On the expansions of analytic functions on convex locally closed sets in exponential series

In the late sixties Leont’ev (see [10]) proved that each analytic function f on a convex bounded domain Q ⊂ C can be expanded in an exponential series _{j ∈ N} c j exp(λ j · ). This series converges absolutely to f in the Fr´echet space A(Q) of all functions analytic on Q, and its exponents λ j are zeroes of an entire function on C which does not depend on f ∈ A(Q). A formula for the coefficients of a some expansion in such exponential series (with the help of a system orthogonal to (exp(λ j · )) j _∈ N ) was obtained only for the analytic functions on the closure of Q. Later similar results for the analytic functions on convex bounded domain Q ⊂ C ^N were obtained by Leont’ev [9], Korobeinik, Le Khai Khoi [3] (if Q is a polydomain) and Sekerin [15] (if Q is a domain of which the support function is a logarithmic potential).

In [4, 5, 11] was investigated a problem of the determination of the coefficients of the expansions of all f ∈ A(Q), where Q is a convex bounded domain in C, in following setting. Let K ⊂ C be a convex set and suppose that L is an entire function on C with zero set (λ j ) j _∈ N and with the indicator H Q + H K , where H Q and H K is the support function of Q resp. of K. By Λ 1 (Q) we denote a Fr´echet space of all number sequence (c j ) j _∈ N such that the series _{j ∈ N} c j exp(λ j · ) converges absolutely in A(Q). In [4, 5, 11] were established the necessary and sufficient conditions under which a sequence of the coefficients (c j ) j _∈ N ∈ Λ 1 (Q) in a representation f = _{j ∈ N} c j exp(λ j · ) can be selected in such way that they depend continuously and linearly on f ∈ A(Q). In other words, in [4, 5, 11] was solved the problem of the existence of continuous linear right inverse for the representation operator R : Λ 1 (Q) → A(Q), c 7→ _{j ∈ N} c j exp(λ j · ). Note that in [4, 5] a formula for continuous linear right inverse for R (if it exists) was not obtained.

In the present article we consider the following situation. Let Q ⊂ C ^N be a bounded convex set with nonempty interior. We assume that Q is locally closed, i. e. Q has a fundamental sequence of compact convex subsets Q _n , n ∈ N. By A(Q) we denote the space of all analytic functions on Q with the topologie of proj _{← n} A(Q _n ), where A(Q _n ) is endowed with natural (LF)-topologie. We put e \ (z) := exp(P N=1 A m z m ), λ, z ∈ C ^N . For an infinite set M C N N , for a sequence (A (k) ) (k) _G M C C N N with | A (k) | ^ ^ as | (k) | ^ ^ we define a locally convex space Л ₁ (Q) of all number sequence (c (k) ) (k) _G M such that the series ^(k^M C ( k ) e \ (k) converges absolutely in A(Q). The representation operator c ^ ^2(k)eM c ( k ) e ^ (k) maps continuously and linearly Л 1 (Q) into A(Q). We solve the problem of the existence of a continuous linear right inverse for R.

In this paper for N >  1 we assume that (A (k) ) ( k )^ M is a subset of zero set of an entire function L on C ^N with “planar zeroes” and with indicator H Q + H K , where H Q and H K are the support functions of Q resp. of some convex compact set K ⊂ C ^N . By [15] such function L exists if and only if the support function of clQ + K is so-called logarithmic potential (for N = 1 a function L exists for each Q and each K). In contrast to [4, 5] here we do not use the structure theory of locally convex spaces. As in [11], we reduce the problem of existence a continuous linear right inverse for the representation operator to one of an extension of input function L to an entire function L on C ^2N satisfying some upper bounds. With the help of L we construct a continuous linear left inverse for the transposed map to R . Using ∂ -technique, we obtain that the existence of such extension L is equivalent to two conditions, namely, to the existence of two families of plurisubharmonic functions, first of which is associated only with Q and second is associated with K and Q . The evaluation of first condition was realized in [21]. For the evaluation of second condition we adapt as in [21] the theory of the boundary behavior of the pluricomplex Green functions of a convex domain and of a convex compact set in C ^N which was developed in [23, 25].

For N = 1 we obtain more complete results. In the first place we prove the criterions for the existence of a continuous linear right inverse for R without additional suppositions on Q and K . Secondly, with the help of a function L as above we give a formula for a continuous linear right inverse for R .

• 1.    Preliminaries

  • 1.1.    Notations. If B ⊂ C ^N , by cl B and int B we will denote the closure and the interior of B , respectively. By int _r B , ∂ _r B we denote the relative interior and the relative boundary of B with respect to a certain larger set. For notations from convex analysis, we refer to Schneider [26].

  • 1.2.    Definitions and Remarks. A convex set Q ⊂ C ^N admitting a countable fundamental system ( Q _n ) _{n ∈} N of compact subsets of Q is called locally closed. Let Q ⊂ C ^N be a locally closed convex set. We will write ω := Q ∩ ∂ _r Q , where ∂ _r Q denotes the relative boundary of Q in its affine hull. By [21, Lemma 1.2] ω is open in ∂ _r Q . We may assume that the sets Q _n are convex and that Q n C Q n+i for all n E N. A convex set Q C C ^N will be called strictly convex at ∂ _r ω if the intersection of Q with each supporting hyperplane to cl Q ⊂ C ^N is compact. If int Q 6 = ∅, Q is strictly convex at ∂ _r ω if and only if each line segment of ω is relatively compact in ω .

By [13, Lemma 3] Q ⊂ C ^N is strictly convex at ∂ _r ω if and only if Q has a fundamental system of convex neigborhoods.

