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\topmatter
\title
Seventh summer St.Petersburg meeting\\ in mathematical analysis
\endtitle
\endtopmatter
\vskip4truecm
\centerline{$\goth {Euler \quad International \quad Mathematical \quad Institute}$}
\vskip2.5truecm
\centerline {{\it ST.~PETERSBURG}}
\vskip2.5truecm
\centerline{June 17--20, 1998}

\def \lin#1{\par\noindent\hangindent=4cm#1\par}
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\def \coffee{\line{}\centerline{\bf Coffee break}\line{}}
\def \lunch{\line{}\centerline{\bf Lunch}\line{}}
\def \dinner{\line{}\centerline{\bf Dinner}\line{}}

\newpage
\line{}\line{}
\centerline {\bf PROGRAM}
\vskip1truecm
\centerline{{\bf WEDNESDAY, June 17}}
\vskip1truecm

\hfil {{\bf Chairman:} {S.~V.~Kislyakov}}
\line{}
\line{}
Smoothness of quasiconformal mappings at a point}
\coffee
Reproducing kernels in weighted
Bergman spaces and the Green function for the weighted bilaplacian}
\line {}
$L_2$@-stable operator semigroups and the Muckenhoupt condition}
\lunch
\vskip1truecm
\hfil {{\bf Chairman:} {H.~Hedenmalm}}
\line{}
Disjoint hyperinvariant subspaces of operators}
\coffee
Invariant and hyperinvariant subspaces of
Volterra operators in spaces of vector-functions}
\line{}
Entire function associated with
analytic proximate order and its applications}
\coffee
Three spheres theorem for harmonic differential norms}
\line{}
Critical constants, spectral hulls and function calculi}
\dinner

\newpage

\centerline{{\bf THURSDAY, June 18}}
\vskip2truecm

\hfil {{\bf Chairman:} {L.~Kerchy}}
\line{}
Regularity conditions for vectorial stationary processes}
\coffee
Invertibility and ciclicity in large Bergman spaces}
\line {}
The resolvent of Wiener--Hopf operator via function model}
\lunch

\vskip1truecm
\hfil {{\bf Chairman:} {V.~I.~Vasyunin}}
\line{}
On some recent results in real interpolation and their applications}
\coffee
Spectra of inner functions and the S.~A.~Vinogradov
problem on $l^p$@-multipliers}
\line{}
\lin{16:30--16:55 \quad {\bf R.~R.~del Rio Castillo.}
On a problem of P.~Hatrman and A.~Wintner}
\coffee
Exposed points in the unit ball of $H^1$}
\line{}
Indexes of shift-invariant subspaces in $l^p$ and $l^p (w_n)$ spaces}
\dinner

\newpage

\centerline{{\bf FRIDAY, June 19}}
\vskip0.5truecm

\hfil {{\bf Chairman:} {V.~V.~Peller}}
\line{}
Generalized Trace Formula for orthogonal polynomials
with asymptotically periodic recurrence coefficients}
\coffee
Kernels of Toeplitz operators: a parametrization via Bourgain's theorem}
\line {}
Relationship between conditions of free
interpolation and shape of region in complex plane}
\lunch

\vskip1truecm

\centerline{{\bf SATURDAY, June 20}}
\vskip0.5truecm

\hfil {{\bf Chairman:} {A.~B.~Aleksandrov}}
\line{}
Metric properties of exceptional sets for
$\delta$@-subharmonic functions in a ball}
\coffee
Zygmund's dichotomy for pluriharmonic Riesz products}
\line {}
On some Fourier multipliers related with the Sobolev spaces}
\lunch
Interpolation involving bounded bianalytic functions}
\coffee
Nevanlina factorization in one more class of analytic functions}
\line{}
A similarity version of the Nagy-Foias model, duality and exact
controllability}
\line{}

