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On Entire Functions of Exponential Type and Indicators of Analytic Functionals

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On Entire Functions of Exponential Type and Indicators of Analytic Functionals ON ENTIRE FUNCTIONS OF EXPONENTIAL TYPE AND INDICATORS OF ANALYTIC FUNCTIONALS BY C. O. KISELMAN The University of Stockholm, Stockholm, Sweden and The Institute for Advanced Study, Princeton, N.J., U.S.A. x O. Summary We shall be concerned with the indicator p of an analytic functional # on a complex manifold U: p(v) = li--m 1 log I/~(et~)l, where V is an arbitrary analytic function on U. More specifically, we shall consider the smallest upper semicontinuous majorant p~ of the restriction of p to a subspace ~ of the analytic functions. An obvious problem is then to characterize the set of functions p~ which can occur as regularizations of indicators. In the case when U =(~a and :~ is the space of all linear functions on 0 a, this set can be described more easily as the set of functions lira lira 1 log [u(tO)l (0.1) 0---'-~ t--~+r162t of n complex variables ~ 6C a where u is an entire function of exponential type in Ca. We shall prove that a function in Ca is of the form (0.1) for some entire function u of exponential type if and only if it is plurisubharmonic and positively homogeneous of order one (Theorem 3.4). The proof is based on the characterization given by Fujita and Takeuchi of those open subsets of complex projective n-space which are Stein manifolds. Our objective in Sections 4 and 5 is to study the relation between properties ofp ~ and existence and uniqueness of :~-supports of #, i.e. carriers of/~ which are convex with respect to :~ in a certain sense and which are minimal with this property (see Section 1 for defiai- I This work was supported by the Swedish Natural Science Research Council and Air Force Office of Scientific Research grant AF-AFOSR-359-64. 2 C.O. KISELMAN tions). An example is that under certain regularity conditions, io~ is convex if and only if # has only one J-support. Section 2 contains a result on plurisubharmonic functions in infinite-dimensional linear spaces and approximation theorems for homogeneous plurisubharmonic functions in C=. The author's original proof of Theorem 3.1 was somewhat less direct than the present one (see the remark at the end of Section 3). It was suggested by Professor Lars HSrmander that a straightforward calculation of the Levi form might be possible. I wish to thank him also for other valuable suggestions and several discussions on the subject. Notation. The complement of a set A with respect to some bigger set which is under- stood from the context is denoted by C A. We write A\B for Afl [J B. The Cartesian product of n copies of A is denoted by A =. The interior of a set A in a topological space is denoted by A ~ its closure by A, and its boundary by ~A. 1~, It, C stand for the set of non-negative integers, real numbers, and complex numbers, respectively. The sets [-~, +~[=RU{-~} and [-~, +~]= It U {-~, + ~} shall be equipped with their natural topologies so that, for instance, [- ~, + ~] is compact. We use a bar to denote complex conjugation of complex numbers and complex-valued functions. The differential operators ~/~zk and a/~2~ are defined for functions of n complex variables by where zz=xk+iyk; xk, yzEit, k=l, ..., n. The space of complex-valued linear forms on (~n is denoted by E or C(t3n) and the value of ~ E C at a point z E (~ is written ~(z) = <z, ~>. There is sometimes no advantage in identifying s with (~n; on other occasions, however, we shall do so by means of the formula <z, ~> =~zj~j. The norm in Cn (and in C when we use coor- dinates there) will always be Euclidean: I zl = (~zt~)1/~. 