Hankel operators on the Bergman spaces of strongly pseudoconvex domains

We characterize functions fL ²(D) such that the Hankel operators H_f are, respectively, bounded and compact on the Bergman spaces of bounded strongly pseudoconvex domains.Research partially supported by a grant of the National Science Foundation.     

Permutations and piecewise-constant approximation of continuous functions of n variables

We consider the problem of approximation of a continuous function f given on an n-dirnensional cube by step functions in the metrics of C and L _p. We obtain exact error estimates in terms of the modulus of continuity of the function f or a special permutation of it. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 7, pp. 907–918, July, 1998. This work was partially supported by the International Soros Foundation for Support of Education Programs in the Natural Sciences (ISSEP) (grant No. EPU051030).     

Sharpening the LYM inequality

The level sequence of a Sperner familyF is the sequencef(F)={f _i (F)}, wheref _i (F) is the number ofi element sets ofF . TheLYM inequality gives a necessary condition for an integer sequence to be the level sequence of a Sperner family on ann element set. Here we present an indexed family of inequalities that sharpen theLYM inequality.Research supported in part by Alexander v. Humboldt-StiftungResearch supported in part by NSF under grant DMS-86-06225 and AFOSR grant OSR-86-0078Research supported in part by NSF grant CCR-8911388Research supported in part by OTKA 327 0113     

On the nondegeneracy of constant mean curvature surfaces

We prove that many complete, noncompact, constant mean curvature (CMC) surfaces are nondegenerate; that is, the Jacobi operator Δ_f + | A_f |² has no L² kernel. In fact, if ∑ has genus zero with k ends, and if f (∑) is embedded (or Alexandrov immersed) in a half-space, then we find an explicit upper bound for the dimension of the L² kernel in terms of the number of non-cylindrical ends. Our main tool is a conjugation operation on Jacobi fields which linearizes the conjugate cousin construction. Consequences include partial regularity for CMC moduli space, a larger class of CMC surfaces to use in gluing constructions, and a surprising characterization of CMC surfaces via spinning spheres. R.K. partially supported by NSF grants DMS-0076085 at GANG/UMass and DMS-9810361 at MSRI, and by a FUNCAP grant in Fortaleza, Brazil. J.R. partially supported by an NSF VIGRE grant at Utah. Received: January 2005; Accepted: June 2005     

Complexity of Bezout’s Theorem VII: Distance Estimates in the Condition Metric

We study geometric properties of the solution variety for the problem of approximating solutions of systems of polynomial equations. We prove that given the two pairs (f _i ,ζ _i ), i=1,2, there exist a short path joining them such that the complexity of following the path is bounded by the logarithm of the condition number of the problems. Research was partially supported by MTM2007-62799 and by an NSERC discovery grant.     

Polynomial preserving maps on certain Jordan algebras

LetB andQ be associative algebras and letS be a Jordan subalgebra ofB. Letf(x ₁,…,x _m ) be a (noncommutative) multilinear polynomial such thatS is closed underf. Letα:S→Q be anf-homomorphism in the sense that it is a linear map preservingf. Under suitable conditions it is shown thatα is essentially given by a ring homomorphism. An analogous theorem forf-derivations is also proved. The proofs rest heavily on results concerning functional identities andd-freeness. The second author was partially supported by a grant from the Ministry of Science of Slovenia.     

A note on holomorphic approximation in Banach spaces

Given a complex Banach space X and a holomorphic function f on its unit ball B, we discuss the problem whether f can be approximated, uniformly on smaller balls, by functions g holomorphic on all of X. Research partially supported by NSF grant DMS0700281.     

A characterization of the Morrey-Campanato spaces

In this paper, we give a new characterization of the Morrey–Campanato spaces by using the convolution _tB*f(x) to replace the minimizing polynomial P_Bf of a function f in the Morrey-Campanato norm, where is an appropriate Schwartz function.D.G. Deng and L.X. Yan are partially supported by NSF of China and the Foundation of Advanced Research Center, Zhongshan University. X.T. Duong and L.X. Yan are supported by a Discovery grant from Australia Research Council.     