\newpage

\centerline {\bf LIST OF PARTICIPANTS}
\vskip1truecm

\redefine\b{\bigskip}
\define\y{\newline\phantom{AAAAAA}}
\define\n{\noindent}

\n {\bf ABAKUMOV, Evgeny Valerievich}
\y Universit\'e de Marne-la-Vall\'ee
\y 35 Bd\. de Port Royal, 75013 Paris, FRANCE
\y email: abakumov\@math.univ-mlv.fr, abakumov\@pdmi.ras.ru

\b

\n {\bf ALEKSANDROV, Aleksei Borisovich}
\y St.Petersburg Division of the Steklov Mathematical Institute
\y Fontanka 27, St.Petersburg 191011, RUSSIA
\y phone: 7(812)2955992 (home), 7(812)3107164 (office), fax: 7(812)3105377
\y email: alex\@pdmi.ras.ru

\b

\n {\bf ANISOVA, Elizaveta Grigorievna}
\y Moscow State University, Dept\. of Mechanics and Mathematics
\y (graduated; probably since November 1998:
\y St.Petersburg State University, Dept\. of Mathematics and Mechanics)
\y phone: 7(812) 173-49-38
\y email: liza\@IZ1341.spb.edu

\b

\y Samara State University, Dept\. of Mathematics
\y Acad\. Pavlov street, 1, Samara 443011, RUSSIA
\y phone: (8462) - 345431, fax: (8462) - 345417
\y email: astashkn\@ssu.samara.ru

\b

\n {\bf BORICHEV, Aleksandr Aleksandrovich}
\y Universit\'e Bordeaux-I, UFR de Math\'ematiques
\y 351, cours de la Lib\'eration, 33405 Talence Cedex, FRANCE
\y phone: (33) 556 84 26 15, fax: (33) 556 84 69 29
\y email: borichev\@math.u-bordeaux.fr

\b

\n {\bf DEL RIO CASTILLO, Rafael Rene}
\y Deleg. Coyoacan, 04510 Mexico D.F., MEXICO
\y phone: (525) 6223595, fax: (525) 6223596
\y email: delrio\@servidor.unam.mx

\b

\n {\bf DOUBTSOV, Evgeny Sergeevich}
\y Dept\. of Mathematics and Mechanics, St.Petersburg State University
\y Bibliotechnaya pl. 2, Staryi Petergof, 198904 St. Petersburg, RUSSIA
\y email: es\@dub.pdmi.ras.ru

\b

\n {\bf DYAKONOV, Konstantin Mikhailovich}
\y Steklov Institute of Mathematics, St.Petersburg Branch
\y Fontanka 27, St.Petersburg, 191011, RUSSIA
\y phone: (812) 598-21-41, fax:  (812) 346-27-58
\y email:  dyakonov\@pdmi.ras.ru

\b

\n {\bf DYN'KIN, Evsei Matveevich}
\y Dept\. of Mathematics, Technion, 32000, Haifa, ISRAEL
\y phones: +972-4-8225450 (home), +972-4-8294023 (office)
\y fax:  +972-4-8324654
\y email: dynkin\@tx.technion.ac.il

\b

\y Moscow State Civil Engineering University, Dept\. of Higher  Mathematics
\y Yaroslavskoe shosse 26, 129337 Moscow, RUSSIA
\y phone: (095)183-30-38
\y email: eiderman\@orc.ru

\b

\n {\bf FARFOROVSKAYA, Yulia Borisovna}
\y State University of Telecommunication
\y Moyka 61, St.Petersburg 191186, RUSSIA
\y phone: (812) 271-64-90
\y email: rabk\@sut.ru

\b

\n {\bf GAMAL, Maria Feliksovna}
\y St.Petersburg Division of the Steklov Mathematical Institute
\y Fontanka 27, St.Petersburg 191011, RUSSIA
\y phone: 7(812)1666279 (home), 7(812)3107164 (office), fax: 7(812)3105377
\y email: gamal\@pdmi.ras.ru

\b

\n {\bf GLUSKIN, Efim Davidovich}
\y Tel Aviv University
\y Ramat Aviv, 69978, Tel Aviv, ISRAEL
\y phones: 972-3-6408816, 6408043, fax: 972-3-6409357
\y email: gluskin\@math.tau.ac.il

