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Journal of Siberian Federal University. Mathematics & Physics 2008, 1(4), 432-434
УДК 517.55
Some Problems in the Theory of Analytic Multifunctions
Azimbaj A.Sadullaev*
Uzbekistan National University, VuzGgorodok, Taskkent, 700174, Uzbekistan
Received 10.08.2008, received in revised form 20.09.2008, accepted 05.10.2008 In this paper we give a series of open problems in the theory of pseudoconcave sets.
Keywords: pseudoconcave set, set-valued function.
A set-valued function (multifunction) F : z ^ Sz, where z G D С Cn, Sz С Cw, is called an analytic multifunction, if the graph S = {(z,w) G Cn+1 : z G D, w G Sz} is pseudoconcave in G = D x C, i.e., the set S is closed and the set G/S is pseudoconvex in the neighborhood of S.
This definition is motivated by the fact that the graph of an analytic and algebraic function S = {a0(z)wk + a1(z)wfc-1 + ... + ak(z) = 0} where k > 1, aj(z) G O(D), j =0,1,..., k, are analytic multifunctions in D.
Analytic multifunctions (or, equivalently, pseudoconcave sets) are natural object study in the theory of analytic extension, analytic structures of singular sets of holomorphic functions etc. Functional properties and the analytic structure of such functions were investigated in the works of Oka [1], Nishino [2], Yamaguchi [3], Slodkowski [4]-[7], Berndtsson-Ransford [8], Alexander and Wermer [9]-[12], and others.
We used analytic multifunctions in the rational approximation theory, in the problems of extension of holomorphic and pluriharmonic functions [13]-[14]. Analyticity of an arbitrary multifunction F : z ^ Sz is connected with plurisubharmonicity (Psh) of, V(z, w) = — ln p(w, Sz) where p(w, Sz) is the distance from the point w to the set Sz, at a fixed z G D. The function — ln p(w, Sz) is easily determined and convenient for the investigation of F.
If z ^ Sz is analytic, i.e., if S is pseudoconcave, then the function V(z,w) is Psh in G \ S. The converse is not clear, because the function — lnp(w, Sz) in contrast with — lnp((z,w),Sz) does not, in general, tend to +то at (z, w) ^ S.
Example 1. We consider a u(z) G Sh(B) in the unit disk B С C : u < 0 and let {u(z) = —то} be dense on B, u(0) = —1. Then S = {|w| < eu(z)} is bounded and analytic in B x C С C2.
We have V(z, w) = — lnp(w, Sz) = — ln(|w| — e"(z)) and if w0 = —, zv ^ 0 at v ^ то,
1
where u(zv) = —то, then V(zv, wo) = — ln — = —то.
2e
For n = 1, S is analytic if and only if — ln p(w, Sz) G Psh(D) (see Slodkowski [4]). On the other side, for S = (C2 x {|w| = 1}) U ({0} x {|w| < 1}) С C3 the function — lnp(w, Sz) G Psh(C3/S), but S is not pseudoconcave, i.e., z ^ Sz is not analytic (at a point {0} x {|w| < 1}).
The class of multifunctions S, when S is pluripolar in Cn+1 is more interesting in complex analysis. For this kind of "good" multifunctions the following theorem is valid:
Theorem 1 (A. Sadullaev [15]). A "good" multifunction S С D x C is analytic if and only if the function V(z, w) = — ln p(w, Sz) G Psh(G/S).
*e-mail: [email protected] © Siberian Federal University. All rights reserved
Azimbaj A.Sadullaev
Some Problems in the Theory of Analytic Multifunctions
We have the following series of open problems for "good" analytic multifunctions
1. Is the analog of the Hartogs theorem true for a "good" analytic multifunction: If F : z ^ Sz, z € D C Cn, is separately analytic in the variables zi, z2,..., zn, is then F analytic in D?
2. Let z ^ Sz be a multifunction such that Sz is polynomially convex, or, more generally, G/S is connected. Is z ^ Sz analytic if and only if V(z,w) € Psh(G/S)?
