equal to one on K and the support of p is a compact subset of M. It is clear that a(pDe.) = p' aDO. + ap 1\ DO., and a simple calculation using (lO) shows that !)(pDe.) = p' !)DO. + opJDO. with the notation as in (23). Consequently, from the induction hypothesis and the preceding observations, the differential forms a(pDfJ.) and l)(pDO.) also converge in the norms (22) as

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Pseudoconvexity and Dolbeault Cohomology

165

6 tends to zero. Since these latter differential forms have compact support, they even converge in the norm I . II corresponding to K = M in (22), and from Lemma 3 for the special case u = v = w = 0 it follows that functions arising by applying O/OZk to any coefficient of the differential forms pD(). also converge in this norm as 6 tends to zero. Upon taking complex conjugates, precisely the same arguments applied to the operators 0 and 15 also show that the functions arising by applying O/OZk to any coefficient ofthe differential forms pD. also converge in this norm as 6 tends to zero, or alternatively the same result follows upon noting as an easy double application of (12) that lIof/ozkll = I of/ozk I for any CCX) function f with compact support. An evident consequence of this is that for any linear partial differential operator D': tCli q - 1 --+ tCli q - 1 of order ~ v + 1 with constant coefficients, the differential forms D'(). converge in the norm (22) for the given compact set K as 6 tends to zero, since p(Z) = 1 whenever Z E K. The set K was arbitrary, and since lim..... o D'(). = ()D E "If"oP,q-l, it follows that () E "II'/:·t; that demonstrates the desired induction step and thereby concludes the proof.

The result of the preceding theorem can of course be restated as follows.

The Dolbeault cohomology groups of a pseudoconvex open subset M £; or of a pseudoconvex Riemann domain M satisfy .Jft'p,q(M) = 0 whenever q > O.

10, COROLLARY.

The argument in this section follows that in Hormander [32].

en

p Pseudoconvexity and Holomorphic Convexity

The results of the preceding section can now be applied to show that pseudo convexity and holomorphic convexity are indeed equivalent conditions. To begin with the slightly easier case of open subsets of en, the relevant results can be summarized as follows.

1. THEOREM. (i) (ii) (iii) (iv)

For an open subset D £

en the following conditions are equivalent:

D is a domain of holomorphy. D is holomorphically convex. D is pseudoconvex. yt'o,q(D) = 0 for 0 < q < n.

Proof. The equivalence of conditions (i) and (ii) was demonstrated in Theorem GS. It is obvious that (ii) implies (iii), as noted at the beginning of section M; it follows from Corollary 010 that (iii) implies (iv); and it follows from Theorem G14 that (iv) implies (i), thereby completing the circle of implications and concluding the proof.

Since it has been demonstrated that pseudoconvexity and hoi om orphic convexity are equivalent conditions for an open subset D £ en, and that both are in turn equivalent to the condition that D be a domain ofholomorphy, there is a wide variety of criteria available for demonstrating that an open subset D £ e" is a domain of holomorphy, including both the criteria discussed in section G and the criteria for pseudoconvexity discussed in Section M. In addition, various properties of pseudoconvexity discussed in section M extend immediately to holomorphic convexity. Although the extensions are trivial, the final results are sufficiently important to warrant repeating here explicitly.

2. COROLLARY.

If D is an open subset ofe" and if each point A E aD has an open neighborhood UA such that UA (') D is a domain of holomorphy, then D is also a domain of holomorphy. Proof.

M8.

This is an immediate consequence of the preceding theorem and Theorem

P Pseudoconvexity and Holomorphic Convexity

167

This shows that the property that an open subset of en be a domain of holomorphy is purely a local property of the boundary of that set, a very important result in the theory of holomorphic functions of several variables. As already mentioned, the problem of demonstrating the local character of the condition that an open subset of en be a domain of holomorphy is usually called the Levi problem. 3. COROLLARY (theorem of Behnke and Stein). If Dv are domains of holomorphy in en with Dv ~ Dv+1' then their union D = Dv is also a domain of holomorphy.

Uv

Proof. M6.

This is an immediate consequence of the preceding theorem and Theorem

Theorem 1 is important not only for showing the equivalence of hoi om orphic convexity and pseudo convexity for subsets D ~ en, with such consequences as illustrated in the two preceding corollaries, but also for showing the equivalence of holomorphic convexity and the condition that Je°,q(D) = 0 for 0 < q < n, with such consequences as will be illustrated in the next two corollaries. The vanishing of these Dolbeault cohomology groups for open polydiscs was used quite essentially in the discussion in sections E and F, and it is of comparable utility in studying function theory in general domains of holomorphy. A simple illustrative example is the following. 4. COROLLARY. If D ~ en is a domain of holomorphy, h: D -. e is a nonsingular holomorphic mappingfromD intoe, and M ~ Dis thecomplexsubmanifold M = {Z E D: h(Z) = O}, then for any function f holomorphic on M there exists a function 9 holomorphic in D for which glM = f Proof.

This is an immediate consequence of the preceding theorem and Theorem

E7. It is possibly worth noting that the preceding corollary is not true for arbitrary subsets D ~ en. For instance, if D is the complement of the closed unit ball in e 2 and M = {Z ED: Z2 = O}, then M is really just the complement of the closed unit disc in the plane of the complex variable Z l ' There are on M holomorphic functions that cannot be extended to holomorphic functions in the entire plane of the variable Z 1, and such a function cannot be extended to a holomorphic function in D, since by Hartogs's extension theorem, Theorem E6, any function holomorphic in D necessarily extends to a function holomorphic in all of e 2 • In the situation of Corollary 4 the hypothesis that the submanifold M can be defined by a single function holomorphic throughout all of e was very definitely used in the proof of Theorem E7 but is not really essential. Although a considerably more general result will be demonstrated later, it is perhaps worth noting here as another consequence of Theorem 1 how this hypothesis can be eliminated at least to some extent. Indeed, a complex submanifold M of dimension n - 1 in a domain of holomorphy D £; en can always be written as the zero locus of a nonsingular holomorphic mapping h: D -. en provided there is no topological obstruction to doing so. To make clear what is meant by a topological obstruction in this context,

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note that there obviously exist a covering of D by open subsets ~ and nonsingular holomorphic mappings hj: ~ --. e such that M n ~ = {Z E ~: hj(Z) = O}. The quotients hij = hdhj are nowhere vanishing holomorphic functions in the intersections Ui n Uj and satisfy hiiZ)hji(Z) = 1 whenever Z E Ui n ~ and hij(Z)hjk(Z)hki(Z) = 1 whenever Z E Ui n ~ n Uk' so they form a multiplicative analogue of the Cousin data of Theorem E7. If there does exist a nonsingular holomorphic mapping h: D --. e such that M = {Z ED: h(Z) = O}, then the quotients gi = hdh are holomorphic nowhere vanishing functions in the sets Ui and hij = gdgj in Ui n ~. To say that there is no topological obstruction just means that there exist at least Coo nowhere vanishing functions gi in the sets Ui such that hij = gdgj in Ui n ~. There do not always exist such functions gi' so the properties of the multiplicative Cousin data are not totally parallel to the properties of the additive Cousin data considered earlier. It is constructive to try to reduce the multiplicative Cousin data to additive Cousin data by taking logarithms, and thereby to reinterpret the topological obstruction involved; but that will be left for the interested reader to pursue, since the same analysis in more general terms will appear in Volume III after the introduction of sheaf theory.

5. COROLLARY. If M is a complex submanifold of dimension n - 1 in a domain of holomorphy D s en, then there exists a nonsingular holomorphic mapping h: D --. e such that M = {Z ED: h(Z) = O} provided that there is no topological obstruction. Proof. As noted, there exist a covering of M by open sets ~, which will here be assumed simply connected, and nonsingular holomorphic mappings hj: ~ --. e such that M n ~ = {Z E ~: hj(Z) = O}. The assumption that there is no topological obstruction implies that there also exist Coo nowhere vanishing functions gj in the sets ~ such that gdgj = hdhj in Ui n ~. Since the sets ~ are simply connected, it is possible to choose single-valued branches of the functions log gj in each ~, and evidently log gi - log gj is holomorphic in Ui n ~. It is therefore possible to introduce a well-defined a-closed Coo differential form f/J on D by setting f/J(Z) = a log gi(Z) whenever Z E Ui' It follows from Theorem 1 that Jlt'0,1 (D) = 0, so there is a Coo function 9 in D such that ag = f/J. Now the functions /; = e-gg i are then hoi om orphic and nowhere vanishing in Ui' and /;/jj = gdgj = hdhj in Ui n ~. The function h defined by setting h(Z) = hi(Z)//;(Z) whenever Z E Ui is then a holomorphic function in D defining the submanifold M as desired, thus concluding the proof.