\b

\y South--Ukrainian Pedagogical University,
\y Staroportofrankovskaya 26, Odessa 270020, UKRAINE
\y email: rector\@tm.odessa.ua,
\y $\quad$ Dmitriy.Kalyuzhniy\@p25.f61.n467.z2.fidonet.org

\b

\n {\bf HEDENMALM, H\aa kan}
\y Mathematics Department, Lund University
\y Box 118, 22100 Lund, SWEDEN
\y phone: +46462220464, fax: +46462224213
\y email: haakan\@maths.lth.se

\b

\y St.Petersburg Division of the Steklov Mathematical Institute
\y Fontanka 27, St.Petersburg 191011, RUSSIA
\y phone: 7(812)2473879 (home), 7(812)3107164 (office), fax: 7(812)3105377
\y email: kapustin\@pdmi.ras.ru

\b

\n {\bf KARGAEV, Pavel Petrovich}
\y St.Petersburg State University, Dept\. of Mathematics
\y Bibliotechnaya pl.2, Stary Petergof, St.Petersburg 198904, RUSSIA

\b

\n {\bf KERCHY, Laszlo}
\y University of Szeged, Bolyai Institute
\y Aradi vertanuk tere 1, H-6720 Szeged, HUNGARY
\y fax: (36) (62) 326-246
\y email: kerchy\@math.u-szeged.hu

\b

\n {\bf KISLYAKOV, Sergei Vitalievich}
\y St.Petersburg Division of the Steklov Mathematical Institute
\y Fontanka 27, St.Petersburg 191011, RUSSIA
\y phone: 7(812)3107164 (office), fax: 7(812)3105377
\y email: skis\@pdmi.ras.ru

\b

\n {\bf KOTOCHIGOV, Aleksandr Mikhailovich}
\y Sankt-Petersburg Electro-Technical University
\y Ul. Professora Popova, 5, St.Petersburg 197367, RUSSIA
\y fax: 7(812) 346-2758
\y email: kam\@kam.spb.su

\b

\n {\bf KRUGLIAK, Natan}
\y Yaroslavl State University, RUSSIA
\y email: natan\@univ.uniyar.ac.ru

\b

\y Moscow State Institute of Elerctronics and Mathematics (MGIEM)
\y Dept\. of Mathematical Analysis
\y Bolshoy Trehsvjatitelskii 3/12, Moscow 109028, RUSSIA
\y fax: (095) 9162807, phones: office (095) 2357187, home (095) 3941080
\y email: ma\_miem\@orc.ru (for V.~V.~Lebedev)

\b

\n {\bf MAERGOIZ, Lev Sergeevich}
\y Krasnoyarsk State Academy of Architucture and Civil Engineering
\y pr. Svobodny, 82, Krasnoyarsk, 660036, RUSSIA
\y phone: (3912)49-45-75
\y email: root\@maergoiz.krasnoyarsk.su

\b

\n {\bf MAKAROV, Boris Mikhailovich}
\y St.Petersburg State University, Dept. of Mathematics
\y Bibliotechnaya pl.2, Stary Petergof, St.Petersburg 198904, RUSSIA
\y phone: (812) 251 58 98 (home), (812) 428 42 11 (office)
\y email: bmmak\@bmmak.usr.pu.ru

\b

\n {\bf MALAMUD, Mark  Mikhailovich}
\y Mathematics Dept., Donetsk State University
\y Universitetskaya str. 24, Donetsk 340055, UKRAINE
\y fax: +380(622)927112, phone: (0622)371897
\y email:    mmm\@univ.donetsk.ua

\b

\y St.Petersburg State University, Dept\. of Mathematics and Mechanics
\y Bibliotechnaya pl.2, Petrodvorets, St.Petersburg, RUSSIA
\y phone: 7(812) 245-43-14
\y email: mal\@math.sch239.spb.ru