• 1.3.    Convention. For the sequel, we fix a bounded, convex and locally closed set Q ⊂ C ^N with 0 in its nonempty interior and with a fundamental system of compact convex

subsets Q n C Q n+i , n E N. By (^ n ) _{n G} N we shall denote some fundamental system of compact subsets of ω = Q ∩ ∂ r Q.

K will always denote a compact convex set in C ^N .

• 1.4.    Notations. For each convex set D ⊂ C ^N we denote by H D the support function of D, i. e. Hd (z) := sup _{w G} D Re h z, w i , z E C N . Here h z, w i := P N=1 z j W j . We put H n := H Q n , n ∈ N.

Let e λ (z) := exp h λ, z i , λ, z ∈ C ^N . For a locally convex space E by E _b ⁰ we denote the strong dual space of E.

• 1.5.    Function spaces. We set | z | := h z, z) ¹/² , z E C N ; U (t, R) := { z E C N : | t - z | < R } , t E C N , R> 0; U := U (0,1). For all n,m E N let E nm := A ^ (Q n + ml U ) denote the Banach space of all bounded holomorphic functions on Q n + 1 U , equipped with the sup-norm. We consider the spaces A ( Q _n ) = _{m ∈ N} E _n,m of all functions holomorphic in some neighborhood of Q _n , n ∈ N, and endow them with there natural inductive limit topology. By A ( Q ) we denote the vector space of all functions which are holomorphic on some neighborhood of Q . We have A ( Q ) = _{n ∈ N} A ( Q _n ), and we endow this vector space with the topology of A ( Q ) := proj _{← n} A ( Q _n ). This topology does not depend on the choice of the fundamental system of compact sets ( Q _n ) _{n ∈} N . If Q is open, A ( Q ) is a Fr´echet space of all holomorphic functions on Q .

For all n, m ∈ N let

An,m := If E A(CN): kf kn,m := sup |f (z)| exp ( - Hn(z) - |z|/m) < ro) z∈CN and

A Q := ind proj A n,m . n → _{← m}

• 1.6.    Duality. The ( LF )-space A ( Q ) := ind _{n →} A ( Q _n ) ⁰ _b and A Q are isomorphic by the Laplace transformation

F : A ( Q ) ⁰ → A Q , F( ϕ )( z ):= ϕ ( e z ) , z ∈ C ^N .

In addition ( LF )-topology of A ( Q ) ⁰ equals the strong topology.

The assertion has been proved in [21, Lemma 1.10] (see Remark after 1.10, too)

If we identify the dual space of A ( Q ) with A Q by means of the bilinear form h· , ·i , then h e _λ , f i = f ( λ ) for all λ ∈ C ^N and all f ∈ A Q .

• 1.7.    Sequence spaces. Representation operator. Let M ⊂ N ^N be an infinite set and (A (k) ) (k) _G M C C N be a sequence with | A (k) | ^ ro as | (k) | ^ ro . For all n,m E N we introduce the Banach spaces

^Л n,m ^{(Q) :}= ^^c = ^(c (k) ) ( k )G M ^{C C :} ^2 | c (k) | exp (H n ^(A (k) ^{) + | A} (k) ^{| /m}) <  ^ГО p

(k) G M

Kn,m(Q) := ^c = (c(k))(k)GM C C : sup |c(k)| exp ( - Hn(A(k)) -|A(k)|/m) < ro^ (k)GM and put

Л 1 (Q n ) := indЛ n_Jm (Q), Л 1 (Q) := projЛ 1 (Q n ), K ^ (Q) :=indproj K n,m (Q). m →                _{← n}                n → _{← m}

We note that the series ^( k ^^ M c ( k ) e\_w converges absolutely in A(Q) if and only if c E Л 1 (Q) (see [2,Ch.I, §§ 1,9]).

The operator R(c) := ^Z( k) _G M c ( k ) e \ (k) maps continuously and linearly Л 1 (Q) in A(Q). We call R the representation operator. By Korobeinik [2], if R : Л 1 (Q) ^ A(Q) is surjectiv, (e ^ ( k ) ) (k) _G M is called an absolutely representing system in A(Q).

Let e ( k ) := (§ ( k ) , ( m ) ) (m) G M , (k) E M , where S^m ) is the Kronecker delta.

• 1.8.    Duality. ( i ) The transformation ^ ^ (^(e ( k ) )) (k) _G M is an isomorphism of (LF )- space Л ₁ (Q) ' := ind n , Л ₁ (Q _n ) b onto K ^ (QY The duality between Л ₁ (Q) and K ^ (Q~) is defined by the bilinear form h c,d) := ^2(k) _G M c ( k ) d ( k ) .

• (ii)    A transposed map R' : Aq ^ K ^ (Q) to R : Л ₁ (Q) ^ A(Q) is the restriction operator ^f ^ ^(f(\ k) )) (k) G M •

• ( iii )    R has a continuous linear right inverse if and only if R ⁰ has a continuous linear left inverse.

C The assertions (i) and (ii) were in [13, Lemma 6] proved.

(iii): This can be proved in the same way as ( i ) ⇒ ( ii ) in [21, Lemma 1.12]. (We note that we can not assume in advance the surjectivity of R .) B

• 1.9.    Notations. Let S := { z ∈ C ^N : | z | = 1 } . For a convex set D ⊂ C ^N , γ ⊂ D and A ⊂ S we define

Sγ(D) := {a ∈ S : Rehw, ai = HD(a) for some w ∈ γ} and

F A ( D ) := { w ∈ D : Re h w, a i = H D ( a ) for some a ∈ A } .

We will write S _Y := S _Y (Q), A := S f _a ( k ) (K), S o := S \ S ^ •

Definition 1.10. (a) Given an open subset B ⊂ S and a compact convex set K ⊂ C ^N . K is called smooth in the directions of the boundary of B if for each compact set κ ⊂ B the compact set К := Sf _k (k ) (K ) is still contained in B .

