The dual of the space of holomorphic functions on locally closed convex sets

Publ. Mat. 49 (2005), 487–509 THE DUAL OF THE SPACE OF HOLOMORPHIC FUNCTIONS ON LOCALLY CLOSED CONVEX SETS Jos´ e Bonet, Reinhold Meise and Sergej N. Melikhov Abstract Let H(Q) be the space of all the functions which are holomorphic on an open neighbourhood of a convex locally closed subset Qof CN, endowed with its natural projective topology. We characterize when the topology of the weighted inductive limit of Fr´echet spaces which is obtained as the Laplace transform of the dual H(Q)0of H(Q) can be described by weighted sup-seminorms. The behaviour of the corresponding inductive limit of spaces of continuous functions is also investigated. 1. Introduction Martineau investigated in [15] the spaces H(Q) of analytic functions on a convex nonpluripolar set Qin CNin the case that Qadmits a countable fundamental system of compact sets. This setup covers nonpluripolar compact convex sets and convex open sets in CNand convex open sets in RN. In the latter case H(Q) coincides with the space of all real analytic functions on Q. The strong dual H(Q)0 bof each of these spaces can be canonically identified, via the Laplace transform, with a weighted (LF)-space V H(CN) of entire functions on CN, i.e. H(Q)0 bis isomorphic to a Hausdorff countable inductive limit of Fr´echet spaces of entire functions defined by weighted sup-seminorms. See the details below. The description of the topology of such weighted inductive limits of spaces of holomorphic functions has been investigated thoroughly in recent years since the work of Ehrenpreis [10] on analytically uniform spaces. Similar questions on the projective description were later investigated by various authors (see [1]). 2000 Mathematics Subject Classification. Primary: 46E10; Secondary: 46A13. Acknowledgement: The research of J. Bonet and R. Meise was partially supported by MEC and FEDER Project MTM2004-02262 and AVCIT Grupo 03/050. S. N. Melikhov thanks the support by DAAD and by Russian Foundation of Basic Research (project no. 02-01-00372). 488 J. Bonet, R. Meise, S. N. Melikhov Many results about the projective description of weighted (LF)-spaces of entire functions were obtained in [4], [5], [6], [7]. In particular, we derived in [6] necessary as well as sufficient conditions for the projective description to hold algebraically or topologically for the space of Laplace transforms of H(Q)0 bwhere Qis a bounded locally closed convex subset of CN. In the present paper we continue these investigations for such sets Qwhich are not necessarily bounded. As in [6] we have algebraic and topological projective descriptions if Qis strictly convex at the relative boundary of Q∩∂rQ(see Definition 2.4 and Theorem 3.1). In Theorem 3.11 we show that this condition is necessary for the topological projective description if we assume in addition that Qadmits a neighbourhood basis of domains of holomorphy. To obtain this result, we use a method different from that of our earlier work. The new idea is to apply the approach that was used in [5]. We use a result of Mal’tsev [14] which implies that if a convex, locally closed set Qhas a neighbourhood basis of domains of holomorphy and there is a supporting hyperplane to the closure of Qwith non-compact intersection with Q, then there is a differential operator P(D) on