\b

\n {\bf MIKHAILOVA, Ekaterina Mikhailovna}
\y Ross\. Univ\. Druzhby Narodov, Dept\. of Math\. Analysis
\y ul\. Ordzonikidze 3, 117198 Moscow, RUSSIA
\y phone: 134-36-38 (home), 952-35-83 (office)

\b

\n {\bf NIKOLSKI, Nikolai Kapitonovich}
\y St.Petersburg Division of the Steklov Mathematical Institute
\y Fontanka 27, St.Petersburg 191011, RUSSIA, and
\y Universit\'e Bordeaux-I, UFR de Math\'ematiques
\y 351, cours de la Lib\'eration, 33405 Talence Cedex, FRANCE
\y phone (Bordeaux): +33 556846061; fax: +33 556846959
\y email: nikolski\@math.u-bordeaux.fr

\b

\y St.Petersburg State University, Physical Department
\y Ul'yanovskaya 1, Petrodvorets, St.Petersburg 198904, RUSSIA
\y fax: (812)4287240, phones: office (812)4287579,  home (812)4275711

\b

\n {\bf OSILENKER, Boris Petrovich}
\y Moscow State Civil Engineering University, Mathematical Department
\y Yaroslavskoe Shosse 26, 129337 Moscow, RUSSIA
\y phone: (095)2846749
\y email: borosilr\@orc.ru

\b

\n {\bf PATSEVICH, Elena Leonardovna}
\y Novgorod State University, Div. of Mathematical Analysis
\y Novgorod, RUSSIA
\y phone: (8162)-110245  (home)
\y email: patsev\@pdmi.ras.ru

\b

\n {\bf PAVLOV, Boris Sergeevich}
\y Dept. of Mathematics, the University of Auckland
\y Auckland, NEW ZEALAND
\y and: private box 92019, Auckland, NEW ZEALAND
\y phone: 64-9-3737599 ext 8783; fax: 64-9-3737457
\y email: pavlov\@math.auckland.ac.nz

\b

\y Kansas State University, USA
\y phone: (913) 537-0314 (USA), (812)230-6092 (Russia)
\y email: peller\@math.ksu.edu, peller\@pdmi.ras.ru

\b

\n {\bf PETROV, Andrei Nikolaevich}
\y St.Petersburg State University, Dept. of Mathematics
\y Bibliotechnaya pl. 2, Stary Petergof, St.Petersburg 198904, RUSSIA
\y phone: (812) 427 99 90 (office), fax: (812) 427 54 14
\y email: petrov\@agspbu.spb.su

\b

\n {\bf PETUKHOV, Alexander}
\y St.Petersburg Technical University
\y phone: 277 24 82
\y email: petukhov\@pdmi.ras.ru

\b

\y Massachusetts Institute of Technology
\y 77 Mass. Ave., Cambridge, MA 02139, USA
\y phone: (617)-253-4385, fax: (617)-253-4358
\y email: agp\@math.mit.edu

\b

\n {\bf ROGINSKAYA, Maria Martynovna}
\y St.Petersburg State University, Dept. of Mathematics and Mechanics
\y Bibliotechnaya pl.2, Stary Petergof, St.Petersburg 198904, RUSSIA
\y phone: 7(812)5842444 (home)
\y email: rog\@post.tepkom.ru

\b

\y Laboratory of Complex Systems Theory, Institute for Physics,
\y Saint Petersburg State University, 198904 Saint Petersburg, RUSSIA
\y phone: 007 (812) 526-25-00, fax: 007 (812) 428-66-49
\y email: roma\@rvr.stud.pu.ru

\b

\n {\bf RUDELSON, Mark}
\y Dept\. of Mathematics, Texas A \& M University
\y College Station TX 77843-3368, USA
\y phone: 409-694-1716, fax: 409-845-6028
\y email: Mark.Rudelson\@math.tamu.edu

\b

\n {\bf RUKSHIN, Sergei Evgenievich}
\y Herzen Russian State Pedagogical University
\y St.Petersburg, Moika, 48
\y phone: 7(812) 245-43-14
\y email: serger\@math.sch239.spb.ru

