The ultrametric corona problem

Let K be a complete ultrametric algebraically closed field and let A be the K-Banach algebra of bounded analytic functions in the disk D: |x| < 1. Let Mult(A, ∥ · ∥) be the set of continuous multiplicative semi-norms of A, let Mult _m (A, ∥ · ∥) be the subset of the ϕ ∈ Mult(A, ∥ · ∥) whose kernel is a maximal ideal and let Mult _a (A, ∥ · ∥) be the subset of the ϕ ∈ Mult _m (A, ∥ · ∥) whose kernel is of the form (x − a)A, a ∈ D ( if ϕ ∈ Mult _m (A, ∥ · ∥) \ Mult _a (A, ∥ · ∥), the kernel of ϕ is then of infinite codimension). We examine whether Mult _a (A, ∥ · ∥) is dense inside Mult _m (A, ∥ · ∥) with respect to the topology of simple convergence. This a first step to the conjecture of density of Mult _a (A, ∥ · ∥) in the whole set Mult(A, ∥ · ∥): this is the corresponding problem to the well-known complex corona problem. We notice that if ϕ ∈ Mult _m (A, ∥ · ∥) is defined by an ultrafilter on D, then ϕ lies in the closure of Mult _a (A, ∥ · ∥). Particularly, we show that this is case when a maximal ideal is the kernel of a unique ϕ ∈ Multm(A, ∥ · ∥). Particularly, when K is strongly valued all maximal ideals enjoy this property. And we can prove this is also true when K is spherically complete, thanks to the ultrametric holomorphic functional calculus. More generally, we show that if ψ ∈ Mult(A, ∥ · ∥) does not define the Gauss norm on polynomials (∥ · ∥), then it is defined by a circular filter, like on rational functions and analytic elements. As a consequence, if ψ ∈ Multm(A, ∥ · ∥) \ Multa(A, ∥ · ∥) or if φ does not lie in the closure of Mult _a (A, ∥ · ∥), then its restriction to polynomials is the Gauss norm. The first situation does happen. The second is unlikely. The text was submitted by the authors in English.     

The Kakeya maximal operator with a special base

LetM _ℛ f be the Kakeya maximal function in d-dimensional Euclidean space with some base ℛ, consisting of cylinders of eccentricity N. The inequality ∥M _ℛ f∥ _d ⩽c(logN)^ε∥ is shown for a base ℛ satisfying a direction condition, where ε and c are constants depending only on d. to the memory of Professor Ruilin Long The author is partly supported by the Grants-in-Aid for Scientific Research, The Ministry of Education, Science and Culture, Japan.     

Inner Radius of Univalence for a Strongly Starlike Domain

 The inner radius of univalence of a domain D with Poincaré density ρ _D is the possible largest number σ such that the condition ∥ S _f ∥ _D  = sup _{w∈ D} ρ _D (w) ⁻²∥ S _f (z) ∥ ≤ σ implies univalence of f for a nonconstant meromorphic function f on D, where S _f is the Schwarzian derivative of f. In this note, we give a lower bound of the inner radius of univalence for strongly starlike domains of order α in terms of the order α.     

Inner Radius of Univalence for a Strongly Starlike Domain

 The inner radius of univalence of a domain D with Poincaré density ρ _D is the possible largest number σ such that the condition ∥ S _f ∥ _D  = sup _{w∈ D} ρ _D (w) ⁻²∥ S _f (z) ∥ ≤ σ implies univalence of f for a nonconstant meromorphic function f on D, where S _f is the Schwarzian derivative of f. In this note, we give a lower bound of the inner radius of univalence for strongly starlike domains of order α in terms of the order α. The author was partially supported by the Ministry of Education, Grant-in-Aid for Encouragement of Young Scientists, 11740088. A part of this work was carried out during his visit to the University of Helsinki under the exchange programme of scientists between the Academy of Finland and the JSPS. Received November 26, 2001; in revised form September 24, 2002 Published online May 9, 2003     

Superstability of the generalized orthogonality equation on restricted domains

Chmielinski has proved in the paper [4] the superstability of the generalized orthogonality equation |〈f(x), f(y)〉| = |〈x,y〉|. In this paper, we will extend the result of Chmielinski by proving a theorem: LetD _n be a suitable subset of ℝⁿ. If a function f:D _n → ℝⁿ satisfies the inequality ∥〈f(x), f(y)〉| |〈x,y〉∥ ≤ φ(x,y) for an appropriate control function φ(x, y) and for allx, y ∈ D _n, thenf satisfies the generalized orthogonality equation for anyx, y ∈ D _n.     

Uniqueness of the uniform norm and adjoining identity in Banach algebras

LetA _e be the algebra obtained by adjoining identity to a non-unital Banach algebra (A, ∥ · ∥). Unlike the case for aC*-norm on a Banach *-algebra,A _e admits exactly one uniform norm (not necessarily complete) if so doesA. This is used to show that the spectral extension property carries over fromA to A _e . Norms onA _e that extend the given complete norm ∥ · ∥ onA are investigated. The operator seminorm ∥ · ∥_op onA _e defined by ∥ · ∥ is a norm (resp. a complete norm) iffA has trivial left annihilator (resp. ∥ · ∥_op restricted toA is equivalent to ∥ · ∥).     