H(Q) which is not surjective. Bierstedt and Bonet [2] extended the projective description techniques from (LB) to (LF) spaces of continuous functions using Vogt’s approach [21] to Palamodov and Retakh theory of (LF)-spaces. The problem of topological projective description for the spaces V C(CN) of continuous functions corresponding to the Laplace transform of the space H(Q)0 bhas a positive answer by a result of Bierstedt, Meise, and Summers [3]. The algebraic coincidence of the (LF)-space V C(CN) and its projective hull is characterized in Theorem 4.5 by the condition that Qis strictly convex at the relative boundary of Q∩∂rQ. 2. Notation and Preliminaries 2.1. Definition. A subset Qof CNis called locally closed if for each z∈Qthere is a closed neighbourhood Uof zin CNsuch that Q∩Uis closed. Every open subset and every closed subset of CNis locally closed. For a convex set Q⊂CNthe symbols intrQand ∂rQdenote the relative interior and the relative boundary of Qwith respect to the affine hull of Q. For example, if 0 ∈Q, the affine hull of Qis the real linear span of Q. We write ω:= Q∩∂rQ. By ∂rωwe denote the relative boundary of ωwith respect to ∂rQ. 2.2. Remark. By [6, Lemma 1] and [18, Lemma 1.2], the following assertions are equivalent for a convex subset Qof CN: (i) Qis locally closed. Spaces of Holomorphic Functions 489 (ii) Qadmits a countable fundamental system of compact subsets. (iii) Qis the union of the relative interior intrQof Qand a subset ω of ∂rQwhich is open in ∂rQ. 2.3. General Assumption. Throughout this article Qdenotes a locally closed convex set and (Qn)n∈Na fixed increasing fundamental sequence of compact convex sets in Q. 2.4. Definition ([18, Definition 1.3]).A locally closed convex set Qis called (C-)strictly convex at the relative boundary of ωif the intersection of Qwith each supporting (complex) hyperplane to the closure Qof Q is compact. If the interior of Qis empty, the set Qis strictly convex at the relative boundary of ωif and only if Qis compact. If the interior of Q is not empty, Qis (C-)strictly convex at the relative boundary of ω if and only if each line segment (of which the C-linear affine hull belongs to some supporting hyperplane of Q) of ω=Q∩∂rQis relatively compact in ω. By [6, Proposition 2], a locally closed convex set Q is strictly convex at the relative boundary of ωif and only if Qhas a neighbourhood basis of convex domains. For example Qis strictly convex at the relative boundary of ωif Qis open or compact. Let Q⊂CNbe a convex locally closed set such that Q:= {x∈CN= R2N|x1≥f(x2, . . . , x2N),(x2, . . . , x2N)∈R2N−1}for a convex function f:R2N−1→R. If the function fis strictly convex in the sense of H¨ormander [12, p. 56], then Qis strictly convex at the relative boundary of ω. If Qis closed and there is a unbounded interval in R2N−1on which fis affine then Qis not strictly convex at its relative boundary. 