Weights andL Φ-boundedness of the Poisson integral operator

In this paper, we get a necessary and sufficient condition on the weights (μ,v) for the Poisson integral operator to be bounded fromL _Φ(R ⁿ, v(x)dx) to weak-L _Φ(R ₊ ⁿ⁺¹ ,dμ), where Φ is anN-function satisfying the Δ₂-condition. We also find a necessary and sufficient condition on the weights (μ,v) for the Poisson integral operator to be bounded fromL _Φ(R ⁿ,v(x)dx) toL _Φ(R ₊ ⁿ⁺¹ ,dμ) under some additional condition. Partially supported by NNSF of P.R. China     

Additive Maps Preserving Nilpotent Operators or Spectral Radius

Let X be a （real or complex） Banach space with dimension greater than 2 and let B0（X） be the subspace of B（X） spanned by all nilpotent operators on X. We get a complete classification of surjective additive maps Ф on B0（X） which preserve nilpotent operators in both directions. In particular, if X is infinite-dimensional, we prove that Ф has the form either Ф（T） = cATA^-1 or Ф（T） = cAT＇A^-1, where A is an invertible bounded linear or conjugate linear operator, c is a scalar, T＇ denotes the adjoint of T. As an application of these results, we show that every additive surjective map on B（X） preserving spectral radius has a similar form to the above with ｜c｜ = 1.     

On some properties of Musielak-Orlicz spaces and their subspaces of order continuous elements

Some criterions in order thatl ¹ embeds complementably inE ^Φ(μ) and inL ^Φ(μ) are given. It is also proved that every idealL inL ^Φ(μ) such thatI _Φ(x/‖x‖Φ)=1 for anyxεL/{0} is contained inE ^Φ(μ).     

推广的Volterra复合算子的本性范数

本文研究单位圆盘上Bergman型空间到Zygmund型空间上的一类推广的Volterra复合算子.利用符号函数ϕ和g刻画这类算子的有界性、紧性,并计算其本性范数.     

E Φ andh Φ fail to beM-ideals inL Φ andl Φ in the case of the Orlicz norm

It is well known that in the case of the Luxemburg normE ^Φ (resp.h ^Φ) is anM ideal inL ^Φ (resp.l ^Φ), see [1], [9], [15] and [6]; [17] and [18]. It is proved in this paper that in the case of the Orlicz normE ^Φ (resph ^Φ) is anM-ideal inL ^Φ (resp.l ^Φ) iff Φ satisfies the suitable Δ₂ or Φ^*(a(Φ^*)), where a(Φ^*) is linear on the interval [0,u]} and Φ^* denotes the function complementary to Φ in the sense of Young. It is also proved that any linear continuous regular (i.e. order continuous) functional ξ_υ overE ^Φ (resp.h ^Φ) generated byv∈ L(h ^Φ*) (resp.v∈ L(h ^Φ*)) which attains its norm on the unit sphereS(E ^Φ) (resp.S(h ^Φ)), has a unique norm-preserving extension toL ^Φ (resp.l ^Φ). Finally, it is proved thatL ^Φ (resp.l ^Φ) has the property that any linear continuous regular functional ξ_υ overE ^Φ (resp.h ^Φ) has a unique norm-preserving extension toL ^Φ (resp.l ^Φ) iff Φ orE ^Φ satisfies the suitable Δ₂ and in the second case Φ^* attains the value 1.     

A combinatorial property of convex sets

A known result in combinatorial geometry states that any collection P _n of points on the plane contains two such that any circle containing them contains n/c elements of P _n , c a constant. We prove: Let Φ be a family of n noncrossing compact convex sets on the plane, and let S be a strictly convex compact set. Then there are two elements S _i , S _j of Φ such that any set S' homothetic to S that contains them contains n/c elements of Φ, c a constant (S' is homothetic to S if , where λ is a real number greater than 0 and is a vector of ). Our proof method is based on a new type of Voronoi diagram, called the ``closest covered set diagram' based on a convex distance function. We also prove that our result does not generalize to higher dimensions; we construct a set Φ of n disjoint convex sets in such that for any nonempty subset Φ _H of Φ there is a sphere S _H containing all the elements of Φ _H , and no other element of Φ. Received December 7, 1994, and in revised form March 25, 1996.     