Note that the condition is fulfilled if dK is of class C 1 .

• (b) A convex compact set K ⊂ C ^N is called not degenerate in the directions of B ⊂ S , if K is not contained all in the supporting hyperplane { z ∈ C ^N Re h z, a i = H K ( a ) } of K for each a ∈ B .

Note that the condition is fulfilled if int K 6 = ∅.

Remark 1.11. (a) Under the hypotheses of the Definition 1.10 (a) the following holds: Let S 1 C S be an open neighborhood of S \ B (with respect to S). For к := S \ S 1 , the set к is a compact subset of B. Hence if S ₂ C S \ K is compact, we have S ₂ П к = 0 and thus S ₂ C S 1 . (Otherwise it would follow that К П S 2 = 0.)

• (b) Let K have 0 as an interior point. K is smooth in the directions of the boundary of B if and only if the convex set intK ⁰ U w ' is strictly convex at d r w ' , where w ' := dK ⁰ П Г(В), K ⁰ := { z E C N | Hk (z) 6 1 } and Г(В) := { tb 1 1 > 0 } .

• 2. Conditions of existence of a continuous linear right inverse for the representation operator

• 2.1.    Notations, Definitions and Remarks. (a) Let f be an entire functions of exponential type on C ^N . By h ^∗ _f we denote the (radial) indicator of f , i. e.

h _f ^∗ ( z ) := lim sup(lim sup log | f ( rz ⁰ ) | /r ) for all z ∈ C ^N .

z'^z   r ^ + ro

• (b)    An entire function f of exponential type on C is called function of completelly regular growth (by Levin–Pflu¨ger), if there is a set of circles U(µ j , r j ), j ∈ N, with | µ j | → ∞ as j ^ го , such that lim R .^ R P^ j | r j = 0 and outside of U^ n U (^ j ,r j ) the following asymptotic equality holds:

log I f (z) l = h f (z) + o( | z | ) as | z | ^ го .

By Krasichkov–Ternovskii [6], in Definition (b) we can choose the exclusive circles U (µ j , r j ) so that they are mutually disjoint.

• (c)    By Gruman [18] an entire function f of exponential type on C ^N is called function of completelly regular growth, if for almost all a ∈ S the function f (az) of one complex variable has completelly regular growth on C.

• (d)    There are other definitions of the functions of completelly regular growth of Azarin [1] and of Lelon, Gruman [8, Ch. IV, 4.1]. By Papush [14], if f is an entire function on C ^N with “planar” zeroes, i. e. the zero set { z ∈ C ^N : f(z) = 0 } of f is the union of the hyperplanes { z ∈ C ^N : h z, a k i = c k } , a _k ∈ S, c _k ∈ C, k ∈ N, all these definitions (for f ) are equivalent. From this and from [7, 22] it follows that an entire function f on C ^N with “planar” zeroes has completely regular growth on C ^N if and only if f is slowly decreasing on C ^N .

• 2.2.    A special entire function. A structure of the exponents A (k) . (a) Below we shall exploit an entire function L on C ^N of order 1, which satisfies the following conditions: (i) The zero set V (L) of L is a sequence of pairwise distinct hyperplanes P k := { z ∈ C ^N : h a k z i) = C k } , k 6 N, where | a k | = 1 and C k = 0. If for k 1 < k ₂ < ... <  k N the intersection P k 1 П P k 2 П ... П P k _N is not empty, then it consits of a single point A (k) , where (k) denotes multiindex (k i ,k 2 ,..., k N ). Further M is the set of the such multiindexes (k). Moreover, L (k) (A (k) ) = 0, where L_w(z) := L(z)/l_w(z) and l_w(z) := lj =i« a k j , z ) — C k ₃ ), (k) 6 M . (ii) L is a function of completely regular growth with indicator H Q + H K .

We recall the some definitions and results from Sekerin [15].

• (iii)    | L (k) (A (k) ) | =exp(H Q (A (k) ) + Hk (A_w )+ o( | A _fe) | >) as | (k)M го .

We write l k (z) := h a k , z i - c k , z ∈ C ^N , k ∈ N.

• (b)    (i) By [15, Theorem 1], for each f 6 A _int q+k the Lagrange interpolation formula holds:

  ^{f (A)} = X '    ^f ^ ^{A 6} C N

  
  
  
  

where the series converges uniformly on compact sets of C ^N . From (1) it follows that (A (k) ) (k) _G M is the uniqueness set for A int q+k , i. e. from f 6 A(C ^N ), h f (z) <  H q (z) + Hk (z) for all z ∈ C ^N \{ 0 } it follows that f ≡ 0.

• (ii)    There is a function a(z) = o( | z | ) as | z | ^ го such that | L (k) (z) | 6 exp(H Q (z) + H K (z) + α(z)) for all z ∈ C ^N and all (k) ∈ M.

• (c)    A plurisubharmonic function u on C ^N will be called a logarithmic potential if there exists a Borel measure ^ > 0 on [0, го ) x S ^N such that for every R 6 (0, го ) there is a pluriharmonic function u R on U(0, R) with

  u(z) =

  
  

  log | t - h z, w i| dµ(t) + u R (z)

  
  

  for all z ∈ U (0, R).

  
  

[0,R] x S ^N

By [15] for a bounded convex domain D with 0 ∈ D the support function HD is a logarithmic potential for example if D is a polydomain, a ball, an ellipsoid, a polyhedra with symmetric faces, and in the case of C2 , if D = D1 + iD2 , where D1 and D2 are any centrally symmetric convex domains in R2; if D is symmetric with respect to 0 and cl D is a Steiner compact set (see Matheron [19, § 4.5]).