Bounded characteristic functions and models for noncontractive sequences of operators

The Sz.-Nagy-FoiaŞ functional model for completely non-unitary contractions is extended to completely non-coisometric sequences of bounded operatorsT = (T₁,...,T _d) (d finite or infinite) on a Hilbert space, with bounded characteristic functions. For this class of sequences, it is shown that the characteristic function θ_T is a complete unitary invariant. We obtain, as the main result, necessary and sufficient conditions for a bounded multi-analytic operator on Fock spaces to coincide with the characteristic function associated with a completely non-coisometric sequence of bounded operators on a Hilbert space. Research supported in part by a COBASE grant from the National Research Council. The first author was partially supported by a grant from Ministerul Educaţiei Şi Cercetarii. The second author was partially supported by a National Science Foundation grant.     

Kähler–Ricci Flow With Small Initial Energy

In this paper, we prove that the K?hler–Ricci flow converges to a K?hler–Einstein metric when E ₁ energy is small. We also prove that E ₁ is bounded from below if and only if the K-energy is bounded from below in the canonical class. The first named author is partially supported by a NSF grant, while the third author was partially supported by a NSF supplement grant.     

The category of a map and the grade of a module

Letf: Y → X be a continuous map between connectedCW complexes. The homologyH _* (F) of the homotopy fibre is then a module over the loop space homologyH _* (ΩX). Theorem:If H _* (F; R) and H _* (ΩX; R) are R-free (R a principal ideal domain) then for some H _* (ΩX; R)-projective module P=P_>0 and for some m ≤ cat f: Some applications are also given. Research partially supported by a NATO travel grant. Research partially supported by an NSERC operating grant. U.A. au CNRS 751.     

On the global theory of projective mappings

We consider the theory of constant rank projective mappings of compact Riemannian manifolds from the global point of view. We study projective immersions and submersions. As an example of the results, letf:(M, g) → (N, g′) be a projective submersion of anm-dimensional Riemannian manifold (M, g) onto an (m−1)-dimensional Riemannian manifold (N, g′). Then (M, g) is locally the Riemannian product of the sheets of two integrable distributions Kerf _* and (Kerf _*)^⊥ whenever (M, g) is one of the two following types: (a) a complete manifold with Ric ≥ 0; (b) a compact oriented manifold with Ric ≤ 0. Translated fromMatematicheskie Zametki, Vol. 58, No. 1, pp. 111–118, July, 1995. This work was partially supported by the Russian Foundation for Basic Research grant No. 94-01-0195.     

On the irreducibility of sums of rational functions with separated variables

A rational function withn≥3 separated variables, i.e. a sumf ₁(X ₁)+f ₂(X ₂)+…+f _n (X _n ) with nonconstantf _i , is with exception of a special case in characteristicp always irreducible, i.e. has an irreducible numerator. This theorem was first proved by Schinzel. A different proof in characteristic 0 was given by Fried. We carry this proof to all characteristics. As an application we determine all rational functionsf with an addition formula of typef(x)+f(y)=f(h(x,y)) for some rational functionh. This work was partially supported by a grant from G.I.F. (German Israeli Foundation for Scientific Research and Development). I thank Wolfgang Ruppert and Moshe Jarden for helpful comments on a preliminary version of this paper. The paper was written during a stay at the Institute for Advanced Studies at the Hebrew University in Jerusalem. The author thanks the Institute for the warm hospitality which made the stay so pleasant.     