Von Neumann-Jordan Constants of Absolute Normalized Norms on C^n

In this note, we give some estimations of the Von Neumann-Jordan constant C _{N J} (∥·∥_ψ) of Banach space (ℂ ⁿ , ∥·∥_ψ), where ∥·∥_ψ is the absolute normalized norm on ℂ ⁿ given by function ψ. In the case where ψ and φ are comparable, n=2 and C _{N J} (∥·∥_ψ)=1, we obtain a formula of computing C _{N J} (∥·∥_ψ). Our results generalize some results due to Saito and others. Received May 11, 2002, Accepted November 20, 2002 This work is partly supported by NNSF of China (No. 19771056)     

Convergence of an explicit finite volume scheme for first order symmetric systems

Summary.  This paper is devoted to the derivation of a O(h ^1/2) error estimate for the classical upwind, explicit in time, finite volume scheme for linear first order symmetric systems. Such a result already existed for the corresponding implicit in time finite volume scheme, since it can be interpreted as a particular case of the space-time discontinuous Galerkin method but the technique of proof, used in that case, does not extend to explicit schemes. The general framework, recently developed to analyse the convergence rate of finite volume schemes for non linear scalar conservation laws, can not be used either, because it is not adapted for systems, even linear. In this article, we propose a new technique, which takes advantage of the linearity of the problem. The first step consists in controlling the approximation error ∥u−u _h ∥ _L2 by an expression of the form <ν _h , g>−2<μ _h , gu>, where u is the exact solution, g is a particular smooth function, and μ _h , ν _h are some linear forms depending on the approximate solution u _h . The second step consists in carefully estimating the error terms <μ _h , gu> and <ν _h , g>, by using uniform stability results for the discrete problem and regularity properties of the continuous solution. Received December 20, 2001 / Revised version received January 2, 2001 / Published online November 27, 2002 Mathematics Subject Classification (1991): 65N30     

Toeplitz condition numbers as an <Emphasis Type="Italic">H</Emphasis><Superscript>∞</Superscript> interpolation problem

The condition numbers CN(T) = ∥T∥ · ∥T⁻¹∥ of Toeplitz and analytic n × n matrices T are studied. It is shown that the supremum of CN(T) over all such matrices with ∥T∥ ≤ 1 and the given minimum of eigenvalues r = min |λ_i| > 0 behaves as the corresponding supremum over all n × n matrices (i.e., as (Kronecker)), and this equivalence is uniform in n and r. The proof is based on a use of the Sarason-Sz.Nagy-Foias commutant lifting theorem. Bibliography: 2 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 355, 2008, pp. 173–179.     

Minimal entropy and simplicial volume

We prove that MinEnt (Y) ∥Y∥ = MinEnt(X) ∥X∥, for manifolds Y whose fundamental group is a subexponential extension of the fundamental group of some negatively curved, locally symmetric manifold X. This is a particular case of a more general result holding for an arbitrary representation ρ : π₁ (Y) →π₁ (X), which relates the minimal entropy and the simplicial volume of X to some invariants of the couple (Y, ker (ρ)). Then, we discuss some applications to the minimal volume problem and to Einstein metrics. Received: 23 December 1998     

Norms on unitizations of banach algebras revisited

Let A be an algebra without unit. If ∥ ∥ is a complete regular norm on A it is known that among the regular extensions of ∥ ∥ to the unitization of A there exists a minimal (operator extension) and maximal (ℓ₁-extension) which are known to be equivalent. We shall show that the best upper bound for the ratio of these two extensions is exactly 3. This improves the results represented by A. K. Gaur and Z. V. Kovářík and later by T. W. Palmer. The second author was partially supported by the grant No. 201/03/0041 of GAČR.     

L^p-gradient Estimates of Symmetric Markov Semigroups for 1 〈 p ≤ 2

For 1 〈 p ≤2, an L^p-gradient estimate for a symmetric Markov semigroup is derived in a general framework, i.e. ‖Γ^/2（Ttf）‖p≤Cp/√t‖p, where F is a carre du champ operator. As a simple application we prove that F1/2（（I- L） ^-α） is a bounded operator from L^p to L^v provided that 1 〈 p 〈 2 and 1/2〈α〈1. For any 1 〈 p 〈 2, q 〉 2 and 1/2 〈α 〈 1, there exist two positive constants cq,α,Cp,α such that ‖Df‖p≤ Cp,α‖（I - L）^αf‖p,Cq,α（I-L）^（1-α）‖Df‖q＋‖f‖q, where D is the Malliavin gradient （[2]） and L the Ornstein-Uhlenbeck operator.     