For each bounded convex domain D ⊂ C with 0 ∈ int D the function H D is a logarithmic potential.

• (d)    By [15, Theorem 5], there exists a function L satisfying the conditions (i)–(iii) in 2.2 (a) if and only if H Q + H K is a logarithmic potential. H Q + H K is a logarithmic potential if H Q and H K are the logarithmic potentials.

• (e)    Let H Q+K = H Q +H K be a logarithmic potential. By [15] the representation operator R : Λ 1 (intQ + K) → A(int Q + K) is surjective. By [13, Theorem 14] R : Λ 1 (Q) → A(Q) is surjective, if Q is strictly convex at ∂ _r ω, K is smooth in the directions of ∂ _r S _ω and not degenerate in the directions of S ω .

Theorem 2.3. Let Q be strictly convex at ∂ _r ω and L be an entire function on C ^N satisfying the conditions 2.2 (a). Then (II) ⇔ (III) ⇒ (I):

• (I)    The representation operator R : Λ 1 (A) → A(Q) has a continuous linear right inverse.

• (II)    There is a positively homogeneous of order 1 plurisubharmonic function P on C ^2N such that P(z,z) >  Hq(z) + H K (z) and ( V n) ( 3 n') ( V s) ( 3 s ' ) with

P(z, ц) 6 H n (z) + | z | /s + Hk (ц) + Hq(^) — H n (^) - l^ /s' ( V z, ц G C N ).

• (III)    There are the plurisubharmonic functions u _t ,v _t , t G S , on C N such that u t (t) >  0, v t (t) > 0 and ( V n) ( 3 n ' ) ( V s) ( 3 s ' ) with

• (a)    u _t (z) 6 H n o (z) — H _n (t) + | z | /s — 1/s ' and

• (b)    v t (ц) 6 Hk (ц) + Hq(^) — H n (^) — Hk (t) - H Q (t) + Н п (t) — Ws + 1/s for all z,^ G C N and all t ∈ S .

• <1 (II) ^ (III). We may choose

u t ( z ) := P ( z, t ) - H Q ( t ) - H K ( t ) ,   v t ( µ ) := P ( t, µ ) - H Q ( t ) - H K ( t )

for all z, µ ∈ C ^N and t ∈ S .

• (III) ⇒ (II). We put

Po(z,^):= (sup (ut(z) + vt(ц) + Hq(z) + Hk(ц))^ , z,^ G CN, where f∗ denotes the regularization of a function f . P0 is the plurisubharmonic function on C2N with

P(z,z) >  Hq(z) + Hk (z) ( V z G S).

By (III) we have: (∀ m) (∃ n0) (∀ s) (∃ r) with ut(z) 6 Hn0(z) — Hm(t) + |z|/s — 1/r for all z G CN and all t G S and (∀n) (∃ m) (∀r) (∃ s0) with vt(ц) + Hq(€) + Hk(t) 6 Hk(ц) + Hq(p-) — Hn(^) + Hm(t) — |ц|/s + 1/r for all µ ∈ CN and all t ∈ S. By adding the last inequalities, we obtain that (∀n) (∃ n0) (∀ s) (∃ s0) with ut(z) + ^(ц) + HQ(t) + HK(t) 6 Hn0(z) + HK(ц) + HQ(^) — Hn(^) + |z|/s — |ц|/s'

for all z,p E C N and t E S . From this it follows that P g satisfies the upper bounds in (II). As P we may choose P (z,p) := (limsup _t ^ + ^ P (tz,tp)/t) * , z,p E C N .

• (III)    ⇒ (I). By (the proof of) [16, Theorem 4.4.3] (see [8, Theorem 7.1], too) there is a L E A(C ²N ) with L(z, z) = L(z) and ( V n) ( 3 n ⁰ ) ( V s) ( 3 s ⁰ ) ( 3 C ): ( V z,p E C N )

  We define

  
  

  l L(z,p) l 6 C exp (H ^ (z) + | z | /s + Hk (p) + H q (p) - H n (p) - | p | /s ⁰ ).

  
  

  _˜

  
  

  _˜

  / X/ X              ^L (k) ^(z)L(z, ^A (k) )

  ^{К 1 <С)(2):}= £.   L(k) ^    '

  
  

  c ∈ K ∞ (Q), z ∈ C ^N .

  
  
  
  

From (2) it follows that the series in (3) converges absolutely in A ₂q+k . (By [21, Remark 1.5] 2Q + K is locally closed and (2Q _n + K) _{n ∈} N is a fundamental system of compact subsets of 2Q + K .) Hence k ₁ maps K ^ (Q) in A ₂ q + k continuously (and linearly). Since, by (2), for all f E Aq and z E C N the function L(z, • f belongs to A _int q + k , by 2.2 (b) for all z E C N

˜

K i (R 0 (f ))(z) = £ ^{L ( k ) ( z ) L}(^z,X ⁽ k ^{) )} f ( _X ( k ) ) = L(z,z)f (z) = L(z)f (z). (k) _G M        ^(кл ^(k))

From here it follows that K i ◦ R ⁰ is the operator of multiplication by L. By [21, Proposition 2.7] there is a continuous linear left inverse K 2 : A 2 Q + K ^ Aq for K i ◦ R ⁰ . The operator к := K i ok₂ is a continuous linear left inverse for R ⁰ .

Now we shall evaluate the abstract condition (III) (b) of Theorem 2.3. The condition (III) (a) was evalueted in [21, Proposition 3.6]. B

We recall some definitions from [23] and [25].

Definition 2.4. If D C C N is bounded, convex and c >  0, let v H D _c be the largest plurisubharmonic function on C N bounded by H D and with v H D _c (z) 6 c log | z | + O(1) as | z | ^ 0. A function C H D : S ^ [0, 00] is defined by

{z E C N : v H D ,_c (z) = H D (z)} = {Aa : a E S, 1/C H D (a) 6 A <  0 }.