Sharp uniform convexity and smoothness inequalities for trace norms

Summary We prove several sharp inequalities specifying the uniform convexity and uniform smoothness properties of the Schatten trace idealsC _p , which are the analogs of the Lebesgue spacesL _p in non-commutative integration. The inequalities are all precise analogs of results which had been known inL _p , but were only known inC _p for special values ofp. In the course of our treatment of uniform convexity and smoothness inequalities forC _p we obtain new and simple proofs of the known inequalities forL _p .Oblatum 7-VII-1993Work partially supported by US National Science Foundation grant DMS 88-07243Work partially supported by US National Science Foundation grant DMS 92-07703Work partially supported by US National Science Foundation grant PHY90-19433 A02     

Uniqueness theorems for Korenblum type spaces

For a scale of spaces X of functions analytic in the unit disc, including the Korenblum space, and for some natural families ɛ of uniqueness subsets for X, we describe minorants for (X, ɛ), that is, non-decreasing functions M: (0, 1) → (0, ∞) such that f ∈ X, E ∈ ɛ, and log |f(z)| ≤ −M(|z|) on E imply f = 0. We give an application of this result to approximation by simple fractions with restrictions on the coefficients. The first author was partially supported by the ANR project DYNOP. The second author was partially supported by the Research Council of Norway, grant 160192/V30.     

Approximation by non-negative algebraic polynomials

Theorem 1 gives an estimate for the approximation of a continuous functionf by polynomials resulting from the convolution off with non-negative algebraic polynomialsp _n . Jackson's theorem can be deduced from it by choosing a particularp _n whose second Chebyshev-Fourier coefficient is sufficiently close to –1.Work supported in part by the Atomic Energy Commission under contract U.S. AEC AT (11-1) 1469, and in part by the National Science Foundation under grant NSF-GJ-812.     

Length functions of lemniscates

We study metric and analytic properties of generalized lemniscates E _t (f)={z:ln|f(z)|=t}, where f is an analytic function. Our main result states that the length function |E _t (f)| is a bilateral Laplace transform of a certain positive measure. In particular, the function ln|E _t (f)| is convex on any interval free of critical points of ln|f|. As another application we deduce explicit formulae of the length function in some special cases.The author was supported the Göran Gustafsson foundation and grant RFBR no. 03-01-00304.The author was supported by Russian President grant for young doctorates no. 00-15-99274 and grant RFBR no. 03-01-00304. Mathematics Subject Classification (2000):30E05, 42A82, 44A10     

On the value distribution of a certain type of differential polynomials

Some quantitative estimations on the value distribution of the functionafg ⁿ(n2) are obtained, whereg is a linear differential polynomial inf anda is a small function off.The research was partially supported by a U.G.C grant of Hong Kong     

Asymptotic expansion and extrapolation for Bernstein polynomials with applications

Given a real functionf C ^2k [0,1],k 1 and the corresponding Bernstein polynomials {B _n (f)} _n we derive an asymptotic expansion formula forB _n (f). Then, by applying well-known extrapolation algorithms, we obtain new sequences of polynomials which have a faster convergence thanB _n (f). As a subclass of these sequences we recognize the linear combinations of Bernstein polynomials considered by Butzer, Frentiu, and May [2, 6, 9]. In addition we prove approximation theorems which extend previous results of Butzer and May. Finally we consider some applications to numerical differentiation and quadrature and we perform numerical experiments showing the effectiveness of the considered technique.This work was partially supported by a grant from MURST 40.     

When is the union of two unit intervals a self-similar set satisfying the open set condition?

Let U _λ be the union of two unit intervals with gap λ. We show that U _λ is a self-similar set satisfying the open set condition if and only if U _λ can tile an interval by finitely many of its affine copies (admitting different dilations). Furthermore, each such λ can be characterized as the spectrum of an irreducible double word which represents a tiling pattern. Some further considerations of the set of all such λ’s, as well as the corresponding tiling patterns, are given. The first author was partially supported by the RGC grant and the direct grant in CUHK, Fok Ying Tong Education Foundation and NSFC (10571100). The second author was partially supported by NSFC (70371074) and NFSC (10571104).