2.5. Spaces of holomorphic functions. For an open set D⊂CN, we denote by H(D) the space of all holomorphic functions on Dwith its standard Fr´echet topology. For a compact subset Kof CN, the space of all functions which are holomorphic on some open neighbourhood of K is denoted by H(K) and it is endowed with its natural inductive limit topology. We denote by H(Q) the vector space of all functions which are holomorphic on some open neighbourhood of the locally closed convex set Q. Let (Qn)n∈Nbe an increasing fundamental sequence of compact convex sets in Q. Since the algebraic equality H(Q) = ∩n∈NH(Qn) holds, we endow H(Q) with the projective topology of H(Q) := projnH(Qn). This topology does not depend of the choice of the fundamental system (Qn)n∈N. See more details in [18, pp. 296–299]. In the case that Qis a locally closed convex subset of RN, the space H(Q) is a space 490 J. Bonet, R. Meise, S. N. Melikhov of real analytic functions. In particular, if Qis an open convex subset of RN, then H(Q) = A(Q), where A(Q) denotes the space of all real analytic functions on Q. The projective description of the Fourier Laplace transform of H(Q)0 bwas studied by Ehrenpreis in [9] and by the authors in [7]. 2.6. Definition. For each set D⊂CNwe denote by HD:CN→ R∪{∞} the support function of D:HD(z) := supw∈DRehz, wi,z∈CN. Here hz, wi:= PN j=1 zjwj. For each n∈Nlet Hn:= HQnbe the support function of the convex compact set Qn. 2.7. Problem of the projective description. Next, we recall the necessary notation for weighted inductive limits, see [3] and [4], and we state the problem of projective description. For notation concerning locally convex spaces we refer the reader to [17]. We denote by V= (vn,k)n,k∈Na double sequence of strictly positive upper semicontinuous weights on CN,N∈N, such that vn+1,k(z)≤vn,k(z)≤vn,k+1(z), z ∈CN, for each n, k∈N. The weighted inductive limit of Fr´echet spaces V H(CN) of entire functions associated with Vis defined by (2.1) V H(CN) := ind n→proj ←k H(vn,k,CN), where the steps H(v, CN) are defined, for a positive weight von CN, as the Banach space of entire functions H(v, CN):={f∈H(CN)| ||f||v:= sup z∈CN v(z)|f(z)|<∞}. The space V H(CN) is a Hausdorff (LF)-space. In order to describe its topology by means of weighted sup-seminorms, Bierstedt, Meise, and Summers [3] associated with Vthe system Vof all those weights v:CN→[0,∞[ which are upper semicontinuous and have the property that for each nthere are αn>0 and k=k(n) such that v≤αnvn,k on CN. The projective hull of the weighted inductive limit is defined by HV (CN):={f∈H(CN)| ||f||v:= sup z∈CN v(z)|f(z)|<∞for all v∈V}, endowed with the Hausdorff locally convex topology defined by the system of seminorms {||.||v|v∈V}. The projective hull is a complete locally convex space and V H(CN) is contained in its projective hull with continuous inclusion. Spaces of Holomorphic Functions 491 The problem of projective description is to determine conditions under which (1) the spaces V H(CN) and HV(CN) coincide algebraically, or (2) the space V H(CN) is a topological subspace of its projective hull HV (CN). A positive answer to question (2), i.e., whether V H(CN) is a topological subspace of its projective hull, is of particular importance, because when the answer is positive it permits to describe the topology of the weighted (LF)-space of holomorphic functions by means of weighted supseminorms. In this article we are interested in the weight functions vn,k(z) := exp(−Hn(z)− |z|/k), n, k ∈N, z ∈CN, where |z|:= (PN j=1 |zj|2)1/2. By [18, Lemma 1.10], the Laplace transform F(ϕ)(z) := ϕ(exph·, zi), z ∈CN, is a linear topological isomorphism from the strong dual H(Q)0 bof H(Q) onto V H(CN). We denote by V0the set of all weights vsuch that there are unbounded increasing sequences (k(n))n∈N⊂Nand (αn)n∈N⊂]0,∞[ with v(z) = inf n∈Nexp(−Hn(z)− |z|/k(n) + αn) for all z∈CN. It is easy to see that every weight in V0is contained in Vand that every element in Vis estimated by a weight in V0. 