If 0 ∈ int D and if C > 0, let v _H ^∞ _{D ,C} be the largest plurisubharmonic function on C ^N bounded by H D and with v _H ^∞ _{D ,C} (z) 6 C log | z | +O(1) as | z | → ∞ . A function C _H ^∞ ₀ : S → [0, ∞ ] is defined by

{z E C N : v ^ D ,c (z) = H D (z)} = {Aa : a E S, 0 6 A 6 1/C H D (a)} .

Instead C H D and CHI D we shall write briefly C D resp. C ^ .

Proposition 2.5. Let Q be strictly convex at the ∂ _r ω and suppose that 0 ∈ int K. For N > 1 assume that K is smooth in the directions of ∂ _r S _ω . The following are equivalent:

• (i)    There are plurisubharmonic functions v _t (t ∈ S) on C ^N with v _t (t) > 0 such that: ( ∀ n) ( ∃ n ⁰ ) ( ∀ s) ( ∃ s ⁰ ) with

v _t 6 H K + H Q - H _n - | · | /s ⁰ - H K (t) - H Q (t) + H _n 0 (t) + 1/s ( ∀ t ∈ S).

• (ii)    1/C K is bounded on some neighborhood of S g and C I is bounded on each compact subset of S _ω .

C (i) ⇒ (ii). Choose n0 according to (i) for n = 1. On So we have Hn0 < HQ. Thus there are a neigborhood S of So and some ε > 0 with Hn0 + ε 6 HQ on S . We put v := sup(vt + HK (t)) ∗. tes

This function is plurisubharmonic on CN with v > HK on S and satisfies: (∀ n) (∃ n0) (∀ s) (∃ s0 ) such that v 6 HK + | · |/n + max{-HQ(t) + Hn0} + 1/s.

t ∈ S

Since H _n 0 6 H Q , this gives v 6 H K on C ^N . The bounds for n = 1 give v(0) 6 - ε.

From [25, 2.14] it follows that 1/C K is bounded on S.

Let κ ⊂ Sω . We define v := st∈uκp(vt + HK (t)) ∗ .

This function is plurisubharmonic on C ^N with v > H K on κ and satisfies: ( ∀ n) ( ∃ n ⁰ ) ( ∀ s) ( ∃ s ⁰ ) such that v 6 H K + H Q - H _n - | · | /s ⁰ + 1/s 6 H K + H Q - H _n + 1/s. This shows that v 6 H K .

Now choose n with κ ⊂ S _{ω n} , i. e. with H Q = H _n on κˆ. Choose n ⁰ > n according to (i). Choose s ⁰ for s = 1. Then there is a neighborhood κ˜ of κ in S such that

HQ - Hn - | · |/s0 6 -| · |/(2s0) on Γ(κ˜) and thus v 6 HK - | · |/(2s0) + 1 on Γ(κ˜).

In order to reach our claim that C _K ^∞ is bounded on κ, we need an estimate like the previous one on all C ^N (not only on the particular cone). For this purpose we are going to modify v . First note that, if N = 1, it follows from what we have already proved that ∂K has to be of class C ¹ (see [20, 2.10, 2.14]). For N >  1 we use our special hypothesis. For this reason we may assume that we have constructed v for the set κ ˆ instead of κ .

Define

L ( z ) := sup Re h w, z i ,   z ∈ C ^N .

w ∈ F κ

The positively homogeneous function L satisfies L 6 H K on C ^N , and L = H K on κ . If L ( a ) = H K ( a ), there is w ∈ F _κ with Re h w, a i = H K ( a ), hence a ∈ S F _κ . Thus L < H on S outside the compact set κ ˆ. We replace v by v ˜ := v/ 2 + L/ 2 and obtain v ˜ 6 H K on C ^N , v ˜ = H K on κ and v ˜ < H K outside a neighborhood of the origin. By [23, 2.1] this shows that C _K ^∞ is bounded on κ .

• (ii)    ^ (i). By the hypothesis, 1/C K is bounded on some neighborhood S of S q . Hence there is c > 0 such that the plurisubharmonic function v H _{K c} equals H K on S . Let n ∈ N. Since H _n <  ^Iq on S q , there is a co^mpact neighborhood Sn of S q ^^ith ^in <  ^^Q on S n . ^We may assume S n C S n -i C ... C S i C S . Since C K is bounded on S \ S n , there is C n > 0 with v n := v U k ,C n = Hk on S \ S n₊2 .

Again for N = 1 it follows from (ii) that dK is of class C ⁱ . For N >  1 we apply the extra hypothesis to obtain (as in the first part of the proof) a positively homogeneous function L _n bounded by H on C N , which equals H on к = S n+i , and such that L _n <  H outside the compact set S n+i C S _n (see Remark 1.11 (a)). Then the plurisubharmonic function v n := v H _K ,c /2 + L n /2 satisfy v n 6 Hk on C N , v n = Hk on S n+i , V n 6 (H k + L n )/2 <  Hk on S \ S _n .

Fix n E N. Since vn 6 (Hk + Ln)/2 < Hk + Hq — Hn on S, and since vn(0) < 0, there is n˜ with vn 6 Hk + Hq — Hn - D/2 - 1/n on CN, were D:=HK+HQ-Hn-(HK+Ln)/2=(HK-Ln)/2+HQ-Hn.