It is worth observing explicitly here that the proofs of the preceding two corollaries really rest solely on the condition that Jlt'0,1 (D) = O. Therefore, both corollaries actually hold for any complex manifold D for which Jlt'0,1 (D) = o. Examples. It is easy to see that the preceding corollary is not true for arbitrary open subsets of en when n> 1. For instance, let D be the union of the open ball DI = {ZEe 2 : IIZII
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169

1/2 ~:::-----T-t 1/41--""*--+-\1 I

-+-__:-7_-=-"I-,-_ _~=--..... Izd v'3/2 Figure 10

{Z E Dl : Z2 = O}, M (') D2 = {Z E D2 : Z2 = !}, M (') D3 = 0. There is no topological obstruction to defining M by a single Coo function in D. Indeed, ifj: ~ -+ ~ is any Coo function for which ~(r) = 0 for 0 ~ r ~ and ~(r) = ! for .J3/2 ~ r ~ !, then the function g defined by g(Z) = Z2 - ~(lZll) is a Coo function in D such that M = {Z ED: g(Z) = O}, and g is indeed even holomorphic in Dl U D2. However, there cannot exist any nonsingular holomorphic mapping h: D -+ ([: defining the submanifold M, for by Hartogs's extension theorem any function h holomorphic in D must extend to a holomorphic function in the entire ball B(O; !) containing D, and any such function vanishing on M must obviously vanish in addition on the submanifold {Z E Dl : Z2 =!} u {Z E D2 : Z2 = O} of D. Even if D s ([:n is a domain of holomorphy, though, there may exist topological obstructions to defining a complex submanifold M of dimension n - 1 by a single function hoI om orphic throughout D, as in the following example due to Oka. LetD s ([:2 be the product of two open annuli of the form D = {Z E ([:2:1 < IZjl < afor j = 1, 2}, so that D is a domain ofholomorphy in ([:2, and let N s D be the complex submanifold of dimension n - 1 defined by N = {Z ED: Zl - Z2 - 1 = O}. For Zj = Xj + iYj' note that ZEN precisely when ZED and Xl = x 2 + 1, Yl = Y2. There are clearly no points ZEN for which Yl = Y2 = 0, since the condition that ZED then reduces to 1 < IXjl < i and there are no values Xj satisfying these inequalities and the equality Xl = X2 + 1. Consequently, N = MuM where M and M are complex submanifolds of D of dimension n - 1 and Yl = Y2 > 0 whenever Z E M, while Yl = Y2 < 0 whenever Z E M, with the submanifold M consisting evidently of the complex conjugates of points in M. Suppose that there does exist a nonsingular holomorphic mapping h: D -+ ([: for which M = {Z ED: h(Z) = O}. The restriction of this function h to the product of the unit circles can be viewed as the Coo doubly periodic function hO(tl' t 2) = h(e 21tit " e21ti/2) of the two real variables (tl' t 2) E ~2, and it is easy to see that this function vanishes only when tl = 2n/3 + 2nn l , t2 = 4n/3 + 2nn2 for arbitrary integers n l and n2. Now since the function ho

t

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12

2n

4n/3

~

G

1'0

~

I'

--4---~----~----~--~ll

o

2n/3

2n

Figure 11

is doubly periodic, then if')' denotes the boundary ofthe square WI' t 2 )} E ~2: 0;£ tl ;£ 2n,O ;£ t2 ;£ 2n} as indicated in Figure 11, it is clear that L d log ho = O. On the other hand, since d log ho is a Coo closed differential I-form everywhere in this closed square except at the point To = (2n/3, 4n/3), it follows that L d log ho = JYod log ho, where ')'0 is a small circle around the exceptional point To. However, the integral d log ho is nonzero; that can be seen either by direct calculation or by noting that it represents the change in the argument of the function h around the loop ')'0' and since ')'0 is homotopic to a loop in a complex plane transverse to M (the local fundamental group of the complement of the submanifold M evidently being infinite cyclic), the non vanishing of this integral reduces to a familiar result about holomorphic functions of one variable. This contradiction shows that M cannot be defined by a single holomorphic function in all of D after all. While the results of Corollaries 4 and 5 are possibly of some interest, they are clearly rather special cases, and more general results along the same lines would be much more interesting. Such results will be established later. The extension from submanifolds to more general analytically interesting subsets requires some further local analysis of hoI omorphic functions, which will be taken up in Volume II, while the most useful and convenient tool for handling these and related matters is the theory of sheaves, which will be considered in Volume III. Before leaving this topic however, one more point should be made. It may have been noticed that Corollary 010 implies that .1{'0'4(D) = 0 for 0 < q ;£ n whenever D is a pseudoconvex subset of en, while on the other hand, by Theorem G14 it is only necessary to assume that .1{'0,4(D) = 0 for 0 < q < n to show that an open subset D £ en is a domain of holomorphy. That leaves the Dolbeault group .1{'O,n(D) in a somewhat anomalous position, and quite rightly so. Actually .1{'o,n(M) = 0 for any noncompact connected n-dimensional complex manifold M. This really involves only the noncompactness of M and has nothing to do with any complex analytic properties, such as being a domain of holomorphy when M is an open subset of en. The assertion can easily be reduced to known results from the theory of elliptic partial differential equations. For the special case in which M is an open subset of en or more generally a Riemann domain over en, the auxiliary result needed is that for any Coo function f on M there

Lo

P Pseudoconvexity and Holomorphic Convexity

171

exists a Coo function 9 on M such that f = ilg. Proofs can be found in many treatises on partial differential equations, or in the paper by B. Malgrange (Annales de l'Institut Fourier, Grenoble, Vol. 6, 1956, pages 271-355) in which the general assertion was first proved. To apply this auxiliary result, note that a Coo differential form f/J = f dZ 1 /\ ••• /\ dZn is necessarily a-closed and can always be written as f/J = ilg dz 1 /\ ... /\ dZn for some Coo function g. If ifJ = 9 dz 1 /\ •.. /\ dzn , a simple calculation using formulas (9) and (10) from the preceding section shows that f/J = a!:JifJ, and consequently .1t'0.n(D) = 0 as desired. A similar argument works on arbitrary noncompact manifolds, although to introduce the relevant operators !:J and it is necessary to use some Riemann geometry. The operator in the general case is a second-order elliptic system of partial differential operators acting on arbitrary differential forms given explicitly by = a!:J + !:Ja. The auxiliary result is that any Coo differential form f/J on M can be written as f/J = DifJ for some Coo differential form ifJ having the same bidegree as f/J. If f/J E 8~n, then af/J = aifJ = 0 automatically; hence, f/J = a!:JifJ as before. Then for the case of Riemann domains, the proof of the analogue of Theorem 1 is slightly more complicated than the proof of Theorem 1 itself, since some of the auxiliary results used there have not yet been shown to hold for Riemann domains. Indeed, it is convenient to have delayed demonstrating these results until this point so that the powerful machinery developed in the preceding section can be applied. At the same time there remain from the discussion in section I some questions about the extent to which it is necessary to assume that holomorphic functions separate points. It is convenient first to consider the one-dimensional case.

°

°

°

6. THEOREM.

Anyone-dimensional Riemann domain is holomorphically convex and has the property that holomorphic functions separate points.

Proof. Anyone-dimensional Riemann domain M is pseudoconvex by Theorem N7, and it then follows from Corollary 010 that .1t'0.1 (M) = O. The projection mapping P: M -+ e is nonsingular, so by Theorem E7 any holomorphic function on the submanifold p-1(a) £ M, for any point a E M, extends to a holomorphic function on all of M. Although that theorem was actually just stated for open subsets of en, it was noted after the proof that it applies as readily to complex manifolds, the only critical assumption being that .1t'0.1 (M) = O. The set p- 1(a) is, of course, just a discrete subset of M, so the conclusion is that there is a holomorphic function on M taking any prescribed values at these points; thus, holomorphic functions do separate the points of M. To show that M is holomorphically convex, it is clearly sufficient just to show that for any discrete sequence of points {a.} in M there exists a holomorphic function f E (!}M such that lim suP. If(a.)1 = 00. For that purpose it is in turn sufficient to show that there is a nonsingular holomorphic mapping h: M -+ e vanishing at the points avo since it will then follow from Theorem E7 as before that there actually exists a holomorphic function f E (!}M taking any preassigned values at the points a •. Finally by Corollary 5 it is enough to show that there is no topological obstruction to the existence of such a mapping h, in the sense made precise in the discussion preceding the statement of that corollary, which also clearly applies to any complex manifold M for each which .1t'0. 1 (M) = O. There are various

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ways of showing that there is no topological obstruction, depending on the amount of topological machinery brought to bear. At this stage it may be most appropriate to use only the most intuitive topological results, to show that there is a Coo function g on M such that g is a nonsingular holomorphic mapping near av vanishing at a v and nowhere else. It is quite evident then that there is no topological obstruction in the sense used here. Choose a simple path on M passing through all of the points a.. and observe that a smoothly bounded open neighborhood of this path is homeomorphic to the right half-plane {z E C: Re z > O}, in such a manner that the points a v correspond to the odd half-integers (2v + 1)/2 and the homeomorphism is actually holomorphic near the points avo The function cos nz when composed with this homeomorphism will then clearly have the desired properties within the chosen neighborhood of the path through the points avo On the boundary of that neighborhood, the function behaves as cosh ny and hence is real-valued and strictly positive and can readily be extended as a real-valued and strictly positive function to the rest of the manifold M. That then suffices to conclude the proof. Since a holomorphically convex Riemann domain is also a Riemann domain of holomorphy by Theorem 16, an immediate consequence of the preceding result is that anyone-dimensional Riemann domain is a Riemann domain ofholomorphy. For arbitrary dimensions the basic result is the following. 7. THEOREM.

