Zbigniew Slodkowski

Weak solutions for the Levi equation and envelope of holomorphy

We consider the Dirichlet problem for the Levi equation in C2 and we prove an existence theorem of weak solutions (i.e., in the sense of viscosity). As an application we obtain the existence of the envelope of holomorphy for certain compact 2-manifolds.

Christensen zero sets and measurable convex functions

A notion of measurability in abelian Polish groups related to Christen- sens Haar zero set is studied. It is shown that a measurable homomorphism or a measurable Jensen convex function defined on a real linear Polish space is continuous.

Local maximum property and q-plurisubharmonic functions in uniform algebras

Abstract There is proven a formula expressing higher order Shilov boundaries of the tensor products of uniform algebras in terms of boundaries of the factor algebras. Main step: Cartesian product of k- and l-maximum sets is a (k + l + 1)-maximum set. (Definition: locally closed X⊂Cn is a k-maximum set if polynomials have local maximum property on intersections of X with (n-k)-dimensional complex planes.) Other properties and characterizations of k-maximum sets are given, e.g., X⊂Cn is a k-maximum set iff each k-plurisubharmonic function has local maximum property on X.

Polynomial hulls with convex fibers and complex geodesics

Let X be a compact subset of {|z| = 1} × Cn with convex fibers. Several equivalent conditions are obtained that characterize polynomially convex hulls Y = X with nonempty interior. Under the latter assumption, it is shown that every (z0, w0) ϵ ∂ Y and such that ¦z0¦ < 1 is contained in a closed n-dimensional complex submanifold of d × Cn which is tangent to Y along an analytic disc. We show, as an application, that some results of Lempert on complex geodesies in convex domains are direct consequences of properties of polynomial hulls.

On bounded analytic functions in finitely connected domains

A new proof of the corona theorem for finitely connected domains is given. It is based on a result on the existence of a meromorphic selection from an analytic set-valued function. The latter fact is also applied to the study of finitely generated ideals of H°° over multiply connected domains. Introduction. In this paper, which is a direct continuation of [18], we study applications of analytic multifunctions to some topics in function theory on finitely connected domains, related mainly to the corona problem. Concerniag analytic multifunctions (which are certain set-valued functions, cf. Definition 1.2), §§1-3 of [18] form sufficient background for our purposes here. The reader is also referred there for the information about the origin of the approach employed in this paper. However, basic definitions and references are provided below. Our basic new result (Theorem 1.4) associates to every analytic multifunction defined in a finitely connected domain some meromorphic functions with poles at the critical points of Greens function of this domain. (Some improvements of this fact, in tXhe special case of an annulus, are discussed in §3.) This main technical result is applied in §2 to help obtain a new, simple proof of the well-known corona theorem for finitely connected domains. A novel feature of this proof (as compared with e.g. Forelli [9], Stout [19], Gamelin [7]) is that it does not use the corresponding result for the unit disc. The same methods are used in §4, where we consider the question, when some power of g from H°°(G) belongs to the ideal Af1 + Af2 + + Afn. We obtain generalizations to multiply connected domains of a result due to T. Wolff (Theorems 4.1 and 4.2), as well as its refinements in the case of the unit disc (Corollary 4.4 and Example 5.3). All proofs given in this paper extend to finite Riemann surfaces. 1. A meromorphic selection theorem. The main result of this section relates some meromorphic vector-valued functions to analytic multifunctions defined in finitely connected domains. First, we recall some definitions. DEFINITION 1.1 (FOLK). A locally compact set Z c Cn is a maximum set if for every compact set N c Z and for every analytic function f defined in a neighborhood of N the inequality maxN If l < maxazN If l holds. DEFINITION 1.2 [16, 18]. Anuppersemicontinuousset-valuedcorrespondence z K(z): G 2C, where G c C is open and all K(z) are nonempty and Received by the editors January 16, 1986. Presented to the Society, Special Session on Convex Analysis, New Orleans, January 11, 1986. 1980 Mathematics Subject Classifiratson (1985 Ren). Primary 32A80, 32E20, 46J15. (2)1987 American Mathematical Society 0002-9947/87

COMPLEX INTERPOLATION OF NORMED AND QUASINORMED SPACES IN SEVERAL DIMENSIONS. I

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Operators with Closed Ranges in Spaces of Analytic Vector-Valued Functions

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Approximation of analytic multifunctions

A variety of complex interpolation methods for families of normed or quasi-normed spaces, parametrized by points of domains in complex homo- geneous spaces, parametrized by points of domains in complex homogeneous spaces, is developed. Results on existence, continuity, uniqueness, reiteration and duality for interpolation are proved, as well as on interpolation of oper- ators. A minimum principle for plurisubharmonic functions is obtained and used as a tool for the duality theorem.

Pseudoconvex classes of functions. III. Characterization of dual pseudoconvex classes on complex homogeneous spaces

Abstract Let A + ( P , X ) denote the Banach space of X -valued analytic functions on a polydisc P ⊂ C n with absolutely convergent Taylor series. Main result : Let T ( z ) be an analytic family of bounded operators from X to Y . Assume that all T ( z ) have closed ranges depending continuously on z . Then the “multiplication” operator T : A + (P, X) → A + (P, Y) , induced by T ( z ), has closed range. (Equivalent characterization of such operator-valued functions are given.) This result makes it possible to construct Banach spaces of sections of some infinite-dimensional analytic sheaves. The construction is functorial and has certain exactness properties which help to study analytic perturbations of the joint spectrum of J. L. Taylor.

Natural extensions of holomorphic motions

On definit des generalisations a valeurs ensembles des fonctions analytiques par une forme du principe du maximum local. On les identifie aux limites des suites decroissantes de multifonctions dont les graphes sont localement couverts par ceux des applications analytiques univalentes