The continuity of superposition operators on some sequence spaces defined by moduli

Let λ and μ be solid sequence spaces. For a sequence of modulus functions Φ = (ϕ k) let λ(Φ) = {x = (x _k ): (ϕk(|x _k |)) ∈ λ}. Given another sequence of modulus functions Ψ = (ψ_k), we characterize the continuity of the superposition operators P _f from λ(Φ) into μ (Ψ) for some Banach sequence spaces λ and μ under the assumptions that the moduli ϕk (k ∈ ℕ) are unbounded and the topologies on the sequence spaces λ(Φ) and μ(Ψ) are given by certain F-norms. As applications we consider superposition operators on some multiplier sequence spaces of Maddox type. This research was supported by Estonian Science Foundation Grant 5376.     

On the length of commutators in Chevalley groups

Let G = G(Φ,R) be the simply connected Chevalley group with root system Φ over a ring R. Denote by E(Φ,R) its elementary subgroup. The main result of the article asserts that the set of commutators C = {[a, b]|a ∈ G(Φ, R), b ∈ E(Φ, R)} has bounded width with respect to elementary generators. More precisely, there exists a constant L depending on Φ and dimension of maximal spectrum of R such that any element from C is a product of at most L elementary root unipotent elements. A similar result for Φ = A _l , with a better bound, was earlier obtained by Sivatski and Stepanov.     

Amemiya Norm Equals Orlicz Norm in Musielak-Orlicz Spaces

Let （Ω,μ） be a a-finite measure space and Φ ： Ω × [0,∞） → [0, ∞] be a Musielak-Orlicz function. Denote by L^Φ（Ω） the Musielak-Orlicz space generated by Φ. We prove that the Amemiya norm equals the Orlicz norm in L^Φ（Ω）.     

On the classification of bases in <Emphasis Type="Italic">P</Emphasis><Subscript>k</Subscript> according to the decidability of the completeness problem for automata

The completeness problem for bases of the form Φ ∪ ν, where Φ ⊆ P _k and ν is a finite system of automaton functions, is considered. Previously, the problem for k = 2 was solved by the author; it was also shown that there is an algorithm for determining the completeness of the system Φ∪ν when [Φ] = Pk. The paper is concerned with the case where [Φ] is the maximal (precomplete) class in P _k . The problem of completeness for systems Φ ∪ ν is shown to be undecidable if Φ is embedded in a Slupecki class and algorithmically decidable if Φ contains the class preserving all constants. Thus, the bases in P _k , k ≥ 3, can be classified according to their ability to guarantee the decidability of the completeness problem for automaton functions.     

Shellability and higher Cohen-Macaulay connectivity of generalized cluster complexes

Let Φ be a finite root system of rank n and let m be a nonnegative integer. The generalized cluster complex Δ^m(Φ) was introduced by S. Fomin and N. Reading. It was conjectured by these authors that Δ^m(Φ) is shellable and by V. Reiner that it is (m + 1)-Cohen-Macaulay, in the sense of Baclawski. These statements are proved in this paper. Analogous statements are shown to hold for the positive part Δ₊^m(Φ) of Δ^m(Φ). An explicit homotopy equivalence is given between Δ₊^m(Φ) and the poset of generalized noncrossing partitions, associated to the pair (Φ, m) by D. Armstrong.     

Stably free modules over rings of Laurent polynomials

Let Φ be a finite group and let A be the group algebra of a free abelian group over a field k. We show that, in general, A[Φ] admits nontrivial stably free modules. By contrast, if A is the group algebra of a finitely generated free group then A[Φ] has stably free cancellation.     

The modal logic of continuous functions on the rational numbers

Let L^\square°{{\mathcal L}^{\square\circ}} be a propositional language with standard Boolean connectives plus two modalities: an S4-ish topological modality □ and a temporal modality ◦, understood as ‘next’. We extend the topological semantic for S4 to a semantics for the language L^\square°{{\mathcal L}^{\square\circ}} by interpreting L^\square°{{\mathcal L}^{\square\circ}} in dynamic topological systems, i.e., ordered pairs 〈X, f〉, where X is a topological space and f is a continuous function on X. Artemov, Davoren and Nerode have axiomatized a logic S4C, and have shown that S4C is sound and complete for this semantics. S4C is also complete for continuous functions on Cantor space (Mints and Zhang, Kremer), and on the real plane (Fernández Duque); but incomplete for continuous functions on the real line (Kremer and Mints, Slavnov). Here we show that S4C is complete for continuous functions on the rational numbers.     