(i) (ii) (iii (iv)

For a Riemann domain M over

en the following conditions are equivalent:

M is holomorphically convex. M is a Riemann domain of holomorphy. M is pseudoconvex. Yfo·q(M) = 0 for 0 < q < n.

In any of these cases holomorphic functions separate points on M.

Proof.

First, that (i) implies (ii) was demonstrated in Theorem 16. Next, to show that (ii) implies (iii), suppose that M is a Riemann domain of holomorphy, so that there exists a holomorphic function f E (!)M such that PM,R(f; A) = bM,R(A) for any polyradius R and any point A E M. Recall from Definition 11 that if P: M --+ C n is the projection mapping and A E M is any point of M, then fA = f 0 (PI~M(A; eRW 1 is a well-defined holomorphic function in an open neighborhood of P(A) in en whenever e is sufficiently small, and the radius of convergence PM,R(f; A) is the supremum of those positive real numbers b such that the power series expansion of the function fA about the point P(A) converges in ~(P(A); bR). For a fixed polyradius R, note that the restriction of fA to a complex line through P(A) in the direction of the vector R is the holomorphic function fA(P(A) + zR) of the single variable z near z = 0 and hence has a power series expansion of the form fA(P(A)

+ zR) = I v

cv(A)zV

P Pseudoconvexity and Holomorphic Convexity

173

and that the radius of convergence of this power series is precisely PM,R(f; A). It follows from Hadamard's formula for the radius of convergence that .

1

-log PM R(f; A) = hm sup -loglc.(A)1 ,

•

v

Now the Taylor coefficients c. are quite evidently holomorphic functions on M, and since the modulus of a holomorphic function is plurisubharmonic, it follows that (1/v) 10glc.(A)1 are plurisubharmonic functions on M. It also follows readily from the Cauchy inequalities that these functions are locally uniformly bounded from above, and an application of Theorem L9 shows that -log PM,R(f; A) is nearly plurisubharmonic. Then since bM,R(A) is continuous on M, it finally follows that -log bM,R(A) = -log PM,R(f; A) is an upper semicontinuous nearly plurisubharmonic function on M and hence is actually plurisubharmonic on M. Theorem N3 then shows that M is pseudoconvex as desired. That (iii) implies (iv) follows from Corollary 010. Finally, that (iv) implies (i) will be demonstrated by induction on the dimension of the Riemann domain. For a one-dimensional Riemann domain, hypothesis (iv) is vacuous, but by Theorem 6 anyone-dimensional Riemann domain is holomorphically convex. For the induction step consider a Riemann domain M of dimension n > 1 satisfying (iv), and suppose that the desired result has been demonstrated for Riemann domains of dimensions < n. It will first be shown that for any compact subset K £; M and any poly radius R, necessarily bM,R(K) = bM,R(KM). If that is not the case, there exist a compact subset K £; M and polyradius R such that bM,R(K) > bM,R(K M); hence, there is a point A E KM for which bM,R(A) < bM,R(K) = b. Now for any function f E (!)M the argument of Lemma G4 shows that the power series expansion about the point P(A) E en of the function fA = f 0 (PIL\M(A; bM,R(A)R»-l actually converges in the polydisc L\(P(A); bR) and therefore that PM,R(f; A) ;;:;; b > bM,R(A), where P: M --+ en is the projection mapping and the radius of convergence PM,R(f; A) is as in Definition 11. By Lemma 14 that means that the Riemann domain M admits a properly larger holomorphic extension E, with a projection mapping that will also be denoted by P. Choose a point BEaM nE and an (n - I)-dimensional linear subspace L £; en tlirough P(B). The submanifold E' = P-i(L) £; E is clearly a Riemann domain with the projection P' = PIE': E' --+ L ~ en- i , as is the submanifold M' = P-i(L) £; M, and it can be assumed that L is so chosen that BEaM' n E' as well. Just as in the proof of Theorem G14 it can be shown that -*,o,n(M') = 0 for 0 < q < n - 1, so by the inductive hypothesis M' is holomorphically convex and hence is a domain of holomorphy. There must consequently exist a holomorphic function f E (!)M' that cannot be extended as a holomorphic function to E'. However, since -*,o,l(M) = 0, it follows from Theorem E7 that f extends to a holomorphic function on M, and hence necessarily extends further to a holomorphic function on E :2 E'. That is a contradiction, from which it follows as desired that bM,R(K) > bM,R(KM) for any compact subset K £; M and polyradius R. From this result it is not at all difficult to show that M is holomorphically convex. Indeed, if M is not holomorphically

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convex, it follows from Lemma 19 that there exist a compact subset K £; M and a point Zo E C n such that p-l(ZO) n KM contains infinitely many distinct points A v. Now if Ll £; C n is an (n - I)-dimensional linear subspace passing through Zo and Ml = P-l(Ld, then as already noted Ml is a Riemann domain for which ;¥,O,q(Md = 0 whenever 0 < q < n - 1, and any holomorphic function on Ml extends to a holomorphic function on M. Then if L2 £; Ll is an (n - 2)-dimensional linear subspace passing through Zo and M2 = P-l(L 2 ), the same argument shows that M2 is a Riemann domain for which ;¥,O,Q(M2 ) = 0 whenever 0 < q < n - 2, and any holomorphic function on M2 extends to a holomorphic function on Ml and thence to a holomorphic function on M. Continuing in this way shows eventually that any function on the discrete subset P-l(ZO) extends to a holomorphic function on M. In particular, therefore, there exists a holomorphic function f E (!)M such that limv If(Av)1 = 00 where {Av} = P-l(Zo)nK M; but that is impossible, since If(Av)1 ~ IlfilK by the definition of the holomorphically convex hull K M. That contradiction shows that M is holomorphically convex as desired. Note incidentally that in proving that (iv) implies (i) it was actually demonstrated that for any point Zo E P(M) there exists a holomorphic function on M taking any preassigned values at the points P-l(ZO)' Therefore, holomorphic functions separate points on any Riemann domain satisfying (iv), and with that the proof is concluded. Just as for open subsets ofC n so also for Riemann domains do the criteria for and properties of pseudoconvexity carryover to provide criteria for and properties of hoI om orphic convexity. A Riemann domain is a Riemann domain ofholomorphy precisely when it is holomorphically convex, without the assumption that holomorphic functions separate points. Moreover, since the Dolbeault cohomology groups in positive dimensions vanish on Riemann domains of holomorphy, Corollaries 4 and 5 also hold for Riemann domains of holomorphy. Finally, the other criteria of Theorem 113 also hold without the necessity of assuming that holomorphic functions separate points, as follows.

8. THEOREM.

For a Riemann domain M the following conditions are equivalent:

(i) M is a Riemann domain of holomorphy. (ii) For any discrete sequence of points Av E M there exists a function f E (!)M for which lim supv If(Av)1 = 00. (iii) M admits no properly larger holomorphic extension. (iv) For any polyradius R and any compact subset K £; M, necessarily bM,R(K) = bM R(K M )· (v) There exists a fixed polyradius R such that bM,R(K) = bM,R(KM)for any compact subset K £; M. (vi) dM(K) = dM(K M) for any compact subset K £; M.

Proof. That (i) implies (iv) follows from Theorem 17. It is, of course, obvious that (iv) implies (v), and it is a purely formal consequence of the definition of these distance functions as in Corollary G6 that (iv) implies (vi). Essentially the same proof shows that either (v) or (vi) implies (i); so for this purpose let d denote either bM,R

P Pseudoconvexity and Holomorphic Convexity

175

for some fixed polyradius R or dM, and suppose that d(K) = d(KM) for any compact subset K ~ M. It will then be demonstrated that M is pseudoconvex in the sense of Hartogs, which by Theorems N3 and N7 implies that M is a Riemann domain of holomorphy. If ~: [0, 1] x A(O; 1) -+ en is a continuous mapping from the compact subset [0, 1] x .:\(0; 1) ~ IR x e into en such that F is holomorphic in .:\(0; 1) for each fixed point of [0, 1], and if G: ([0, 1) x A(O; 1» u (1 x 0.:\(0; 1» -+ M