Choose n0 with Hq — Hn0 6 1/n on Sn+1. Then for each s there is s0 with D/2 > | • |/s0 on CN such that vn 6 HK + HQ — Hn — I • |/s0 — HQ(t) + Hn0(t) + 1/s (Vt E Sn+1)-

For the functions v _n ^∞ we get: Choose n ⁰ (in addition) so large that H Q = H _n 0 on S \ S _n +2 . For each s we choose s ⁰ (in addition) so large that vn³ 6 Hk - ЫА0 + 1/s (see Definition 2.4). This gives

< ^{6 H} K ^{+ H} Q ^{— H} n ^{— 1} • ^{| /s} 0 ^{— H} Q ^{(t) + H} n 0 ^{(t) + 1/s ( V} t ^{E S} \ ^S n+2 )-

Note that v0 > ... > vn > vn+1 and that v0 6 - - - 6 v0 6 vnO+r That is why for each l ∈ N the following holds: (∀n) (∃ n0) (∀ s) (∃ s0) with v0 6 HK + HQ — Hn — 1 • |/s0 — HQ(t) + Hn0(t) + 1/s (vt E Sn+1), and (∀n) (∃ n0) (∀ s) (∃ s0) with v0 6 HK + HQ — Hn — 1 • |/s0 — HQ(t) + Hn0 (t) + 1/s (V t E S\Sn+2)

By the construction, lim l _→∞ v _l ⁰ =: v _∞ ⁰ exists and defines a plurisubharmonic function with v _∞ ⁰ = H K on S 0 .

For t ∈ S \ S 2 define v˜ _t := v ₁ ^∞ . For t ∈ S l+1 \ S l+2 we put v˜ _t := v _l ⁰ /2 + v _l ^∞ /2. For t ∈ S 0 we define v˜ _t := v _∞ ⁰ . Obviously v˜ _t (t) = H K (t) for all t ∈ S.

Let t E Si+1\Si+2. For n 6 l and n0, s and s0 as above we get vt 6 (Hk + Hq — Hn — | • |/s0 — HQ(t) + Hn0(t) + 1/s)/2 + Hk/2-

By the strict convexity of Q at ∂rω (see [21], the proof of Proposition 3.6), there is n00 such that (HQ + Hno )/2 6 Hnoo and thus (HQ — Hno )/2 > HQ — Hn. This gives vt 6 HK + HQ — Hn — 1 • |/(2s0) — HQ(t) + Hn00(t) + 1/(2s)-

For n > l and n0, s and s0 as above we get vt 6 Hk/2 + (Hk + Hq — Hn — | • |/s0 — HQ(t) + Hn0(t) + 1/s)/2-

As above we get the desired estimate.

For t ∈ S 0 =    _{l ∈ N} S _l , we see as in the first part of the previous arguing that v˜ _t = v _∞ ⁰

satisfies these estimates for all n (6 l = ro ).

For t ∈ S \ S 2 , as in the second part of the arguing just done, we see that these estimates hold for all n (> l = 1).

Finally we put v t := v t — Hk (t), t E S and are done. B

Remark 2.6. Let Q be strictly convex at the ∂ _r ω. By [21, Proposition 3.6] the following are equivalent:

• (i)    There are plurisubharmonic functions u _t (t ∈ S) on C ^N with u _t (t) > 0 such that: ( ∀ n) ( ∃ n ⁰ ) ( ∀ s) ( ∃ s ⁰ ) with

u _t (z) 6 H _n 0 (z) - H _n (t) + | z | /s - 1/s ⁰ ( ∀ z ∈ C ^N , t ∈ S).

• (ii)    C q is bounded on some neighborhood of S ₀ and 1/Cq is bounded on each compact subset of S _ω .

• 3. The case of one complex variable

Theorem 2.7. Let Q be strictly convex at the ∂ _r ω and suppose that 0 ∈ int K and L is a function as in 2.2 (a). For N > 1 assume that K is smooth in the directions of ∂S _ω . If C _Q ^∞ and 1/C K are bounded on some neighborhood of S ₀ , 1/C q and Ck are bounded on each compact subset of S ^ then the representation operator R : Л 1 (Q) ^ A(Q) has a continuous linear right inverse.

C The assertion hold by Theorem 2.3, Proposition 2.5 and Remark 2.6. B

The equivalent conditions of Theorem 2.7 are fulfilled if ∂Q and ∂K are of H¨older class C ^1,л for some A > 0. They are not fulfilled if Q or K is a polyedra, and for N = 1 if dQ or ∂K has a corner [24].

In this section we consider the case N = 1 for which the results of the previous sections can be improved.

Convention 3.1. Further L is an entire function on C satisfying following conditions:

• (i)    The zero set of L is a sequence of pairwise distinct simple zeros λ k , k ∈ N, such that | A k | 6 | A k₊i | for each k E N.

• (ii)    L is a function of completely regular growth with indicator H Q + H K .

• (iii)    The asymptotic equality holds:

| L ⁰ ( λ k ) | = exp( H Q ( λ k ) + H K ( λ k ) + o ¯( | λ k | )) as k → ∞ .

Such function L exists (see for example [10]).

Leont’ev (see [10]) introduced an interpolating function, which is defined with the help of an entire function of one complex variable. Leont’ev’s interpolating function is a functional from A (cl Q + K ) ⁰ \ A ( Q ) ⁰ for every K (if Q 6 = cl Q ). With the help of an entire function of two complex variables we give the analogous definition of an interpolating functional from A ( Q ) ⁰ .

Definition 3.2. Let L be an entire function on C ² such that L( ^ , ц) E Aq for each µ ∈ C. Q -interpolating functional we shall call a functional

t

Q l (z,^,f) := F ^{- 1}

Ес,ц))^ I^{f (t} — £^)exp^(z£) d^y 0

z,µ ∈ C , f ∈ A ( Q ) ,

where the integral is taken along the interval [0 , t ].

We show certain properties of Ql .