\b

\n {\bf SHAMOYAN, Faizo Agitovich}
\y Bryansk State Ped\. University
\y Bejizkaya 14, 241036 Bryansk, RUSSIA
\y phone: 46-64-87
\y email: sham\@bgpi.bitmcnit.bryansk.su

\b

\n {\bf SHIMORIN, Sergei Mikhailovich}
\y Department of Mathematical Analysis of St.Petersburg University
\y Bibliotechnaja 2, St.Petersburg 198904, RUSSIA
\y phone: 7(812) 428-42-11
\y email: shi\@math.lgu.spb.su

\b

\n {\bf SHIROKOV, Nikolai Alekseevich}
\y St.Petersburg Electro-Technical University
\y ul. Professora Popova, 5, St.Petersburg 197367, RUSSIA
\y fax: 7(812) 346-2758, phone:  580 81 90 (home)
\y email: kam\@kam.spb.su (for Shirokov)

\b

\n {\bf SOLYNIN, Alexander Yurievich}
\y V.~A.~Steklov Mathematical Institute,  Russian Academy of Sciences
\y Fontanka 27, 191011,  St.Petersburg,   RUSSIA
\y phone: (7)-(812)-3104736, fax: (7)-(812)-3105377
\y email: solynin\@pdmi.ras.ru

\b

\n {\bf VASYUNIN, Vasily Ivanovich}
\y St.Petersburg Division of the Steklov Mathematical Institute
\y Fontanka 27, St.Petersburg 191011, RUSSIA
\y phone: 7(812)2135903 (home), 7(812)3107164 (office), fax: 7(812)3105377
\y email: vasyunin\@pdmi.ras.ru

\b

\n {\bf VIDENSKII, Ilya Viktorovich}
\y St.Petersburg Electro-Technical University
\y ul. Prof.~Popova 5, St-Petersburg, 197367, RUSSIA
\y phone: 7(812)234-8918, fax: 7(812)346-2758
\y email: ilya\@viden.pdmi.ras.ru

\b

\n {\bf VLASOV, Victor Valentinovich}
\y Moscow Inst\. of Physics and Technology (MIPT), Dept\. of Math\.
\y Institutsky per. 9, Dolgoprudny, Moscow region, 141700, RUSSIA
\y phones: (095) 308-83-80 (home), (095) 408-81-72 (office)
\y email: vlasov\@math.mipt.ru

\b

\n {\bf WOJCIECHOWSKI, Michal}
\y Institute of Mathematics of Polish Academy of Sciences
\y ul. Sniadeckich 8, 00-950 Warszawa, POLAND
\y fax: (48)(22)6293997
\y email: miwoj\@impan.gov.pl

\b

\y Dept\. of Mathematics and Mechanics, St.Petersburg State University
\y Bibliotechnaya pl., 2, Stary Peterhof, 198904 St.Petersburg, RUSSIA
\y phone: (812) 4287063, fax (812) 4287039
\y email: dm\@yakub.niimm.spb.su

\newpage
\centerline {\bf ABSTRACTS}
\vskip1truecm
\define \bb {\bigbreak\bigskip}

{\bf E.~V.~Abakumov.} {\it Indexes of shift-invariant subspaces
in $l^p$ and $l^p (w_n)$ spaces}. (Joint work with A\. Borichev)

For $l^p$@-spaces ($p>2$) and for a wide classe of weighted $l^p$@-spaces,
we give a very elementary construction of shift-invariant subspaces
of indexes $2, 3, 4, ... , \infty$.

\bb

{\bf E.~G.~Anisova.} {\it Quadrics of codimention $4$ in $\Bbb C^7$
and their automorphisms}.

We study nondegenerate $(4,3)$-quadrics, where by a $(k,n)$-quadric
we mean a quadratic model for a real $CR$-submanifold
(of $CR$-dimension $n$ and codimension $k$) of the complex space. We
find all (up to biholomorphic equivalence) $(4,3)$-quadrics
that have nonlinear holomorphic automorphisms. For these 9 quadrics we
calculate the algebras of infinitesimal automorphisms.