(1)

is a continuous mapping such that P 0 G = F where P: M -+ en is the projection mapping, then clearly K = G([O, 1] x 0.:\(0; 1» is a compact subset of M and KM contains the entire image of the mapping (1). But since d(KM) = d(K) > 0 by hypothesis the mapping G obviously extends to a continuous mapping G: [0, 1] x A(O; ) -+ M. That means that M is pseudoconvex in the sense of Hartogs, and consequently it has been demonstrated that conditions (i), (iv), (v), and (vi) are equivalent. Next, to show that (i) implies (ii), note that by Theorem 7 if a Riemann domain M satisfies (i), then Mis holomorphically convex; but the proof of Theorem G7 then shows that M also satisfies (ii). It is obvious that (ii) implies (iii). Finally to demonstrate that (iii) implies (i), suppose to the contrary that M is a Riemann domain satisfying (iii), but that M is not a Riemann domain of holomorphy. Since it has already been shown that (i) is equivalent to (iv), there must be some compact set K ~ M and polyradius R for which bM R(K) > bM R(K M). There is thus a point A E KM for which bM,R(A) < bM,R(K) =~. Then fr~m Lemma G4 it follows that PM,R(f; A) ~ b for every f E (I)M, and that in turn by Lemma 4 shows that M admits a properly larger holomorphic extension, contradicting the hypothesis that M satisfies (iii). With that contradiction it has been shown that conditions (i), (ii), and (iii) are also equivalent, thereby concluding the proof. As a final comment it is perhaps worth noting that if M is a Riemann domain for which holomorphic functions separate points, then M is a Riemann domain of holomorphy precisely when M = E(M). This is really just a special case of the equivalence of conditions (i) and (iii) in the preceding theorem, the hypothesis that holomorphic functions separate points being required here in order that the envelope of holomorphy E(M) be well defined.

a Plurisubharmonic and Holomorphic Functions

Plurisubharmonic functions played an important role in the preceding discussion of domains ofholomorphy, and they have many more relations to and consequences for hoi om orphic functions, just two of which will be discussed briefly here. One of these is a useful extension of Riemann's removable singularities theorem. It was noted at the beginning of section M that If I is plurisubharmonic whenever f is holomorphic. Actually it follows immediately from Jensen's inequality in several variables, Theorem A8, that loglfl is plurisubharmonic whenever fis hoi om orphic. This is a somewhat stronger result, since the real exponential function is convex and monotonically increasing, so by Theorem KS(d) the plurisubharmonicity ofloglfl implies that of If I = exp loglfl. The zero locus of f is the set of points at which the function loglfl takes the value -00. The major role played by the zero sets of holomorphic functions thus suggests introducing the following notion. 1. DEFINITION. A subset X of an open set D ~ en is called a pluripolar set in D if for each point A ED there are an open polydisc A(A; R A ) and a plurisubharmonic function UA in A(A; R A ) such that U A is not identically equal to -00 and

..."

Note that a pi uri polar set X in D is necessarily a proper subset in each connected component of D; indeed, it follows immediately from Theorem K6 that X is a subset of D of Lebesgue measure zero. The pluripolar sets can be viewed as playing for plurisubharmonic functions roughly the same role that thin sets play for hol0 7 morphic functions. Actually since loglfl is plurisubharmonic whenever f is holomorphic, it is evident that any thin set in D is also a pluripolar set in D. The converse assertion is false, for the class of pi uri polar sets is considerably more extensive than the class of thin sets. For example, suppose that Xl' X 2, X 3, ... are pluripolar sets in a connected open subset D ~ en, and that Xy ~ {Z ED: uy(Z) = - oo} where U y are plurisubharmonic functions that are not identically equal to -00 in D; then X = Uy Xy is a pluripolar set in D. Indeed choose a sequence of open subsets Dy ~ D such that Dy is a compact subset of D, that Dy ~ Dy+ 1 , and that Uy Dy = D; and choose a point A E D such that uy(A) # - 00 for all indices v, noting that there

Q Plurisubharmonic and Holomorphic Functions

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certainly exists such a point since u. is equal to - 00 only on a set of Lebesgue measure zero. By adding a suitable real constant to u. it can be assumed that u.(Z) ~ 0 whenever Z = A or ZED., since A u D. is compact and hence the mapping u. is bounded from above on that set. By multiplying u. by a suitable positive real constant, it can also be assumed that 0 ~ u.(A) ~ - 2-'. Now the mappings v. = U 1 + ... + u. are plurisubharmonic functions on D, and whenever v ~ J.l this sequence of functions is monotonically decreasing on the subset DIl , since u. ~ 0 on DIl ~ D. by construction. The limit function u = lim. v. = L. u. is therefore a plurisubharmonic function in D, and this function is not identically equal to - 00 since 0 ~ u(A) = L. u.(A) ~ L. - 2-' = - 1. Since moreover it is evident that X. ~ {Z ED: u(Z) = - oo} for each index v, the same is true for the union X = U. X. and hence X is a pluripolar subset of D as desired. In particular, if f. are holomorphic functions that are not identically equal to zero in a connected open set D ~ en and if X. ~ {Z ED: f.(Z) = OJ, then the union X = U.X. is a pluripolar set in D but is not necessarily a thin set in D. Thus, any countable subset of an open set D ~ e is a pluripolar set, but can only be a thin set if it is discrete. Even a countable dense set is a pi uri polar set, so clearly the closure of a pluripolar set is not necessarily a pluripolar set. It should be mentioned here in passing that the assumption that the pluripolar sets X. above are defined by global plurisubharmonic functions u. in D is not really essential, since B. 10sefson has shown that any pluripolar set in an open subset D ~ en can be defined in terms of a global plurisubharmonic function in D. The proof of this latter assertion will not be given here. Thin sets have thus far been considered only in connection with holomorphic extension theorems, in particular in Theorem D2. There are analogous results for plurisubharmonic functions and pluripolar sets, but since such results involve extensions of plurisubharmonic functions from the complement of a pluripolar set X in an open subset D ~ en to all of D, and since plurisubharmonic functions have been defined only in open subsets of en, it is apparent that only closed pluripolar sets X playa role in this discussion. A useful preliminary result in this direction is the following. If u: D -+ [ -00,00) is an upper semicontinuous mapping in an open subset D ~ e", if x is a closed subset of D, and if u is a plurisubharmonic function in D - X and u(Z) = - 00 whenever Z E X, then u is necessarily a plurisubharmonic function throughout D.

2. LEMMA.

Proof.

For any nonnegative integer v, introduce the function u. in D defined by u.(Z)

= sup(u(Z), - v)

This function is of course plurisubharmonic in the open set D - X by Theorem KS(d). On the other hand, since u is upper semicontinuous, D. = {Z ED: u(Z) < - v} is also an open subset of D, and since u.(Z) = - v whenever ZED., it follows that u. is also plurisubharmonic in D•. Since X ~ D. by hypothesis the open subsets D - X and D. cover D, and since plurisubharmonicity is a local property it follows that u. is plurisubharmonic throughout D. Now the functions u. are a montonically

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decreasing sequence converging to u at each point of D, so it follows from Theorem K5(e) that u is plurisubharmonic in D as desired; that suffices to conclude the proof. 3. THEOREM. If X is a closed pluripolar set in an open subset D ~ en, then a plurisubharmonic function u in D - X extends to a plurisubharmonic function in all of D if and only if lim SUPz-+A.zeD-XU(Z) < 00 at each point A E X, and the extension is a uniquely determined plurisubharmonic function in D.

Proof. If there is a plurisubharmonic function v in D such that v(Z) = u(Z) whenever ZED - X, then lim sUPZ-+A.ZeD-X u(Z) = v(A) < 00 at each point A E X. Moreover, since X is a subset of Lebesgue measure zero, it follows from Theorem K15 that v(A)

= lim

I~(A; 8)1-

= lim

I~(A; 8)1- 1

£-+0

£-+0

1

Jr4(A;£) v(Z) dV(Z)

r

u(Z) dV(Z)

J(D-X)f""I4(A;£)

at each point A E D; hence, the plurisubharmonic function v is uniquely determined by u. On the other hand, suppose that u is plurisubharmonic in D - X and that lim SUPZ-+A.ZeD-XU(Z) < 00 at each point A E X. Since X is a pluripolar set, for each point A E X there are an open polydisc ~A = ~(A; RA ) and a plurisubharmonic function WA in ~A such that WA is not identically equal to - 00 and X n ~A ~ {Z E ~A: wA(Z) = -oo}. Then for any 8> 0, introduce the mapping v~: ~A --. [-00, +(0) defined by if Z E (D - X) n if Z E X n~A

~A

This mapping is obviously upper semicontinuous in the open subset (D - X) n ~A' while if B E X n ~A it follows from the hypothesis on u and the definitions of WA and VA that lim SUPZ-+BV~(Z) = lim SUPZ-+B.ZeD-X(U(Z) + 8WA(Z» = -00 = v~(B); consequently, v~ is upper semicontinuous throughout ~A- Since v~ is plurisubharmonic in (D - X) n ~A and v~(Z) = -00 whenever Z E X n ~A' it follows from Lemma 2 that v~ is actually plurisubharmonic in all of ~A' The functions 8WA and v~ are therefore both locally Lebesgue integrable in ~A' since they are not identically equal to - 00. The function u must consequently also be locally Lebesgue integrable in ~A' irrespective of its definition on the subset X n ~A of measure zero. Moreover, it is evident that lim£-+o v~ = u in the topology of local L 1 -convergence; hence, by Corollary K17 the mapping u is equal to a plurisubharmonic function almost everywhere in ~A' This plurisubharmonic function coincides with u in (D - X) n ~A and is uniquely determined as demonstrated in the first part of the proof. So since this holds for all points A E X, it follows that u extends to a plurisubharmonic function in D as desired. That suffices to conclude the proof.