Commutators in model spaces

Let θ be an inner function, let K _θ = H ² ⊖ θH ², and let S_θ : K_θ → S_θ be defined by the formula S_θf = P_θzf, where f ∈ K_θ is the orthogonal projection of H² onto K_θ. Consider the set A of all trace class operators L : K_θ → K_θ, L = ∑(·,u_n)v_n, ∑∥u_n∥∥v_n∥ < ∞ (u_n, v_n ∈ K_θ), such that ∑ū_n v_n ∈ H ₀ ¹ . It is shown that trace class commutators of the form XS_θ − S_θX (where X is a bounded linear operator on K_θ) are dense in A in the trace class norm. Bibliography: 2 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 333, 2006, pp. 54–61.     

Integral inequalities with weights for maxiamal operators

Let μ be a measure on the upper half-space R ₊ ⁿ⁺¹ , and v a weight onR ⁿ, we give a characterization for the pair (v, μ) such that ∥M(fv)∥_{L Θ (μ}) ⩽ c ∥f∥_{L Θ (μ}), where Φ is an N-function satisfying Δ₂ condition andMf(x,t), is the maximal function onR ₊ ⁿ⁺¹ , which was introduced by Ruiz, F. and Torrea, J.. Supported by NSFC.     

Sur les a Algèbres Préhilbertiennes Vérifiant ∥a 2∥ ≤ ∥a∥2

Résumé.  Soit A une algèbre réelle sans diviseurs de zéro. On suppose que l’espace vectoriel A est muni d’une norme ∥.∥ préhilbertienne vérifiant ∥a ²∥ ≤ ∥a∥² pour tout . Alors A est de dimension finie dans chacun des quatre cas suivants :

Hydrodynamic limit for ∇φ interface model on a wall

 We consider random evolution of an interface on a hard wall under periodic boundary conditions. The dynamics are governed by a system of stochastic differential equations of Skorohod type, which is Langevin equation associated with massless Hamiltonian added a strong repelling force for the interface to stay over the wall. We study its macroscopic behavior under a suitable large scale space-time limit and derive a nonlinear partial differential equation, which describes the mean curvature motion except for some anisotropy effects, with reflection at the wall. Such equation is characterized by an evolutionary variational inequality. Received: 10 January 2002 / Revised version: 18 August 2002 / Published online: 15 April 2003 Mathematics Subject Classification (2000): 60K35, 82C24, 35K55, 35K85 Key words or phrases: Hydrodynamic limit – Effective interfaces – Hard wall – Skorohod's stochastic differential equation – Evolutionary variational inequality     

Norm Attaining Operators and Pseudospectrum

It is shown that if 1 < p < ∞ and X is a subspace or a quotient of an ℓ_p-direct sum of finite dimensional Banach spaces, then for any compact operator T on X such that ∥I + T∥ > 1, the operator I + T attains its norm. A reflexive Banach space X and a bounded rank one operator T on X are constructed such that ∥I + T∥  > 1 and I + T does not attain its norm. The author would like to thank E. Shargorodsky for his interest and comments.     

L P boundedness of area function associated with operators on product spaces

Let L be the infinitesimal generator of an analytic semigroup on L2 (Rn) with suitable upper bounds on its heat kernels. Assume that L has a bounded holomorphic functional calculus on L2(Rn). In this paper,we define the Littlewood- Paley g function associated with L on Rn × Rn, denoted by GL(f)(x1, x2), and decomposition, we prove that ‖SL(f)‖p ≈‖GL(f)‖p ≈‖f‖p for 1 ＜ p ＜∞.     

Lebesgue constants of Gaussian cardinal interpolation operators from l (ℤ) into L (ℝ)

Summary In recent years, with the attention to the radial-basis function by mathematicians, more and more research is concentrated on the Gaussian cardinal interpolation. The main purpose of this paper is to discuss the asymptotic behavior of Lebesgue constants of the Gaussian cardinal interpolation operator ℒ_λ from l (ℤ) into L (ℝ), that is, ∥ℒ_λ∥₁. We obtain the strong asymptotic estimate of the Lebesgue constants which improves the results of Riemenschneider and Sivakumar in [11].     

Small fractional parts of additive forms

Letf(x)=θ₁ x ₁ ^k +...+θ _s x _s ^k be an additive form with real coefficients, and ∥α∥ = min {|α-u|:uεℤ} denote the distance fromα to the nearest integer. We show that ifθ ₁,…,θ _s , are algebraic ands = 4k then there are integersx ₁,…,x _s , satisfying l ≤x ₁,≤ N and ∥f(x)∥ ≤ N ^E , withE = − 1 + 2/e. Whens = λk, 1 ≤λ ≤ 2k, the exponentE may be replaced byλE/4, and if we drop the condition thatθ ₁,…,θ _s , be algebraic then the result holds for almost all values of θεℝ ^s . Whenk ≥ 6 is small a better exponent is obtained using Heath-Brown’s version of Weyl’s estimate.