\bb

Let
$$r_k(t)\,=\,sign\,\sin{2^k{\pi}t}\,\,(k=0,1,2,..)$$
be the Rademacher system on $[0,1]$. Then the set of functions
$$R(t)=\,\sum_{1\le ic>0 and of P(r)(1-r)^{m-1}\to0. We describe the size of C by estimation of the (m-1)-dimensional Lebesgue measure of C\cap\{x:|x|=r\},\ 00, \ \text{Re}\mu>0$$
is of a great importance in the theory of integral transformations.
But its applications in the theory of entire functions is limited to the
class of functions of finite order and normal type.
The next analogue of Mittag-Leffler function is considered in the
talk:
$$E_\rho(z; V)=\sum\limits_{k=0}^{\infty}\frac{z^k}{\Delta(k+1)},\ z\in \Bbb C; \quad \Delta(\lambda)=\rho\int_{0}^{\infty}e^{-V(t)} t^{\lambda-1}dt,$$
where \ $\rho>0$; $\Delta(\lambda)$ is an analogue of the Euler Gamma-function
associated with a given real analytic function \ $V$ \ such that \
$\rho(t)=\ln V(t)]/\ln t$\
is a proximate order with \ $\rho(t)\to \rho$\ as \
$t\to\infty$. This proximate order exists for any given proximate order \
$\rho_1(t)$\ with \ $\rho(t)\sim \rho_1(t)$. Various properties of \
$E_\rho(z; V)$ \ are investigated. An application is obtained of
those results to discription of the relation between the indicator and
conjugate diagrams  of entire function of proximate order \ $\rho>0$ \ with
nonnegative indicator (an analogue the Polya theorem).

\bb

{\bf M.~M.~Malamud.} {\it Invariant and hyperinvariant subspaces of
Volterra operators in spaces of vector-functions}.

A description of invariant, hyperinvariant and cyclic subspaces
of a Volterra operator $A=J^{\alpha}\otimes B$, i.e., the
tensor product of a positive power of the integration
operator $J$ by a finite matrix $B$, which is
a Jordan cell or a direct sum of Jordan cells is
investigated. Some sufficient conditions for a Volterra
operator to be similar to an operator $A$ are given.

\bb

{\bf E.~Malinnikova.} {\it Three spheres theorem for harmonic
differential forms}.

A generalization of the Hadamard three-circles theorem
to harmonic differential forms is proved.
Let $u$ be a harmonic differential
form on a {\it spherical shell} $\{r_1\le|x|\le r_2\}\subset\Bbb R^n$.
By $\|u\|_R$ we denote the $L^2$ norm of the form $u$
over the sphere $\{|x|=R\}$.
Then the following inequality holds
$$\|u\|_r\le\|u\|_{r_1}^\alpha\|u\|_{r_2}^{1-\alpha},$$
where $r_10$ such that for
each $x$ in $F$ the ball with the center at $(-x)$ and of radius
$\varepsilon |x|$ does not intersect $F$. Then $F$ is a Riesz set.

Some new examples of Riesz sets will be shown.

\bb

{\bf R.~Romanov.} {\it  $L^p$ estimate related to the evolution
operator for the Friedrichs model}.
(Joint work with S. N. Naboko)

The talk is devoted to an estimate of the evolution operator
for the nonselfadjoint dissipative Friedrichs model of rank 1. Let $L$ be
a rank $1$ perturbation of the selfadjoint multiplication operator $A$,
let $Z_t = e^{ iLt }$ be the corresponding evolution semigroup, and let
$U_t = e^{ iAt }$. We deal with the
comparison operator $U_{ - t } Z_t - I$. It
is shown that some $L^p$@-estimate related to this operator is sharp.
The proof of the estimate leans upon the functional model.

\bb

{\bf M.~Rudelson.} {\it Isotropic random vectors}.