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179

en,

then

4. COROLLARY. If X is a closed pluripolar set in a connected open subset D D - X is also connected.

~

Proof. If D - X = U 1 U U2 where U 1 , U2 are disjoint nonempty open sets, then the function u in D - X defined by u(z)

-oo ={ 0

is plurisubharmonic in D - X, and lim sUPZ-+A.ZeD-X u(Z) < 00 at each point A E X. It then follows from the preceding theorem that u extends to a plurisubharmonic function in D; but that is impossible, since the extension would be a plurisubharmonic function not identically equal to - 00 in a connected open subset D ~ en but equal to - 00 in a subset U1 ~ D of positive measure, contradicting Theorem K6. That contradiction shows that D - X must be connected as desired. If X is a closed pluripolar set in an open subset D ~ en and f is a uniformly bounded holomorphic function in D - X, then there is a unique holomorphic function jin D such thatj(Z) = f(Z) whenever ZED - X.

5. COROLLARY.

Proof. The functions ± Re fare pluriharmonic and hence plurisubharmonic in D - X, and since both are bounded from above in D - X by assumption it follows from the preceding theorem that both extend to plurisubharmonic functions in D. The extensions are the negatives of one another in D - X, and since X is a subset of D of Lebesgue measure zero, it follows from Theorem KI5(i) that the extensions are the negatives of one another throughout D. Thus, if u is the extension of Re f, then both u and - u are plurisubharmonic in D and hence u is pluriharmonic in D. Similarly, 1m f extends to a pluriharmonic function v in D. Now j = u + iv is a COO function in D and aj = 0 in the dense open subset D - X ~ D, so by continuity aj = 0 in D and hencejis hoI om orphic in D. This provides an extension of the function f as desired, and since D - X is a dense open subset of D, the uniqueness ofj follows immediately from the identity theorem, Theorem A3. That suffices to conclude the proof.

As in the discussion of the corresponding result for thin sets, so also here the theorem can be restated as the assertion that any function that is holomorphic outside a closed pluripolar subset of an open set D ~ en and locally bounded in D has a unique extension to a holomorphicfunction in D. Since pluripolar sets are more general than thin sets, this result is an extension of Theorem D2. An immediate application is the following quite useful and perhaps surprising result. 6. COROLLARY (Rado's theorem). If f is a continuous complex-valued function in an open set D ~ en and if f is holomorphic in the open subset D - X ~ D where X = {Z ED: f(Z) = O}, then f is necessarily holomorphic throughout D. Proof. It follows immediately from the hypotheses that the mapping log If I: D -+ [ - 00, + 00) is upper semicontinuous and that log If I is plurisubharmonic in D - X

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and log If(Z} I = - 00 whenever Z E X. But then by Lemma 2 the function log If I must be plurisubharmonic throughout D. The closed subset X = {Z ED: 10glf(z)l = - oo} is therefore a pluripolar set in D, and it then follows from the preceding corollary that the restrictionflD - X extends to a holomorphic function in D. This extension must coincide with f by continuity, so that f is holomorphic in D as desired. That suffices to conclude the proof. The discussions of plurisubharmonic functions have from time to time involved those of the special form u(Z} = 10glf(Z}1 for a holomorphic functionf; they are a very interesting special class of plurisubharmonic functions, which are in a natural sense extreme points in the convex cone of all the plurisubharmonic functions. This observation is essentially an interpretation of the following simple result. 7. LEMMA. If U1 and U2 are plurisubharmonic functions in an open polydisc .1\(0; R} about the origin in ([:" and if u 1 (Z) + U2(Z} = loglzll, then ui(Z} = Re/;(Z} + ciloglzti where /;(Z) are holomorphic in .1\(0; R} and Ci are positive real constants. Proof. Write .1\(0; R} = .1\(0; r 1 } x .1\(0; R"} C (;1 X (;n-l as usual, and introduce the submanifold V = {Z E .1\(0; R}: ZI = O} c .1\(0; R}. Note that -u 1 (Z} = U2(Z) + Re log liz 1, so that - U 1 is also plurisubharmonic in .1\(0; R} - V; hence, by Theorem K5(h} the function U 1 must actually be pluriharmonic in .1\(0; R} - V, and for the same reasons so, of course, is u 2 • The differential form OUj for j = 1 or 2 is then a closed holomorphic I-form in .1\(0; R} - V, just as in the proof of Theorem K3. If 2niaj is the value of the integral of this differential form oUj along any closed path in .1\(0; R} - V encircling V once, then the function

is evidently a single-valued holomorphic function in .1\(0; R} - V, for any fixed base point A E .1\(0; R} - V. Now d(uj - hj - aj log ZI -hj -li.i log zd = 0 as in the proof of Theorem K3, so that

for some real constant bj. Here Re(aj log ZI} = (Re aj) 10glz11 - (1m aj) arg ZI must be single-valued in .1\(0; R} - V, since uj(Z} and hiZ} are; but that can only be the case when 1m aj = 0, so that aj is also real and (I) For any fixed point Z" E .1\(0; R"} C ([:"-1 the function hj (zl' Z") is holomorphic in the variable ZI in the disc .1\(0; rd except possibly at the origin. However, uj(Zl> Z"} is upper semicontinuous and so is bounded above near the origin in Z 1. Hence, from (I) it is evident that

Q Plurisubharmonic and Holomorphic Functions

181

for 0 < IZ11 < <5(Z") < 1 and some value mAZ"), or equivalently that

for 0 < Iztl < <5(Z") < 1 and some value MAZ"). But that means that ehiZ"Z") is meromorphic at the origin and hence that hj(z l ' Z") is actually holomorphic in the entire disc .::\(0; r 1 ) in the variable Z1' Then in the identity (1) the function hj(Z) is hoi om orphic in the polydisc .::\(0; R), so that 2aj loglz 11 must be plurisubharmonic there; consequently, aj > 0, and that yields the desired assertion. To examine the significance of the preceding result, recall first from the discussion in section K that the pluriharmonic functions in an open subset D s; en form the maximal linear subspace in the cone of plurisubharmonic functions in D. The notion of an extremal point in the latter cone really only makes sense modulo the maximal linear subspace of the cone. Thus, it is necessary either to consider the image cone in the quotient space modulo this maximal linear subspace, or to modify the customary definition somewhat. The latter procedure is more straightforward. So a plurisubharmonic function u in an open subset D s; en will be called extremal in D if u can only be written as a sum u = U1 + U2 of plurisubharmonic functions Uj in D when uj = cju + Vj for constants cj ~ 0 and pluriharmonic functions Vj in D.

8. THEOREM.

Iff is a holomorphic function in an open subset D s; en and the zero locus of f is a connected complex submanifold of dimension n - 1 in D, then loglfl is an extremal plurisubharmonic function in D.

Proof. Suppose that loglfl = U 1 + U 2 for some plurisubharmonic functions uj in D. These functions uj are of course pluriharmonic outside the zero locus V of the function f, as was observed in the proof of the preceding lemma. In an open neighborhood U of any point of V it is possible to choose a system of local coordinates (z 1, ... , zn) in en such that U is a polydisc in these coordinates and U n V = {Z E U: ZI = O}. Since f vanishes on V, it is evident from a consideration of the power series expansion of f in these coordinates that f(Z) = z Ul (Z) for some integer v > 0 and nowhere-vanishing holomorphic function f1 in U. Now

for any point Z E U, where ( -l/v) loglf1 (Z)I is pluriharmonic in U, so it follows immediately from the preceding lemma that (1/v)u 1 (Z) - (l/v) loglf1(Z)1 - C 1 loglz11 and (1/v)u 2 (Z) - c2 10glz 1 1are pluriharmonic in U for some uniquely determined positive constants cj . Consequently, uj(Z) - cj 10glf(Z)1 are pluriharmonic in U for these constants. This is the case for the same constants cj near any other point Z of V, since Vis connected; thus, uj - cj loglfl is pluriharmonic near Vas well as in the complement of V, and that suffices to conclude the proof. The hypothesis of the preceding theorem is unnecessarily restrictive, simply because the discussion of the zero loci of hoiomorphic functions has not been