Clifford martingale Φ-Equivalence betweenS(f) andf*

TheL ²-norm equivalence between a Clifford martingalef and its square functionS(f) plays an important role in the proof of theL ²-boundedness of Cauchy integral operators on Lipschitz graphs and the CliffordT(b) Theorem [2, 4]. This note generalises the result to the Φ-equivalence between the maximal functionf* andS(f), where Φ is a nondecreasing and continuous function fromIR ⁺ toIR ⁺, of the moderate growth Φ(2u)≤C ₁Φ(u) and satisfies Φ(0)=0.     

Smoothness equivalence properties of univariate subdivision schemes and their projection analogues

We study the following modification of a linear subdivision scheme S: let M be a surface embedded in Euclidean space, and P a smooth projection mapping onto M. Then the P-projection analogue of S is defined as T := P ◦ S. As it turns out, the smoothness of the scheme T is always at least as high as the smoothness of the underlying scheme S or the smoothness of P minus 1, whichever is lower. To prove this we use the method of proximity as introduced by Wallner et al. (Constr Approx 24(3):289–318, 2006; Comput Aided Geom Design 22(7):593–622, 2005). While smoothness equivalence results are already available for interpolatory schemes S, this is the first result that confirms smoothness equivalence properties of arbitrary order for general non-interpolatory schemes.     

Raising approximation order of refinable vector by increasing multiplicity

An algorithm is presented for raising an approximation order of any given orthogonal multiscaling function with the dilation factor a. Let φ(x) = [φ1(x),φ2(x),…,φr(x)]T be an orthogonal multiscaling function with the dilation factor a and the approximation order m. We can construct a new orthogonal multiscaling function φnew(x) = [ φT(x). f3r 1(x),φr 2(x),…,φr s(x)}T with the approximation order m L(L ∈ Z ). In other words, we raise the approximation order of multiscaling function φ(x) by increasing its multiplicity. In addition, we discuss an especial setting. That is, if given an orthogonal multiscaling function φ(x) = [φ1 (x), φ2(x), …, φr(x)]T is symmetric, then the new orthogonal multiscaling function φnew(x) not only raise the approximation order but also preserve symmetry. Finally, some examples are given.     

An equivalence between varieties of cyclic Post algebras and varieties generated by a finite field

In this paper we give a term equivalence between the simple k-cyclic Post algebra of order p, L _p,k, and the finite field F(p ^k) with constants F(p). By using Lagrange polynomials, we give an explicit procedure to obtain an interpretation Φ₁ of the variety V(L _p,k) generated by L _p,k into the variety V(F(p ^k)) generated by F(p ^k) and an interpretation Φ₂ of V(F(p ^k)) into V(L _p,k) such that Φ₂Φ₁(B) = B for every B ε V(L _p,k) and Φ₁Φ₂(R) = R for every R ε V(F(p ^k)).     

Regenerative embedding of Markov sets

Summary.  Given a closed Markov (i.e. regenerative) set in [0,∞), we characterize the laws of the Markov sets which are regeneratively embedded into the latter. Typically, let Φ⁽¹⁾ and Φ⁽²⁾ be two Laplace exponents corresponding to two regenerative laws, and M ⁽²⁾ a Markov set with exponent Φ⁽²⁾. There exists a Markov set M ⁽¹⁾ with exponent Φ⁽¹⁾ which is regeneratively embedded into M ⁽²⁾ if and only if Φ⁽¹⁾/Φ⁽²⁾ is a completely monotone function. Several examples and applications are discussed. Received: 12 April 1996 / In revised form: 12 March 1997     

Spherical means and CR functions on the Heisenberg group

The injectivity of the spherical mean value operator on the Heisenberg group is studied. Whenf ∈L ^P (Hⁿ), 1 ≤p < ∞ it is proved that the spherical mean value operator is injective. When 1 ≤p ≤ 2,f(z, ·) ∈L ^P (ℝ) the same is proved under much weaker conditions in the z-variable. Some extensions of recent results of Agranovskyet al. regardingCR functions on the Heisenberg group are also obtained.     

Surjective partial differential operators on real analytic functions defined on open convex sets

Let P(D) be a partial differential operator with constant coefficients which is surjective on the space A(Ω) of real analytic functions on a covex open set Ω⊂ℝ ⁿ . Let L(P _m ) denote the localizations at ∞ (in the sense of H?rmander) of the principal part P _m . Then Q(x+iτN)≠ 0 for (x,τ)∈ℝ ⁿ ×(ℝ\{ 0}) for any Q∈L(P _m ) if N is a normal to δΩ which is noncharacteristic for Q. Under additional assumptions this implies that P _m must be locally hyperbolic. Received: 24 January 2000