2.8. Remark. Let G⊂CNbe open and convex and let (Gn)n∈Nbe a fundamental sequence of (convex) compact subsets Gnof Gwith Gn⊂ int Gn+1 for all n∈N. We put VG:= (vG,n)n∈N, vG,n(z) := exp(−HGn(z)), z ∈CN, n ∈N. By [12, Theorem 4.7.3] the Laplace transform F(ϕ)(z) := ϕ(exph·, zi), z ∈CN, is a linear topological isomorphism from the strong dual H(G)0 bof H(G) onto the weighted (LB)-space VGH(CN). Moreover, as a consequence of [3, Theorem 1.6], the space VGH(CN) and its projective hull HVG(CN) coincide algebraically and topologically. 492 J. Bonet, R. Meise, S. N. Melikhov 3. Spaces of holomorphic functions In this section we study the problem of topological projective description for the weighted (LF)-space V H(CN). 3.1. Theorem. Let Q⊂CNbe a convex locally closed set. If Qis strictly convex at the relative boundary of ω, then the weighted inductive limit V H(CN)coincides with its projective hull HV (CN)algebraically and topologically. Proof: We first claim that the space V H(CN) coincides algebraically and topologically with proj←G∈G H(G)0 b, where G:= {G|Gis open and convex, G⊃Q}. To show this as in the proof of [6, Theorem 6 (i)], note first that by [6, Proposition 2], we have H(Q) = ind→G∈G H(G) algebraically. If Bis any bounded set in H(Q) then Bis bounded in H(Qn) for each n∈N. Since H(Qn) is a (DFS)-space, there exists a convex open neighbourhood Unof Qnso that B⊂H(Un) and such that the functions in Bare uniformly bounded on Un. Then V:= Sn∈NUnis an open neighbourhood of Q. Since Qis strictly convex at the relative boundary of ω, there is an open neighbourhood Uof Qin Gwhich is contained in V. Clearly, B⊂H(U). From the construction it follows easily that B is bounded in H(U) and consequently bounded in ind→G∈G H(G). Since Qis strictly convex at the relative boundary of ω,H(Q) is bornological by results of Martineau [15] (see [18, Remark 1.9 (b)]). From this and the preceding considerations we get that H(Q) = ind→G∈G H(G) holds topologically and H(Q)0 b= proj←G∈G H(G)0 b. By Remark 2.8 this implies V H(CN) = proj ←G∈G proj ←u∈VG H(u, CN). Next we show that the space HV (CN) is contained in the space projGproju∈VGH(u, CN) and that the inclusion is continuous. We fix a convex open neighbourhood Gof Q, a function κ∈VGand show that there exists v∈Vwith κ≤v. To do so, for every nthere is αn≥1 with κ≤αnexp(−HGn). Since Gis a neighbourhood of Q, for each n there exist k(n) such that Qn+1 k(n)B(0,1) ⊂Gk(n)and, consequently, Hn(z) + |z|/k(n)≤HGk(n)(z) for all z∈CN. Hence κ(z)≤exp(−Hn(z)− |z|/k(n) + log αk(n)) for all z∈CNand n∈N. Moreover, the sequences (k(n))n∈Nand (αn)n∈Ncan be taken increasing and unbounded. If we define v(z) := inf n∈Nexp(−Hn(z)− |z|/k(n) + log αk(n)) for all z∈CN, Spaces of Holomorphic Functions 493 then v∈V0and κ≤v. From here it follows that HV (CN) is continuously embedded in projGproju∈VGH(u, CN). Consequently, the space V H(CN) and its projective hull HV (CN) coincide algebraically and topologically. We prove in Theorem 3.11 below that the strict convexity of Qat its relative boundary is necessary for V H(CN) to be a topological subspace of HV (CN), if Qhas a basis of neighbourhoods which consists of domains of holomorphy. To show this we need some results about entire functions and differential operators of infinite order with constant coefficients. 