1. Basic definitions Let U be a complex analytic manifold. We shall denote by A(U) the space of all ana- lytic functions in U equipped with the topology of uniform convergence on all compact subsets of U. A continuous complex-valued linear form/x on M(U) or, in other words, an element of the dual space M'(U), is called an analytic functional in U. If gEM'(U), the continuity means that for some compact set Kc U and some constant C we have <c sup (1.1) K for all ~ E A(U). ENTIRE FUNCTIONS OF EXPONENTIAL TYPE It is convenient to have a name for sets which are limits of decreasing sequences of compact sets for which (1.1) holds: De/inition 1.1. A carrier o/an analytic/unctional I~E.4'(U) is a compact subset K o/ U such that/or every neighborhood L o/K there is a constant C such that i/~(q~)l <~c supLIq~l/or all q) 6.4(U). In Martineau [8] the word carrier is used with a different meaning. However, Marti- neau's definition and Definition 1.1 coincide if U is a Stein manifold and K is holomorph convex and this will usually be the case in what follows. The main part of this paper is devoted to a study of carriers of an analytic functional # which are minimM with respect to inclusion in the family of all those carriers of #~ which are convex in a certain sense. We proceed to define these convexity properties. De/inition 1.2. Let M be a subset o~ U. We de/ine the supporting/unction H M o/M by HM(q~) = sup Re T(z), ~e.4(U). zeM (The supremum over the empty set is de/ined as - ~.) With obvious conventions on infinite values, H~ is convex and positively homogeneous in .4(U), that is HM(tqJ) = tHM(q~), HM(q~+~p) <HM(qJ) + H~(y~), t >O, T,~pe.4(U). If M is non-empty and relatively compact in U, /-/M is in addition real-valued and con- tinuous. In the special case when U = Cn, the restriction of H M to the linear functions ~: in (~n is the usual supporting function of M and it is well-known that every convex and positively homogeneous real-valued function in s is such a restriction. In general, however, the restrictions of the supporting functions to a subspace :~ of .4(U) form a proper subset of the convex and positively homogeneous functions in ~. Definition 1.3. Let ~ be an arbitrary subset o/.4(U). We de/ine the :~-hull h~M o/a set McU by h~M = {z E U;/or all q~ 6 3, Re q~(z) <~HM(~)}. I/h~M=M we say that M is :~-convex. The mani/old U is called ~-convex i/ h~K is compact /or every compact set Kc U. In other words, h~ M is the largest subset of U whose supporting function coincides in with that of M. If ~c ~.4(U) and McN~ U it is obvious that h~Mch~N. We also have h~hqM=h~M=hqhTM if ~ ~; in particular h~M is ~-convex. Another property of 4 C. O. KISELMAN the :~-hull is that h~( f'l Ms)~ f3 h~Mi; this implies that the intersection of a family of :~-con- vex sets is ~-convex. If U=C ~ and ~=s the space of all linear functions on C~, h~M=h:M is the closed convex hull of M. More generally, if :~ is a finite-dimensional complex linear subspace of A(U) we can choose a basis ~1 ..... zr in if, and then h~M is the inverse image under ~= (~1, ..., ~m) of the closed convex hull of co(M). Another important special case is when = A(U). Then the :~-hnll is the usual holomorph convex hull. In fact, if z is in the holo- morph convex hull of M we have [~v(z)[ ~<supM[~[ for all of CA(U), in particular with r vj, eRe~(z)<~sllpMeRevJ SO that zEh,4(u)M. On the other hand, if zEh~(v)M and ~eA(U), It] = 1, we obtain Re @(z) <~supMRe t~v ~< SUpMI~I and hence [q(z) I ~<SUpM[~ I which means that z belongs to the holomorph convex hull of M. The same proof shows that the real parts in Definitions 1.2 and 1.3 can be replaced by absolute values without affecting h~M if ~ is a subalgebra of j4(U) and in addition either ~ is closed or M is relatively compact in U. De/inition 1.4. A compact subset K o/ U is called an :~-support o] ttEz4'(U ) i/K is an ~.convex carrier o/# and K is minimal with respect to this property, that is, h~L~ K/or every carrier L o/tt such that L c K. By the Zorn lemma, tt has an J-support if (and only if) ft is carried by some J-convex compact set. This is always the case on an :~-convex manifold. The supporting functions of the J-supports of a functional are closely connected with the growth properties of a generalized Laplace transform/2 of #. This relation is the subject of Section 5; we prefer, however, to define/2 now. Definition 1.5. Let ~teA'(U). The (generalized) Laplace trans/orm /2 o/# is de/ined by fi@) =tt(e~), ~ e z4(U), and the indicator o/# (or o]/2) is p(cf )=limt_~+~ log ]/2(tq~) l l/t, t> O. If U = C~, the restriction of/2 to E is the usual Laplace (or Fourier-Borel) transform of # which is an entire function of exponential type in s The name indicator or indicator function is also usually reserved for the restriction of p to s Now suppose that K is a carrier of ft.
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