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carried out far enough yet. The proper hypothesis, in the terminology to be established in Volume II, is that the zero locus of f is an irreducible holomorphic subvariety of the envelope ofholomorphy of D. That means that a dense open subset of the zero locus off is a connected submanifold of dimension n - 1 in the envelope ofholomorphy, so that the argument of the preceding theorem can be applied, while the remainder of the zero locus is a subvariety of smaller dimension and hence a removable singularity set for plurisubharmonic functions. The details will be left as an exercise to be completed when the relevant material from Volume II has been read, since the result will not be used here. The connectivity of the zero locus was used in the proof of the theorem, and irreducibility is essential. If the zero locus is reducible in the envelope of holomorphy, then f = fd2 for some holomorphic functions./j with distinct zero loci, so that loglfl = 10glfl1 + loglf21 and loglfl is not extremal. . There remains the question whether the converse of the preceding theorem holds-that is, whether the extremal plurisubharmonic functions are all of the form loglfl. The interest in extremal functions is, of course, motivated by the KreinMilman theorem, which asserts that under suitable conditions all points in a convex cone can be expressed as convex linear combinations of extreme points of the cone. In the present circumstances, it would be very useful to be able to show directly that arbitrary plurisubharmonic functions can be expressed suitably in terms of the special plurisubharmonic functions of the form loglfl for holomorphic functions f, for that would yield a direct proof of the theorem that pseudoconvexity is the same as holomorphic convexity, thereby finessing the digression through Dolbeault cohomology in section O. It is possible to express arbitrary plurisubharmonic functions in terms of these special plurisubharmonic functions by lattice rather than linear operations, at least in a domain of holomorphy. But present proofs require the equivalence of pseudoconvexity and holomorphic convexity. The interest in such a representation was indicated in the discussion in the book on several complex variables by Bochner and Martin; a proof was outlined by H. J. Bremermann, but the version to be given here follows that of N. Sibony. It may be recalled from the discussion in section L that the supremum of any collection of plurisubharmonic functions locally bounded from above is a nearly plurisubharmonic function v, the upper envelope of which is a plurisubharmonic function v*. The two functions v and v* coincide outside a set of measure zero and so represent the same locally L l-function. 9. THEOREM. If D is a domain of holomorphy in en, the set of plurisubharmonic functions of the form (sup{cj logl./jl} )*, where Cj > 0 are constants and ./j are holomorphic functions in D, is dense in the cone of plurisubharmonic functions in D in the topology of local L l-convergence. Proof.

Considering the cone of plurisubharmonic functions in the topology of local

L l-convergence implicitly excludes those functions that are identically equal to - OCJ on any connected component of D. It follows immediately from Theorem Kll that

the Coo plurisubharmonic functions are dense in this cone of plurisubharmonic functions, as a simple consequence of the monotone convergence theorem for the Lebesgue integral. Hence, it is sufficient just to show that any smooth plurisubharmonic function in D can be approximated arbitrarily well in the topology of

Q Plurisubharmonic and Holomorphic Functions

183

local U-convergence by the special plurisubharmonic functions ofthe statement of the theorem. Actually something a bit stronger will be demonstrated. Consider then a Coo plurisubharmonic function u in the domain D, and in terms of this function introduce the auxiliary open subset

D=

{(Z, w) E D xC: Iwl <

e-U(Z)} £;

cn+l

It will first be demonstrated that D is a domain of holomorphy in cn+l. For this purpose, note that D can be described equivalently as D = {(Z, w) ED xC: IweU(Z) I < I}. The function IweU(Z) I is clearly plurisubharmonic in D x C. Indeed, if w =F 0, that follows immediately from Theorem K8, since the Levi form of Iwe ,,(Z) I = IwIeU(Z) is the direct sum of the Levi forms of the smooth plurisubharmonic functions Iwl and eU(Z), and it is then an evident consequence of Theorem 3 and Theorem KS(c) that this function is also plurisubharmonic when w = O. If v(Z) is a continuous plurisubharmonic exhaustion function for D, as exists by Corollary MS, and K is a compact of D, then {(A,

b)

ED: v(A) ~ sup v(Z) (Z,w)eK

and

Ibeu(A) I ~ sup IweU(Z)I} (Z,w)eK

is clearly a compact subset of D. Thus, D is pseudoconvex and hence by Theorem PI is holomorphically convex. Now consider a compact subset KeD and a value e > O. For any point A E K, the power series 00

g(A, w) =

L

e·u(A)w·

.=0

in the complex variable w clearly has radius of convergence R(A) = e-u(A). This series can thus be viewed as representing a holomorphic function on the submanifold (A x C) n D of the domain of holomorphy D, so by Theorem E7 (in view of what was established in Theorem PI) it can be extended to a holomorphic function g(Z, w) in all of D. At any point ZED the latter function has a power series expansion 00

g(Z, w) =

L

.=0

f.(Z)w·

valid in (Z x C) n D and hence with radius of convergence R(Z) for which R(Z) ~ The coefficients f.(Z) are hoi om orphic functions in D, with the property that f.(A) = e·u(A), of course. As is quite familiar, the radius of convergence is given explicitly in terms of the coefficients of the power series by Hadamard's formula e-u(Z).

_1_) = lim sup yllf.(z)1 R(Z

Consequently,

•

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s~p ~ 10glfv(Z)1

(2)

1 -) = I·1m sup -1 Ioglfv(A)1 u(A) = log - (

(3)

u(Z)

~ log R;Z) =

lim

for all ZED, while

RA

v

v

The assertion (2) can be refined considerably with just a bit more effort. Choose an open neighborhood U of the compact set K having compact closure U c D, and a value r > 0 sufficiently small that ~(Zo,

whenever

r) s; U

Zo E K

and u(Z) ~ u(Zo)

e

+4

whenever

Zo E K

and

Z E ~(Zo; r)

It is a simple consequence of the Cauchy inequalities of Theorem AS that the functions (l/v) 10glfv(Z)1 are uniformly bounded for all v ~ 1 and Z E U. SO choose a constant M for which 1

-loglfv(Z)1 ~ M V

for

v ~ 1 and

Z EU

and u(Z) ~ M

for

Z EU

The functions

are nearly plurisubharmonic on U by Theorem L9, satisfy Igv(Z)1 ~ M for all Z E U, and from their definition are monotonically decreasing in v with limit 1 g(Z) = lim gv(Z) = lim sup -loglfv(Z)1 v v v

Hence from (2), g(Z) ~ u(Z)

The subsets

for

Z EU

Q

U. =

{Z

E

U: g.(Z) < u(Z)

+

Plurisubharmonic and Holomorphic Functions

185

i}

are then measurable subsets of U, and U. ~ U.+ 1 while U. U. = U. There is consequently an index Vo such that the Lebesgue measure IU - U.I of the set U - U. satisfies 6

IU - U.I < 8M (nr2)"

For any point Zo g:(Zo)

E

whenever

v> Vo

K the plurisubharmonic function g:(Z) satisfies

~ 1~(Zo; r)r1

r

g:(Z) dV(Z)

J.1(zo;r)

where dV(Z) is Lebesgue measure in C". This is an immediate consequence of the basic integral inequality J(2), as observed earlier in the proof of Theorem K6, for instance. Then if v > Vo and Zo E K, g.(Zo) - u(Zo) ~ g:(Zo) - u(Zo)

r

~ 1~(Zo; r)I- 1

[g:(Z) - u(Zo)] dV(Z)

J .1(Zo;r)

~ 1~(Zo; r)I~ 1~(Zo; r)I-

1

r

[g.(Z) - u(Z)

+

J .1(Zo;r)

1{

r

iJ

[g.(Z) - u(Z)

J.1(zo;r)nu,

+

Jr

[g.(Z) - u(Z)

.1(Zo;r)n(U-U,)

~ 1~(Zo; r)r1 { r

f

.1(Zo;r)n(U-u,)

iJ

~ dV(Z) +

J.1(zo;r)nu,

+

+

dV(Z)

+

iJ

dV(Z)

dV(Z)}

r

J.1(zo;r)n(u-u,)

6}

-dV(Z)

4

<6 In view of the definition of the function g., this inequality implies that

2M dV(Z)

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Volume I Function Theory

u(Z)

+e>

1 -logl!.(Z)1 v

whenever

v>

Vo

and

Z

E

K

(4)

the desired strengthening of (2). Finally from (3) and (4) it is evident that there is an index v for which u(Z)

1

+ e > -logl!.(z)1 v

for all

Z

E K

for all

Z

E UA

and

1

u(A) - e < -logl!.(A)1 v

Hence, of course, 1

u(Z) - e < -loglJ.(Z)1 V

where UA is some open neighborhood of A. The argument can be applied to each point A E K, and finitely many of the associated open sets UA will cover K. This will yield a finite collection of holomorphic functions jj in D and constants cj for which u(Z)

+ e > sup {cj 10gljj(Z)I} >

u(Z) - e

j

for all points Z E K, showing that u(Z) can be approximated arbitrarily well in the topology of local uniform convergence and hence in the topology of local L1-convergence by functions of the desired form, and thereby completing the proof. As is quite evident, the proof actually yields the stronger assertion that any Coo plurisubharmonic function in D can be approximated uniformly on compact subsets of D by special plurisubharmonic functions of the form SUPj{ cj logljjl}, in which jj are holomorphic in D and Cj ~ 0, with cj > 0 for only finitely many indices. The same is, of course, true for merely continuous plurisubharmonic functions, since they can be approximated similarly by COO plurisubharmonic functions as a consequence of Theorem K 11. The stronger result is not terribly natural, though, since the functions C 10gl!1 may well fail to be continuous in D and so do not lie in a space for which the topology of local uniform convergence is natural. The theorem itself does show that holomorphic and plurisubharmonic functions are really quite closely related, at least in domains of holomorphy; that may make it clearer why pseudoconvexity and holomorphic convexity amount to the same thing. The result of the theorem is not true for arbitrary open subsets of en, though. That is evident from the example at the end of section M of a subset D £: en

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187

such that the envelope of holomorphy of D is properly larger than D, but there are plurisubharmonic functions in D that cannot be extended to plurisubharmonic functions in any properly larger domain. There is a considerable body of further knowledge about plurisubharmonic functions and their role in the study of holomorphic functions, but there are many interesting open questions as well.