3.2. Definition. An entire function fon CNis said to be of order at most one and zero type if for each ε > 0 there is C≥0 with |f(z)| ≤ Cexp(ε|z|) for all z∈CN. We denote by A0the set of all entire functions fon CNof order at most one and zero type. By [12, Theorem 4.7.3], A0=F(H({0})0). By [16, Lemme 15] each nonzero entire function P∈A0is slowly decreasing, i.e., for each ε > 0 there is R > 0 such that for all z∈CN with |z| ≥ Rthere is w∈B(z, ε|z|) with |P(w)| ≥ exp(−ε|w|). Here we denote B(µ, r) := {z∈CN| |z−µ| ≤ r},µ∈CN,r≥0. We will use the following statements about functions in A0. 3.3. Lemma. Let Pbe a nonzero entire function on CNof order at most one and zero type. (i) For each locally bounded function r:CN→[0,∞[satisfying lim|z|→∞ r(z)/|z|= 0, there is a continuous function m:CN→ [0,∞[satisfying lim|z|→∞ m(z)/|z|= 0 and sup |t−z|≤r(z) |P(t)| ≤ exp m(z)for all z∈CN. (ii) There exists a continuous function r:CN→[0,∞[satisfying lim|z|→∞ r(z)/|z|= 0 and sup |t−z|≤r(z) |P(t)| ≥ exp(−r(z)) for all z∈CN. Proof: (i) Since for each ε > 0 there is C > 0 such that |P(z)| ≤ Cexp(ε|z|) for all z∈CN, for the function α(z) := max(0; log |P(z)|) 494 J. Bonet, R. Meise, S. N. Melikhov we have lim|z|→∞ α(z)/|z|= 0. From the inequalities ( sup |t−z|≤r(z) α(t))/|z| ≤ sup |t−z|≤r(z) |t| |z|sup |t−z|≤r(z) α(t) |t| ≤|z|+r(z) |z|sup |t|≥|z|−r(z) α(t) |t| = (1 + r(z)/|z|) sup |t|≥|z|−r(z) α(t) |t| and lim|z|→∞ r(z)/|z|= 0, lim|z|→∞(|z| − r(z)) = ∞it follows that β(z) := sup|t−z|≤r(z)α(t) = o(|z|) as |z|→∞. As mwe can choose any continuous majorant of βwith m(z) = o(|z|) as |z| → ∞. (ii) Since Pis of order at most one and zero type, we can select an increasing unbounded sequence Rn>0 such that for each n∈Nand each z∈CNwith |z| ≥ Rnthere exists w∈B(z, |z|/n) such that |P(w)| ≥ exp(−|w|/n). We put s(z) := 1 if |z|< R2and s(z) := |z|/n if Rn+1 ≤ |z|< Rn+2 and n∈N. For z∈CNwith |z|∈[Rn+1, Rn+2), n∈N, we choose w∈B(z, |z|/(n+ 1)) ⊂B(z, s(z)) such that |P(w)| ≥ exp(−|w|/(n+ 1)). Then sup |t−z|≤s(z) |P(t)| ≥ exp −|z|+|z|/(n+ 1) n+ 1 = exp −n+ 2 (n+ 1)2|z| ≥exp −|z| n= exp(−s(z)). As rwe can choose any continuous majorant of swith lim|z|→∞ r(z)/|z|= 0. 3.4. Lemma. (i) For each v∈Vand every upper semicontinuous function m:CN→Rsuch that lim|z|→∞ m(z)/|z|= 0 the function vexp mbelongs also to V. (ii) For each v∈Vand every locally bounded function r(z): CN→ [0,∞[for which lim|z|→∞ r(z)/|z|= 0 there is u∈Vwith v≤ inf|t−z|≤r(z)u(t)for all z∈CN. Proof: (i) This follows easily from the definition of Vand the condition on m. (ii) It is enough to show the assertion for v∈V0. To do so, fix v(z) = inf n∈Nexp(−Hn(z)− |z|/k(n) + αn)∈V0. Spaces of Holomorphic Functions 495 Since for all n∈Nand ε > 0 the function Hn(z) + ε|z|is subadditive and positively homogeneous, for each n∈Nthere exists Cn≥0 such that sup |t−z|≤r(z) (Hn(t)+|t|/(k(n)+1)) ≤Hn(z)+|z|/k(n)+Cnfor all z∈CN. Moreover, the sequence (Cn)n∈Ncan be taken increasing. The desired function is u(z) := inf nexp(−Hn(z)− |z|/(k(n)+1)+αn+Cn), z ∈CN. 3.5. Lemma. For each nonzero entire function P∈A0the multiplication operator MP:V H(CN)→V H(CN), MP(f) := Pf, is an injective topological homomorphism if V H(CN)is endowed with the topology induced by HV (CN). Proof: It is clear that MPis