Lemma 3.3. (a) Ql ( • ,ц,f ) E Aq for all ц E C and f E A(Q).

• (b)    For all z, ц E C the equality Ql (z, z, e ^ ) = 1(ц, z) holds where a function l E A(C ² ) is such that L ( µ, z ) - L ( z, z ) = l ( µ, z )( µ - z ) .

• (c)    Q ^ (ц, z, • ) E A(Q) for all z, ц E C.

C (a): We fix µ ∈ C, f ∈ A(Q) and a domain G with Q ⊂ G and f ∈ A(G). We choose a contour C in G which contains in its interior the conjugate diagram of L( · , µ). If γ( · , µ) is Borel conjugate of L( · , µ), we have:

t

^Q l ^(z,^^,f) = 2П J y (t,^ I J

f(t - ξ)exp(zξ)dξ dt, z ∈ C.

C        \0                   /

Since the function (t, ^) ^ y (t, ^) (J ot f (t - ^)exp(z^) d^ is continuous by t E C and entire by z, the function Q l (z,^,f) is entire (with respect to z). From direct upper bounds for | Q l (z,^,f) | it follows that Q l (^,f) E A q .

(b): Obvious.

(c): Since the map f ^ Jt f (t — £)exp(z£) d^ , t E Q, is continuous and linear in A(Q) and F ^{- 1} (Q( - ,^)) is a continuous and linear on A(Q), the functional Qq(z,^, • ) is continuous and linear on A(Q), too. B

Lemma 3.4. We assume that a function L, as in 3.2, satisfies in addition the following conditions: L˜ (z, z) = L(z) for each z ∈ C and (∀n) (∃ n0) (∀s) (∃ s0) (∃ C) with lL(z,^)l 6 Cexp (Hn(z) + Hk(д) + HqM — HnM + |z|/s — |^|/s°) (Vz, д E C).

Then n(f) := (Ql (X k , X k , f )/L 0 (X k )) k ^ N , f E A(Q), is continuous linear operator from A(Q) into Л ₁ (Q).

C We define L _k (z) := L(z, λ _k )/(L ⁰ (λ _k )(z - λ _k )), k ∈ N. By using upper bounds for | L | , 3.1 (iii) and 3.3 (b), we obtain, that L k is entire function on C and ( ∀ n) ( ∃ n ⁰ ) ( ∀ s) ( ∃ s ⁰ ) ( 3 C 1 , C ₂ ) such that for all z E U(X k , (1 + | X k | ) ^{- 2} )

^lL k ^{(z) |} 6 ^C 1 ^exp (^H n ^{0 (z)} + H K ^{( X} k ) + ^H Q ^{( X} k ) ^— H n ^{( X} k ) + ^{| z | /s — | X} k | /(s' ^— 1) +^2log(1 + ^{| X} k | ) ^{— log | L} 0 ^(X k ) | ) 6 ^C 2 ^exp (H n ^{0 (z) — H} n ^{( X} k ) + ^{| z | /s — | X} k ^{| /s} 0 )   ^{( V k E N )}-

Applying the maximum principle we get that ( V n) ( 3 n ⁰ ) ( V s) ( 3 s ⁰ ) ( 3 C 3 ) with

\ L k (z) | 6 C 3 exp (H n o (z) — H n (X k ) + | z | /s — | X k | /s ⁰ ) ( V z E C, k E N).

From this it follows that the series P k ^ N C k L k converges absolutely in A q for each c = (c k ) k e N E K ^ (Q) and к : c ^ P k ^ N C k L k is continuous linear operator from K ^ (Q) into A q . We shall find its adjoint operator к ⁰ : A(Q) ^ Л 1 (Q):

h c^,K (e ^j i = h ^K(c),f i = <^X c k L k ^,e^

k ∈ N

= 52 C k L k (^) = 52 C k Q L (X k , X k , e _M )/L ⁰ (X k ) ( V ^ E C, c E Л 1 (Q)). k ∈ N           k ∈ N

Hence к ⁰ (е ^ ) = (Ql (X k , X k , e ^ )/L 0 (X k )) k ^ N , ^ E C. Let C ^N be a space of all number sequence with its natural topologie. The maps κ ⁰ : A(Q) → C ^N and Π : A(Q) → C ^N are continuous and linear. Since the set { e µ : µ ∈ C } is total in A(Q), we have Π = κ ⁰ on A(Q) and Π is continuous and linear from A(Q) into Л 1 (Q). B

Theorem 3.5. (I) Let 0 ∈ int _r K. The following assertions are equivalent:

• (i)    The representation operator R : Л 1 (Q) ^ A(Q) has a continuous linear right inverse.

• (ii)    There is an entire function L on C ² such that L(z,z) = L(z) and ( V n) ( 3 n ' ) ( V s) ( ∃ s ⁰ ) ( ∃ C ) with

| L(z,^) | 6 Cexp (H_n- (z) + Hk M + Hq(^) - H n (^) + | z | /s - Ws )  ( V z, д G C).

• (iii)    Q is strictly convex at d r w, the interior of K is not empty, C q and 1/C K are bounded on some neighborhood of Sq , 1/C Q and C ^ are bounded on each compact subset of S ^ .

• (II)    (iv) If L is a function as in ( ii ) , the operator

n(f) ^ (Ql(Xk, Xk, f )/L'(Xk))k^N, f G A(Q), is a continuous linear right inverse for R.

(v) If П : A(Q) ^ Л 1 (Q) is a continuous linear right inverse for R, then there is a unique function IL as in (ii) such that H(f) = (Ql (X k ,X k ,f )/L ' (X k )) k _G N , f G A(Q).