R Pseudoconvex Sets with Smooth Boundaries

For open subsets of en with smooth boundaries, there are additional useful alternative characterizations of pseudoconvexity and various other properties closely related to pseudoconvexity. Recall that an open subset D £;; ~m is said to have a smooth boundary of class C iffor each point A E aD there are an open neighborhood UA of A in ~m and a C diffeomorphism FA: UA ~ B(O; 1) £;; ~m such that FA(Ua n D) = {X = (x l' ... , xm) E B(O; 1): Xm < O}. A C diffeomorphism is a homeomorphism FA with the property that all the component functions of both FA and the inverse mapping FAl have continuous partial derivatives of all orders ~r. The condition that an open subset D £;; ~m has a smooth boundary of class C will be indicated briefly by writing aD E c. Although this is a significant smoothness condition even for r = 0, the interest here lies more in subsets with boundaries of class C for r ~ 1indeed, usually for r ~ 2. If r ~ 1, it is clear that alternatively an open subset D £;; ~m has a smooth boundary of class C if and only if for each point A E aD there are an open neighborhood UA of A in ~m and a real-valued function PA of class C in UA such that dPA #- 0 in UA and UA n D = {X E UA : PA(X) < O}. The functions PA are called local defining functions of class C for the subset D. Indeed, the condition that dPA #- 0 in UA just means that the matrix of first partial derivatives of the function PA is nonsingular at each point of UA' and is precisely the condition that the function PA can be viewed as one of a system oflocal coordinate functions of class cr at each point of UA • Thus, it is clear that these two characterizations of sets with smooth boundaries of class C for r ~ 1 are actually equivalent. The convention that sets will be described in terms of local defining functions PA by inequalities of the form PA < 0 will be followed systematically here. The particular choice of convention is not important, although there are some minor conveniences in the choice made here, but consistency is important. Note that if PA are local defining functions for a subset D £;; ~m, then their negatives - PA are local defining functions for the complementary subset ~m - D. Note also that if P and P' are any two local defining functions of class C, r ~ 1, for an open subset D £;; ~m in a neighborhood U, then P can be viewed as a local coordinate function near any point of U and the Taylor expansion of p'

188

R Pseudoconvex Sets with Smooth Boundaries

189

in these local coordinates leads to an expression of the form p' = hp, where h is a function of class Cr - 1 and h(X) > 0 at every point X E U. If D s;; IR mis an open subset with boundary aD E C, then the boundary aD is evidently a differentiable submanifold of class C and of dimension rn - 1 contained in IRm. If r ~ 1, this submanifold aD s;; IRm has a well-defined tangent space at each point A E aD, and that tangent space can be viewed either as a linear subset of dimension rn - 1 in IRm passing through the point A or as the translate of that linear subset to the origin in IRm. In the latter case the tangent space at A is thus identified with a linear subspace 1A(aD) of dimension rn - 1 in the real vector space IRm. If p is a local defining function of class C for the subset D in an open neighborhood of A, then as is familiar 1A(aD) = { T

E

IRm :

~ tj ::j (A) = o}

(1)

This is clearly independent of the choice of local defining function. The preceding familiar differential-geometric considerations naturally also apply to open subsets D s;; en upon identifying en with 1R2n. If D s;; en is an open subset with boundary aD E C for r ~ 1, then the tangent space to the real differentiable submanifold aD at a point A E aD can be identified with a real linear subset 1A(aD) of dimension 2n - 1 in the complex vector space en. Any tangent vector TE 1A(aD) is thus viewed as a complex vector T = (tl' ... , tn ) E en, but the set of all these tangent vectors forms a real rather than a complex linear subspace of en. To rewrite the explicit expression (1) for this tangent space in complex terms, note that if p is a local defining function of class C 1 for the subset D in an open neighborhood of the point A, then for tj = uj + iVj it follows that

since p is a real-valued function. Consequently, 1A(aD) = { TE

en: Re ~ tj :~ (A) = o}

(2)

clearly emphasizing the fact that 1A(aD) is only a real linear subspace of en. Since 1A(vD) is not a complex linear subspace, i~(aD) does not coincide with ~(aD) but is another real linear subspace of dimension 2n - 1 in en. But the intersection ~(aD) n i~(aD) is clearly a complex linear subspace-indeed, is evidently a complex linear subspace of dimension n - 1 and the largest complex linear subspace of

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en lying in

1A(aD). This subspace is called the complex part of the tangent space 1A(aD) and is denoted by 1A1 .O(aD). It is clear that T E 1A1 .O(aD) precisely when T E 1A(aD) and iT E 1A(aD). It thus follows immediately from (2) that

(3) clearly showing that 1A1 .O(aD) is a complex linear subspace of en. If D s;; en is an open subset with boundary aD E Cr for r ~ 2, then it is possible to consider in addition to the linear approximation to aD, the tangent space to the submanifold aD s;; en, also the second-order approximation to aD, involving local curvature properties of the submanifold; this too is of considerable interest from the point of view of complex analysis. If p is a local defining function of class cr, r ~ 2, for the subset D in an open neighborhood of a point A E aD, the natural quadratic expression to consider is the Levi form Lp(A; T); however, this form depends on the choice of the local defining function p in a nontrivial way. If p' is another local defining function of class cr, r ~ 2, for the subset D in an open neighborhood of the point A, then as already noted p' = hp, where h is a function of class Cr - 1 and h(Z) > O. Even though the function h may only be of class C 1 , nonetheless since p(A) = p'(A) = 0 clearly

Consequently, Lp'(A; T) = h(A)Lp(A; T)

~ ap _ ah + 2 Re L... tr~-(A) tk = jk

uZj

(A)

(4)

uZk

which is a rather nontrivial relation. However, if the vector T is restricted to lie in the complex tangent space 1A1 .O(aD) defined by (3), then Lp'(A; T) = h(A)Lp(A; T), where h(A) > O. Thus, the restriction of the Levi form Lp(A; T) to vectors T E 1A1.0(aD) is a Hermitian form on the complex vector space 1A1 .O(aD) and is uniquely determined up to a positive scalar factor by the boundary aD alone, independent of the choice of a local defining function for the set D. This class of Hermitian forms on 1A1.O(aD), all of which differ by only a positive scalar factor, will be called the Levi form of the boundary aD at the point A. 1. DEFINITION. An open subset D s;; en with boundary aD E c 2 is pseudoconvex in the sense of Levi if the Levi form of the boundary aD is positive semidefinite at each point A E aD. D is strictly pseudoconvex in the sense of Levi if the Levi form of the boundary aD is actually positive definite at each point A E aD. It is clear that these notions are really well defined, since multiplication by a positive constant preserves the classes of positive semidefinite and positive definite

R Pseudoconvex Sets with Smooth Boundaries

191

Hermitian forms. A strictly Levi pseudo convex set is sometimes also referred to as a strongly Levi pseudoconvex set. As might be guessed from the terminology, this notion of pseudoconvexity is equivalent to the notion discussed in earlier sections, for subsets with sufficiently smooth boundaries, of course. A convenient way to demonstrate this equivalence involves using the distance function dD , which played a prominent role in the earlier discussion of pseudoconvexity, as a local defining function for the set D. If should be observed, however, that the function dD as introduced in Definition G3 and used so far has been defined only in the set D itself. But it is a simple matter to extend this function to one defined throughout e" by setting if Z E aD if z¢i5

(5)

e", and moreover that - dD is the continuous analogue of a local defining function for the subset D near each point of the boundary aD. It only fails to be a local defining function by not having the property that dPA =F 0 near aD, a property that of course it cannot have ifthe boundary is nondifferentiable. However, the extended function -d D is a proper local defining function whenever the boundary of D is sufficiently smooth. It is obvious that the function dD as so extended is continuous in all of

If D ~ e" is an open subset with boundary aD E C 2 , then the extended function - dD is a local defining function of class c 2 for the subset D near each boundary point.