injective and linear. To show that MPis continuous we fix a function v∈V. By Lemma 3.3 (i) there is a continuous function m:CN→[0,∞[ such that lim|z|→∞ m(z)/|z|= 0 and |P(z)| ≤ exp(m(z)) for all z∈CN. Then u:= vexp m∈Vand for all f∈V H(CN) kMP(f)kv= sup z∈CN |P(z)f(z)|v(z)≤ kfku. Consequently, MP:V H(CN)→V H(CN) is continuous if V H(CN) is endowed with the topology induced by HV (CN). To show that the division map Pf 7→ fis continuous for this topology, choose a continuous function r:CN→[0,∞[ according to Lemma 3.3 (ii) and a continuous function m:CN→[0,∞[ for 4raccording to Lemma 3.3 (i). Fix v∈V. By Lemma 3.4 (i) for m+ 2r, the function vexp(m+ 2r) belongs to V, and we can apply Lemma 3.4 (ii) to conclude that there is w∈Vsuch that v(z) exp(m(z)+2r(z)) ≤inf |t−z|≤4r(z)w(t). 502 J. Bonet, R. Meise, S. N. Melikhov hence Hm(a)+1/l ≥ −(log C)/t + min(Hn(a)+1/k, Hµ(a)+1/L). Letting t→ ∞, we obtain (4.2). 4.3. Notation. The convex hull of a set D⊂CNis denoted by conv(D). Let (ωn)n∈Nbe a compact exhaustion of ωsuch that ωn⊂ωn+1 for all n∈N. We can suppose that Qn= conv(ωn∪Kn), where (Kn)n∈Nis a fundamental system of (convex) compact subsets of intrQwith Kn⊂ Kn+1 for each n∈N; see [18, Lemma 1.2]. We define the following sets of supporting directions: Sωn:= {a∈S| ∃ z∈ωn: Rehz, ai=HQ(a)}, n ∈N, Sω:= {a∈S| ∃ z∈ω: Rehz, ai=HQ(a)}. If int Q6=∅, then for a∈Sand n∈Nthe equality Hn(a) = HQ(a) holds if and only if a∈Sωn. Indeed, let a∈Sωn. Then there is z∈ωn with HQ(a) = Rehz, ai. From ωn⊂Qn⊂Qit follows that Rehz, ai ≤ Hωn(a)≤Hn(a)≤HQ(a) and Hn(a) = HQ(a). Conversely, assume that Hn(a) = HQ(a). Since Qn= conv(Kn∪ωn) the equality Hn(a) = max{HKn(a), Hωn(a)}holds. The inclusion Kn⊂int Qimplies that HKn(b)< HQ(b) for each b∈S. Consequently Hωn(a) = HQ(a). Since the set ωnis compact there exists z∈ωnsuch that Hωn(a) = sup t∈ωn Reht, ai= Rehz, ai. Hence HQ(a) = Rehz, aiand a∈Sωn. For a set A⊂Swe put as in [18] FA:= {z∈ω| ∃ a∈A: Rehz, ai=HQ(a)}. The following lemma was obtained in [18, Lemmas 3.4 and 3.5] for a bounded convex locally closed set Q⊂CN. 4.4. Lemma. If a convex locally closed set Q⊂CNis strictly convex at the relative boundary of ω, then the following assertions hold: (i) The set Sωis open in S. (ii) If A⊂Sωis compact then FAis also compact. (iii) (Sωn)n∈Nis a compact exhaustion of Sω. Proof: If int Q=∅, then Qis compact, ω=∂rQand Sω=S. Obviously the assertions (i), (ii) and (iii) hold. If int Qis not empty then we argue as follows: (i): If ω=∂Q then Sω=Sand Sωis open in S. Spaces of Holomorphic Functions 503 Let ω6=∂Q. We fix a∈Sωand z∈ωwith Rehz, ai=HQ(a). We will prove first that there is a neighbourhood Uof ain Ssuch that (4.3) sup c∈U sup w∈(∂Q)\ω Rehw, ci< HQ(a). We suppose the contrary. Then there are an∈S,wn∈(∂Q)\ωsuch that limn→∞ an=aand (4.4) Rehan, wni ≥ HQ(a)−1 nfor all n∈N. Assume that (wn)n∈Nis bounded. We choose a subsequence (wns)s∈N and w∈∂Q with lims→∞ wns=w. Since the set (∂Q)\ωis closed then w∈(∂Q)\ω. From (4.4) it follows that Reha, wi ≥ HQ(a) and consequently Reha, wi=HQ(a). Hence [z, w]⊂∂Q but w /∈Q. This contradicts the strict convexity of Qat the relative boundary of ω. Let now (wn)n∈Nbe unbounded. We select a subsequence (wns)s∈N with lims→∞ |wns|=∞. Without loss of generality we can assume that the sequence vs:= |wns−z|−1(wns−z), s∈N, converges to b∈S. Since Qis convex, for each α > 0 and for