3. Let z ^ Sz be a bounded "good" analytic multifunction in D C Cn, that is, S C D x Cw, S C {|w| < R} be a pseudoconcave, and pluripolar set. Is there for any e > 0 and any K C G, K fl S = 0 a pseudopolynomial Pn (z, w) = ao + ai(z)w + ... + a» (z)wN such that S C {|Pn(z)| N < e} and P|k = 0?
The positive solution of this problem is very important in the theory of approximation by rational functions. For if we try to approximate a holomorphic function f by a thin singular set S, then S is pseudoconcave as a singular set as well. And we need the polynomials PN (z)
for construction of a rational sequence r(z
) = , tending to f (see [14]).
PN(z)
4. Let S C Cn+1 be a pluripolar, pseudoconcave set. Is M = Cn+1/S a parabolic manifold? It means that there exists an exhaustion function such that
- VC the set {z € M : p(z) < C} is compact;
- 3K C M such that (ddcp)n+1 = 0 outside of K.
Note that in the theory of Nevanlinna we use this kind of manifolds. If S is the graph of an analytic function, then the problem has a positive solution.
5. Is the graph of a "good (pluripolar)" analytic multifunction z ^ Sz complete pluripolar set, i.e., 3u € Psh(D x C) : S = {u(z) = -ex)}?
6. If F : z ^ Sz and F' : z ^ S^ are two pluripolar analytic multifunctions such that F = F' in a neighborhood U C D, U = 0 will then F = F' everywere in D?
7. N.Shcherbina has recently proved, that if S = {ao(z) + ai(z)w + ... + a^(z)wk = 0} is continuous and pluripolar, then S is analytic. Is its analog true for an arbitrary continuous multifunction?
References
[1] K.Oka, Note sur les familles de fonctions analytiques multiformes etc. J. Sci. Hiroshima Univ., Ser. A4, (1934), 93-98.
[2] T.Nishino, Sur les ensembles pseudo concaves, J. Math. Kyoto Univ., 1-2(1962), 225-245.
[3] H.Yamaguchi, Sur une uniformite des surfaces constants d'une function entire de deux variables complexes, Math. Kyoto Univ., 13(1973), 417-433.
[4] Z.Slodkowski, Analytic set-valued functions and spectra, Math. Ann., 256(1981), 363-386.
[5] Z.Slodkowski, An analytic set-valued selection and its applications to the corona theorem, to polynomial hulls and joint spectra, TAMS, 294(1986), no. 1, 367-378.
Azimbaj A.Sadullaev
Some problems in the theory of analytic multifunctions
[6] Z.Slodkowski, Analytic multifunctions, q-plurisubharmonic functions and uniform algebras, Contemp. Math., 32(1984), 243-257.
[7] Z.Slodkowski, An open mapping theorem for analytic multifunctions, Studia Math., 122(1997), no. 2, 117-21.
[8] B.Berndtsson, T.J.Ranford, Analytic multifunction, the <9-equation and a proof of the corona theorem, Pacific J. Math., 124(1986), 57-72.
[9] H.Alexander, J.Wermer, Polynomial hulls with convex fibers, Math. Ann., 271(1985), 99109.
10] H.Alexander, J.Wermer, On the approximation of singularity sets by analytic varieties, II, Michigan Math J., 32(1985), 227-235.
11] H.Alexander, J.Wermer, Polynomial hulls with convex fibers, Math Ann., 271(1985), 99109.
12] H.Alexander, J.Wermer, Polynomial hulls of sets with intervals as fibers complex variables, VII(1989), 11-19.
13] A.Sadullaev, E.M.Chirka, Extension of function with polar singularity, Math. Sb., 132(1987), no. 3, 383-390 .
14] A.Sadullaev, Rational approximations and pluripolar sets, Math. Sb., 119:1(1982), 96-118.
15] A.Sadullaev, On analytic multifunctions, Matem. Zametki, 83(2008), no. 5, 715-721 (in Russian).