%\define\e{\varepsilon}
%\define\a{\alpha}
\define\th{\theta}
\define\de{\delta}
\define\om{\omega}
\define\Om{\varOmega}
%\define\G{\varGamma(\omega)}
\define\dist{\text{dist}}
\define\conv{\text{conv}}
\define\etc{, \dots ,}
\define \sumi{\sum_{i=1}^M}
\define\rn{$\Bbb R^n \,$}
%\define\Rm{$\Bbb R^M \,$}
\define\RM{\Bbb R^M}
\define\Rn{\Bbb R^n}
\define\bn{$B^n_2 \,$}
\define\Bn{B^n_2}
\define\nor #1{\left \| #1 \right \|}
\define\enor #1{\Bbb E \, \nor{#1}}
\define\tens #1{#1 \otimes #1}
\define\pr#1#2{\langle {#1} , {#2} \rangle}
\define\ba#1#2{\Cal B_{#2} ( {#1} )}

Let $y$ be a random vector in \rn satisfying
$$\Bbb E \, \tens{y} = id.$$
Let $M$ be a natural number and let $y_1 \etc y_M$ be independent copies
of $y$.
We study the question of approximation of the identity operator by finite
sums of the tensors $\tens{y_i}$.
Using the non--commutative Khintchine inequality we prove that for some
absolute constant $C$
$$\enor{\frac{1}{M} \sumi \tens{y_i} - id} \le C \cdot \frac{\sqrt{\log n}}{\sqrt{M}} \cdot \left ( \enor{y}^{\log M} \right )^{1/ \log M},$$
provided that the last expression is smaller than 1.

We apply this estimate to improve  a result of Bourgain
concerning the number of random points needed to bring a convex body
into a nearly isotropic position.

This research was started when the author had a post doctoral position at
MSRI.
Research at MSRI is supported in part  by NSF grant DMS-9022140.
Research was also supported in part by NSF grant DMS-9706835.

\bb

{\bf S.~M.~Shimorin.} See H.~Hedenmalm.

\bb

{\bf N.~A.~Shirokov.} {\it Nevanlinna factorization in one more class
of analytic functions}.

Let a positive function $\omega$ satisfy the conditions
$$c_1 \left({x \over y}\right)^\alpha \leq {\omega(y) \over \omega(x)} \leq c_2 \left({y \over x}\right)^\alpha \quad \hbox{for} \quad 0 -\infty . Let v be the outer function such that |v||_{\Bbb T}=|f|. If for any z \in {\Bbb D} satisfying (2) estimate (3) holds with a constant c_f not depending on z, then v \in \Lambda^{n-1}Z_\omega. The influence of the presence of an inner factor I in a factorization of f \in \Lambda^{n-1}Z_\omega can be described too. \bb {\bf A.~Yu.~Solynin.} {\it Symmetrization families and maximizing functionals}. For D\subset {\Bbb R}^n, y\in {\Bbb R}^{n-1} let l_D(y)=meas\{x_1:(x_1,y)\in D\}. By {\it symmetrization family} of open sets we mean a nonempty maximal family \Cal D such that$$
l_{D_1}(y)=l_{D_2}(y)
$$for all y\in {\Bbb R}^{n-1} and all D_1,D_2 \in {\Cal D}. Symmetrization families of condensers and functions can be defined in the same manner. A functional F:{\Cal D}\to {\Bbb R} is called {\it maximizing} ({\it minimizing}) if$$
\sup_{D\in {\Cal D}} F(D) \  (\inf_{D\in {\Cal D}} F(D)) \ =F(D_0)
$$for some D_0\in {\Cal D}. Several standart functional, such as conformal radius, harmonic measure, capacities, integral mens of solutions to certain PDE's, etc., can be shown to be maximizing. Recent results due to Dubinin, Solynin, Brock and Solynin show that functional mentioned above are strictly increasing under polarization. This leads to a rather simple proofs of isoperimetric inequalities: If F is a miximizing functional on a symmetrization family \Cal D and if F is strictly increasing under polarization, then$$
F(D)