On entire functions of exponential type and indicators of analytic functionals

Abstract

We shall be concerned with the indicator p of an analytic functional μ on a complex manifold U: $p(\varphi ) = \overline {\mathop {\lim }\limits_{t \to + \infty } } \frac{l}{t}\log \left| {\mu (e^{t\varphi } )} \right|,$ where ϕ is an arbitrary analytic function on U. More specifically, we shall consider the smallest upper semicontinuous majorant p^J of the restriction of p to a subspace £ of the analytic functions. An obvious problem is then to characterize the set of functions p^J which can occur as regularizations of indicators. In the case when U=Cⁿ and £ is the space of all linear functions on Cⁿ, this set can be described more easily as the set of functions $\mathop {\lim }\limits_{\theta \to \zeta } \overline {\mathop {\lim }\limits_{t \to + \infty } } \frac{l}{t}\log \left| {u(t\theta )} \right|$ of n complex variables ζ∈Cⁿ where u is an entire function of exponential type in Cⁿ. We hall prove that a function in Cⁿ 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 of p^J and existence and uniqueness of £-supports of μ, i.e. carriers of μ which are convex with respect to £ in a certain sense and which are minimal with this property (see Section 1 for definitions). An example is that under certain regularity conditions, p^J is convex if and only if μ has only one £-support.

Section 2 contains a result on plurisubharmonic functions in infinite-dimensional linear spaces and approximation theorems for homogeneous plurisubharmonic functions in Cⁿ.

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 Hörmander 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.