C (iv) (and (ii) ⇒ (i)): Let L be a function as in (ii). Then

κ : c →

c k

(k) e N

L ^˜ ( · , λ _k )

L ⁰ ( λ _k )( · - λ _k )

maps continuously (and linearly) K _∞ ( Q ) into A Q . Since for each f ∈ A Q the function fL ( z, · ) belongs to A intQ+K , taking into account the Lagrange interpolation formula (1), we obtain:

˜ ад x f ■•■ L4 LLLLk V,=x f (Xk)L(z.Xk) ,/ k∈N      L (λk)(z - λk) k∈N              L(λk)(z-λk)

= L ( z, z ) f ( z ) = L ( z ) f ( z ) ( ∀ z ∈ C , f ∈ A Q ) .

This implies that κ = Π ⁰ is a left inverse for R ⁰ . By the proof of Lemma 3.4 κ is the adjoint to Π for each function L as in (ii). Hence Π is a right inverse for R .

( i ) ⇒ ( ii ): Let Π be a continuous linear right inverse for R . Then κ := Π ⁰ : K _∞ ( A ) → A Q is a left inverse for R 0 . We put f k := K(e ( k ) ), where e ( k ) := (^ k,n ) _{n G} N , k G N. By Grothendieck’s factorization theorem, for each n there is n ⁰ such that κ maps continuously proj _←m K _n,m ( Q ) in proj _←m A _n 0 _,m . Hence the following holds: ( ∀ n ) ( ∃ n ⁰ ) ( ∀ s ) ( ∃ r ) ( ∃ C ) with

| f k (z) | 6 C exp (H „ o (z) - H _n (X ( k ) ) + | z | /s - | X (k) | /r) ( V z G C, k G N).

For f ∈ A Q let

T z (f)(^) := X ^(4“(z — X k )f k (z)f (X k ), ^ G C.

k ∈ N ^{µ - λ k}

By 2.2 (b) (ii) and (4) the series converges absolutely in A Q and converges uniformly (by µ ) on compact sets of C. Fix z ∈ C. Then T _z ( µf )( µ )) = µT _z ( f )( µ ) for all f ∈ A Q and µ ∈ C. By [12, Lemma 1.7] there is a function a _z ∈ A (C) such that T _z ( f )( µ ) = a _z ( µ ) f ( µ ) for all µ ∈ C, f ∈ A Q . The function L ( z, µ ) := a _z ( µ ), z, µ ∈ C, satisfies the conditions in (ii) (see the proof of (i) ⇒ (ii) in [12, Theorem 1.8] too).

• (iii)    ⇒ (i) holds by Theorem 2.7.

• (i)    ^ (iii): Since the operator R has a continuous linear righ inverse, R : Л 1 (Q) ^ A(Q) is surjectiv. By [13, Theorem 8] the set Q is strictly convex at ∂ _r ω.

Since (i) is equivalent to (ii) there is a function L which satisfies the conditions in (ii). Let P be the (radial) indicator of L, i. e.

Л- ML^z,^) \

P (z, µ) := lim sup                  , z, µ ∈ C.

• t ^+ro        t

Then P is a plurisubharmonic function on C2 satisfying the conditions in (II) of Theorem 2.3. Hence, by Theorem 2.3, there are subharmonic functions vt (t ∈ S) as in (III) (b). We put gt(µ) := |t|vt/|t| (µ/|t|), µ, t ∈ C, t 6= 0. Then gt are subharmonic functions on C such that gt(t) > 0 and (V n) (3 n0) (V s) (3 s0) with gtM 6 HKM + HQM — HnM — HK(t) — HQ(t) + Hn0(t) — M/s0 + |t|/s for all µ, t ∈ C, t 6= 0. If Sω = ∅, the set Q is open. Hence the following holds: (∀ n) (∃ n0) with gt(д) 6 Hk(д) - Hk(t) + |^|/s0 -|t|/s (V^,t G C, t = 0).

Then, by [12, Proposition 1.17], an angle with the corner at 0 doesn’t exist in which the support function HK of K is harmonic. Hence int K = ∅. If Sω = ∅, there is an open (with respect to S) subset A of S such that Hn = HQ on A for large n. Let Γ(A) := {ra : r > 0}. Then for each s there is s0 with gt(д) 6 Hk(д) - Hk(t) + |t|/s - M/s' (V^,t G C, t = 0).

As in [12, Proposition 1.17] from the maximum principle for harmonic functions it follows that the interior of K is not empty.

By Theorem 2.3 , Proposition 2.5 and Remark 2.6 C q and 1/C K are bounded on some neighborhood of S q , 1/C q and C ^ are bounded on each compact subset of S ^ .

(v): By the proof of (i) ⇒ (ii) there is an entire function L satisfying the conditions in (ii) and such that n ⁰ (e(_fc}) = _T,,L⁽,^k\ x for each k G N. Hence П ⁰ (с) = P_tpм c^ _T,,L⁽,^^k\ x ^{(k )}'      L '^k )( Д к )                                              ^k fc N ^L ⁰ (Д к )( . — Д к )

for each c G K _ro (Q) and n(f) = (Q l (A ^ , A ^ , f )/L 0 ( >> k )) k ^ N for all f G A(Q) (see the proof of Lemma 3.4). We shall show uniqueness of such function L. Let L i , L 2 be two such functions. Then L 1 (z,A k ) = L ₂ (z,A k ) for all k G N, z G C. Since { A k : k G N } is the uniqueness set for A intQ+K (see 2.2 (b)) and L i (z, ^ ),L 2 (z, • ) G A intQ+K , we get L i (z, • ) = L 2 (z, • ) for each z G C and, consequently, L ₁ = L ₂ on C ² .

Acknowledgement. The first named author thanks for the support by the Deutscher Akademis-cher Austauschdienst during his stay at the University of Du¨sseldorf in autumn 2005.