all s∈Nsuch that α < |wns−z| the point zα,s := z+αvsbelongs to Q. Hence z+αb ∈Qfor all α > 0. Note that, by (4.4), Rehans, zα,si=1−α |wns−z|Rehans, zi+α |wns−z|Rehans, wnsi ≥1−α |wns−z|Rehans, zi+α |wns−z|HQ(a)−1 ns. (4.5) Passing to the limit for fixed α > 0 as s→ ∞ in (4.5) we obtain that Reha, z +αbi ≥ HQ(a),i.e., Reha, z +αbi=HQ(a). Consequently z+αb ∈∂Q for all α > 0. Since Qis strictly convex at the relative boundary of ω, we have z+αb ∈ωfor each α > 0. Hence the intersection of the supporting hyperplane {w∈CN|Reha, wi=HQ(a)} to Qwith Qis unbounded. This contradicts the strict convexity of Qat the relative boundary of ω. Thus there is a neighbourhood Uof ain S such that the relation (4.3) holds. We will show now that there is a neighbourhood of ain Son which HQ is finite. Assume that there are an∈S,n∈N, such that limn→∞ an=a and HQ(an) = ∞for all n∈N. Then for each n∈Nthere exists wn∈Q with Rehan, wni> n. From |wn| ≥ |han, wni| ≥ | Rehan, wni| > n for all n∈N 504 J. Bonet, R. Meise, S. N. Melikhov it follows that limn→∞ |wn|=∞. We select a subsequence (wns)s∈N and b∈Swhich satisfies lims→∞ |wns−z|−1(wns−z) = b. For zα,s as above we have Reha, zα,si=1−α |wns−z|Reha, zi+α |wns−z|Reha, wnsi =1−α |wns−z|HQ(a) + α |wns−z|Reha−ans, wnsi +α |wns−z|Rehans, wnsi ≥1−α |wns−z|HQ(a)−α |wns−z||a−ans||wns| +αns |wns−z| ≥1−α |wns−z|HQ(a)−α |wns−z||a−ans||wns|. Passing to the limit for fixed α > 0 as s→ ∞ we obtain that Reha, z + αbi ≥ HQ(a). Since z+αb ∈Qwe conclude that Reha, z +αbi=HQ(a) and z+αb ∈ωfor all α > 0. This contradicts the strict convexity of Q at the relative boundary of ω. Thus there is a neighbourhood of ain S on which HQis finite. Since the function HQis convex and positively homogeneous on CNthere exists an open convex neighbourhood Vof a in CNsuch that HQis finite on V. By [12, 2.1.22] HQis continuous on V. Consequently, by (4.3), there is a neighbourhood W⊂Uof ain S with sup w∈(∂Q)\ω Rehw, ci< HQ(c)<∞for all c∈W. The last inequality implies W⊂Sω. This means that Sωis open in S. (ii): We fix a sequence (wn)n∈N⊂FAand select an∈Awith HQ(an) = Rehan, wni. There are a subsequence (ans)s∈Nand a∈A such that lims→∞ ans=a. We choose z∈ωwith Reha, zi=HQ(a). Since HQ<∞on Sωthere is a convex neighbourhood of ain CNon which the convex function HQis continuous. We suppose that (wns)s∈Nis unbounded. Without loss of generality lims→∞ |wns|=∞and the sequence (|wns−z|−1(wns−z))s∈Nconverges to a point b∈S. For each α > 0, for large s∈Nthe point zα,s := z+α|wns−z|−1(wns−z) belongs to Qand the following inequalities Spaces of Holomorphic Functions 505 hold: Reha, zα,si=1−α |wns−z|Reha, zi+α |wns−z|Rehans, wnsi +α |wns−z|Reha−ans, wnsi ≥1−α |wns−z|HQ(a) + α |wns−z|Rehans, wnsi −α |wns−z||a−ans||wns|. (4.6) Since lims→∞ Rehans, wnsi= lims→∞ HQ(ans) = HQ(a) and lims→∞ |a−ans|= 0, passing to the limit for fixed α > 0 as s→ ∞ in (4.6), we obtain that Reha, z +αbi=HQ(a) for each α > 0. As in (i) this contradicts the strict convexity of Qat the relative boundary of ω. Hence the sequence (wns)s∈Nis bounded and we can assume that (wns)s∈Nconverges to w∈∂Q. From the strict convexity of Qat the relative boundary of ωit follows that w∈ω. Thus FAis compact. It is easy to see that each Sωnis closed in S. Now the assertion (iii) follows from (ii). 