2. LEMMA.

Proof. The complex structure in e" plays no role here, and it is even convenient to ignore it altogether; hence, e" will be identified with ~211, with real coordinates X = (Xl' ... , X 211 ) or Y = (YI' ••• , Y211)' In order to prove the desired result, it is sufficient merely to show that the extended function dD is of class C 2 near aD and has a nontrivial differential at each point of aD. For any point A E aD, select some local defining function P of class C 2 for the subset D in an open neighborhood U of the point A. It is familiar from elementary differential geometry that the gradient vector Vp(X) = (ap(X)/ax I , ... , ap(X)/ax 211 ) is normal to the submanifold aD at each point X E U n aD. Indeed, in view of the convention for local defining functions adopted here, Vp(X) is a normal vector directed outward from D. It is also familiar that whenever X E Un aD and Y = X + t· Vp(X) where t E ~ and It I is sufficiently small, then the ball B(Y; r) passing through X is tangent to the submanifold aD at X, and that ball is either entirely contained within D or entirely exterior to D depending on the sign of t. Thus, dD(y) = ±r = ± II Y - XII. Now the mapping F: (U n aD) x ~ -+ ~211 defined by F(X, t) = X + t· Vp(X) clearly has a nonsingular Jacobian matrix whenever t = 0, so if the neighborhood U of A is suitably chosen, then F establishes a C 1 diffeomorphism F: (U n aD) x ( - e, e) -+ U for some sufficiently smalle. Any point Y E U can therefore be written in a unique way in the form Y = F(X, t) = X + t· Vp(X), where X E Un aD and Itl < e. The mapping P: U -+ Un all defined by P(Y) = X is then a C l projection mapping, and if U and e are sufficiently small, dD(Y) = ± II Y - P(Y)II whenever Y E U, for an appropriate

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choice of sign depending on whether Y E D or Y ~ D. It is clear from this that dD is a function of class C 1 in U except perhaps at Un aD, and indeed that when YE U - UnaD,

(6) On the other hand, since P(Y) E aD, necessarily p(P(Y» = 0 for all points Y and upon differentiating it follows that

o=

a

L

-;- p(P( Y» = uYi k

where X=P(Y). But if consequently,

ap(X) a -:1- -;-

uXk

uYi

y~aD,

E

U,

Pk( Y)

then Y-P(Y)=t·Vp(X), where t#O, and

Substituting this into (6) shows that

But that just means that in U - Un aD, the gradient vector of the function dD(Y) is the unit vector parallel to Y - P(Y) = tV p(P(Y» and pointing toward the interior of D and hence that Vp(P(Y» VdD(Y) = -IIVp(P(Y»1I

The vector field VdD(Y) is therefore of class C 1 and is nowhere zero, and that suffices to conclude the proof of the lemma. Another useful auxiliary observation here is the following, which is really just a straightforward calculation. 3. LEMMA. If P is a real-valued function of class C 2 in an open neighborhood of the origin o in en, then as Z approaches 0,

R Pseudoconvex Sets with Smooth Boundaries

193

where P2P(0; Z) is the complex polynomial of degree 2 in the variables Z defined by

and Lp(O; Z) is the Levi form of p at the origin.

Since p is of class C 2 near the origin, it follows from the usual Taylor expansion of the function r at the origin in terms of the real coordinates Xj' Yj in IC" = ~2" that as Z approaches 0, Proof.

p(Z) = p(O)

+ L [aa p (0)· Xj + aa p (0)· Yj] j

Xj

Yj

+ 7)/2 and Yj = (Zj - zj)/2i, then grouping together those terms in the preceding formula involving the same variables Zj and Zk and recalling the definitions of the operators a/azj and a/azk show that the preceding formula can be rewritten as

If Xj = (Zj

p(Z) = p(O)

ap (O)·Zj + aap (O)·Zj ] + L [-a j

Zj

Zj

since p is real-valued, and that suffices for the proof. Applying the two preliminary results leads to the following basic observation.

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4. THEOREM. An open subset D ~ en with boundary aD E it is pseudoconvex in the sense of Levi.

c 2 is pseudoconvex if and only if

Proof. First suppose that D is pseudoconvex, so that by Theorem M3 the function -log d is plurisubharmonic in D, where to simplify the notation, d stands for dD • The function d is of class C 2 in D near the boundary aD-indeed, naturally extends to a function of class C 2 in a full open neighborhood of the boundary aD in en by Lemma 2. So by Theorem K8 the condition that -log d be plurisubharmonic is that its Levi form be positive semidefinite-that is, that

o ~ L( -log d)(Z; A) ~

a2 log d(Z)

jk

aZj aZ k

~ - L.

-

_

ajak

for all points ZED sufficiently near the boundary aD and all vectors A E en. In particular, therefore (7)

whenever ZED is sufficiently near the boundary and the vector A satisfies O. It follows from continuity that (7) holds in the limit for all points Z E aD whenever A E Tz1.0(aD); but that isjust the condition that D be pseudoconvex in the sense of Levi, since - d is a local defining function of class C 2 for the subset D near each boundary point of Lemma 2. Next suppose that D is not pseudoconvex. Since the pseudoconvexity of D is a local property of aD in view of Theorem M8, it follows from Theorem M3 that there must be points in D arbitrarily near the boundary at which the function -log d is not plurisubharmonic, where again d is used in place of d D and denotes the extension of dD to a function defined throughout en. Choose such a point A E D at which d is of class C 2 , as is possible in view of Lemma 2. Since d is not plurisubharmonic at A, it follows from Theorem K8 that

L aj aD(Z)/azj =

d Ljk aa 10g a- (A)cjck = Zk 2

L( -log d)(A; C) = -

-2r < 0

Zj

for some vector C E en and some positive real number r, or equivalently that 0 2 log d(A

at at

+ tC) I = 2r > 0 1=0

It follows from Lemma 3 that the Taylor expansion of the real-valued function log d(A + tC) of the point tEe near the origin can be written in the form

R Pseudoconvex Sets with Smooth Boundaries

log d(A

+ tC) =

log d(A)

for some complex constants sufficiently small 8 > 0, then log d(A

0(

195

+ Re[O(t + pt 2] + 2rltl2 + 0(ltI2) and

p. Consequently, whenever It I < 8 for some

+ tC) ~ log d(A) + Re[O(t + pt 2] + rltl 2

or equivalently (8)

Choose a boundary point BEaD nearest A, so that d(A) = liB - A II, and introduce the holomorphic mapping F: C -+ en defined by F(t) = A

+ tC + (B -

A)e at+ Pt2

Note that F(O) = BEaD, but that whenever 0 < It I < d(A

+ tC) >

8,

then erltl2 > 0, so that by (8),

liB - Allleat+Pt21

and consequently F(t) E D. The composite function d(F(t)) is thus nonnegative for It I < 8; indeed, from the triangle inequality and (8) it follows that whenever It I < 8, then d(F(t)) ~ d(A

+ tC) -

II(B - A)eat+Pt211

The function d(F(t)), which is of class C 2 near t = 0 since F(O) = BEaD and by Lemma 2 the extended function is of class C 2 near aD, thus attains a local minimum at t = 0 but is not 0(ltI2). Therefore

a

at d(F(t))lt=o = 0

a2

at at d(F(t))lt=o > 0

Writing F = (fI' ... , In) and using the complex form of the chain rule show, since !j are holomorphic, that

I

j

iJd -(B)· !j'(0) = 0 iJzj

Since - d is a local defining function for D, this shows that D is not pseudoconvex in the sense of Levi, and thereby concludes the proof. The notion of pseudoconvexity given in Definition 1 was introduced by E. E. Levi (in Annali di Mat. pura ed appl., vol. 17, 1910) and antedates the more general

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notion given in Definition MI. Levi showed that every domain in C n that is pseudoconvex in his sense is at least locally holomorphically convex, thus raising the question whether a locally holomorphically convex domain is actually holomorphically convex-that is to say, whether holomorphic convexity is actually a local property. This problem has subsequently commonly been known as the Levi problem. The solution of the Levi problem discussed earlier here involved the more general notion of pseudoconvexity and so was valid for arbitrary open subsets of Cn (or of Riemann domains) without assuming any boundary regularity. The consideration of domains having smooth boundaries leads to a vast array offurther fascinating questions, such as the boundary behavior of various classes of hoI om orphic functions (including generalizations of the classical HP spaces in the unit disc in one variable) or of holomorphic mappings between smoothly bounded domains, and the possibilities of more general Cauchy integral formulas involving integration on the full boundaries. Even to begin to discuss these questions, which have been actively investigated with a wide variety of major results scattered throughout the literature, would extend the present volume excessively. The reader must be referred to several other treatises dealing with these matters.

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About this book: Introduction to Holomorphlc Functions of Several Variables, Volumes I-III provide an extensive introduction to the Oka-Cartan theory of holomorphic functions of several variables and holomorphic varieties . Each volume covers a different aspect and can be read independently. . • Volume I: Function Theory (ISBN: 0-534-13308-8) • Volume II: Local Theory (ISBN : 0-534-13309-6) • Volume III: Homological Theory (ISBN: 0-534-13310-X)

Other titles from the Wadsworth & Brooks/Cole Mathematics Series Complex Variables, Second Edition Stephen D. Fisher This undergraduate text for majors in engineering and mathematics offers a direct route to the most important topics in the theory and applications of complex variables. Thoroughly updated and revised . Contents: 1. The complex plane . 2. Basic properties of analytic functions . 3. Analytic functions as mappings . 4. Analytic and harmonic functions in applications . 5. Transform methods. 1990. Clothbound. 448 pages . ISBN : 0-534-13260-X.:

ISBN 0-534-13308-8 90000

9 780534 133085