4.5. Theorem. Let Q⊂CNis convex and locally closed. The following are equivalent: (i) The algebraic equality V C(CN) = CV (CN)holds. (i)’ The equality V C(CN) = CV (CN)holds algebraically and topologically. (ii) Qis strictly convex at the relative boundary of ω. Proof: The conditions (i) and (i)’ are equivalent by Theorem 4.1. (i) ⇒(ii): We assume that Qis not strictly convex at the relative boundary of ω. Then there is a∈Ssuch that the intersection of the support hyperplane Πa:= {z∈CN|Reha, zi=HQ(a)}to Qwith Q is not compact. Hence there exists an interval I⊂Q∩Πawhich is not relatively compact in Q. We fix n∈Nwith I∩Qn6=∅. Let m≥n and k∈Nbe arbitrary. There are w∈Iwith w /∈Qmand µ≥mwith w∈Qµ. We have Rehw, ai=HQ(a) = Hm(a) = Hn(a). 506 J. Bonet, R. Meise, S. N. Melikhov Fix b∈Sand c∈Rsuch that the hyperplane Rehz, bi=cseparates w and Qm, i.e. Rehw, bi> c > Hm(b). Let bs:= a+s−1b,as:= |bs|−1bs,s∈N. For all s≥2 Hµ(as)≥Rehw, asi=1 |bs|(Rehw, ai+1 sRehw, bi) >1 |bs|(Hm(a) + Hm(s−1b)) =1 |bs|Hm(bs) = Hm(as). (4.7) Since lims→∞ as=ain CN, we have lim s→∞(Hm(as)−Hn(as)) = Hm(a)−Hn(a) = 0. We fix p∈Nsuch that Hm(ap)−Hn(ap)<1 2k. Then for all l≥2k (4.8) Hm(ap) + 1 l< Hn(ap) + 1 2k+1 l≤Hn(ap) + 1 k. Since, by (4.7), Hµ(ap)> Hm(ap) there is l≥2kwith (4.9) Hm(ap) + 1 l< Hµ(ap). For this l, by (4.8) and (4.9), for every L∈N Hm(ap) + 1 l<min Hn(ap) + 1 k;Hµ(ap) + 1 L. Consequently the conditions (4.2) and, by Lemma 4.2, (wQ), are not satisfied. By [2, Proposition 2.4 and Theorem 2.7], V C(CN) is a proper subset of CV (CN). (ii) ⇒(i): Let Qbe strictly convex at the relative boundary of ω. If int Q=∅then Qis compact and (4.2) (hence (wQ)) holds. Suppose now that int Q6=∅and fix n∈N. We will show at first that there is m∈N such that Hm(a) = HQ(a) for every a∈Swith Hm(a)< Hn(a) + 1 m. We suppose the contrary, i.e. for each m∈Nthere is am∈Swith (4.10) Hm(am)< Hn(am) + 1 mand Hm(am)< HQ(am). Spaces of Holomorphic Functions 507 Since Qm=conv(Km∪ωm) (see Notation 4.3) then Hm=max{HKm ;Hωm} for all m∈N. Because HKm> HKn+ 1/m on Sfor large m∈N, from (4.10) it follows that (4.11) Hn(am) = Hωn(am) for large m∈N. There exists a subsequence (ams)s∈Nwhich converges to a∈S. The equality (4.11) and the continuity of Hnimply that Hn(a) = Hωn(a). By (4.10) am/∈Sωmfor all m∈N. By Lemma 4.4 Sωis open (in S) and (Sωm)m∈Nis a compact exhaustion of Sω. Hence, if a∈Sω, there is l∈Nsuch that ams∈Sωlfor large s. This contradicts ams/∈Sωmsfor all s∈N. Consequently a /∈Sωand Hn(a)< HQ(a). Hence there are w∈Qwith ε:= Reha, wi − Hn(a)>0 and s0such that Rehams, wi> Hn(ams) + ε/2 for all s > s0. We select s∈Nsuch that s > s0,ms>2/ε and w∈Qms. Then Hms(ams)≥Rehw, amsi> Hn(ams)+1/ms, which contradicts (4.10). Hence there is m≥nsuch that the following implication holds: a∈S, Hm(a)< Hn(a) + 1 m⇒Hm(a) = HQ(a). Consequently, condition (4.2) holds with k:= mand L:= lfor each l∈Nand µ≥m. 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Jos´e Bonet: Departamento de Matem´atica Aplicada and Instituto de Matem´atica Pura y Aplicada Universidad Polit´ecnica de Valencia E-46071, Valencia Spain E-mail address:[email protected] Reinhold Meise: Mathematisches Institut Heinrich-Heine-Universit¨at Universit¨atsstraße 1 D-40225 D¨usseldorf Germany E-mail address:[email protected] Sergej N. Melikhov: Department of Mechanics and Mathematics Rostov State University Zorge st. 5 344090 Rostov on Don Russia E-mail address:[email protected] Rebut el 23 de febrer de 2005.