Boolean Functions with small Spectral Norm

Let be a boolean function, and suppose that the spectral norm of f is at most M. Then where and each H _j is a subgroup of . This result may be regarded as a quantitative analogue of the Cohen-Helson-Rudin structure theorem for idempotent measures in locally compact abelian groups. Received: May 2006 Accepted: January 2007     

Cylinder Multipliers Associated with a Convex Polygon

In this paper we prove that sharp weak-type estimates for the maximal operators associated with a cylindric distance function associated with a convex polygon on when 2/3 < p < 1 and , or when p = 2/3 and . This work was supported (in part) by research funds from Chosun University, 2007.     

A priori estimates for the vector p-Laplacian with potential boundary conditions

This paper deals with the solvability of the boundary value problem

On Separation of Points from Additive Subgroups of Banach Spaces by Continuous Characters and Positive Definite Functions

Let G be an additive subgroup of a normed space X. We say that a point is weakly separated (resp. -separated) from G if it can be separated from G by a continuous character (resp. by a continuous positive definite function). Let T : X → Y be a continuous linear operator. Consider the following conditions: (ws) if , then x is weakly separated from G; (ps) if , then x is -separated from G; (wp) if Tx is -separated from T(G), then x is weakly separated from G. By (resp. , ) we denote the class of operators T : X → Y which satisfy (ws) (resp. (ps), (wp)) for all and all subgroups G of X. The paper is an attempt to describe the above classes of operators for various Banach spaces X, Y. It is proved that if X, Y are Hilbert spaces, then is the class of Hilbert-Schmidt operators. It is also shown that if T is a Hilbert-to-Banach space operator with finite ℓ-norm, then .      

Infimal convolution and Muckenhoupt A p (.) condition in variable L p spaces

We study the Hardy-Littlewood maximal operator M on . Under the assumptions that the exponent p satisfies and is constant outside some large ball, we prove that if and only if . Received: 2 June 2006 Revised: 28 November 2006     

Multiplicity of standing wave solutions of nonlinear Schrödinger equations with electromagnetic fields

In this paper, we are concerned with the multiplicity of standing wave solutions of nonlinear Schr?dinger equations with electromagnetic fields

Asymptotic behaviour for a nonlocal diffusion equation on a lattice

In this paper we study the asymptotic behaviour as t → ∞ of solutions to a nonlocal diffusion problem on a lattice, namely, with t ≥ 0 and . We assume that J is nonnegative and verifies . We find that solutions decay to zero as t → ∞ and prove an optimal decay rate using, as our main tool, the discrete Fourier transform.      

An asymptotic method for calculating limits of maximal means

For a continuous almost periodic function , we show that the function

Aluthge Transforms of Complex Symmetric Operators

If denotes the polar decomposition of a bounded linear operator T, then the Aluthge transform of T is defined to be the operator . In this note we study the relationship between the Aluthge transform and the class of complex symmetric operators (T iscomplex symmetric if there exists a conjugate-linear, isometric involution so that T = CT*C). In this note we prove that: (1) the Aluthge transform of a complex symmetric operator is complex symmetric, (2) if T is complex symmetric, then and are unitarily equivalent, (3) if T is complex symmetric, then if and only if T is normal, (4) if and only if T ² = 0, and (5) every operator which satisfies T ² = 0 is necessarily complex symmetric. This work partially supported by National Science Foundation Grant DMS 0638789.     

Dicritical holomorphic flows on Stein manifolds

We study holomorphic flows on Stein manifolds. We prove that a holomorphic flow with isolated singularities and a dicritical singularity of the form on a Stein manifold with , is globally analytically linearizable; in particular M is biholomorphic to . A complete stability result for periodic orbits is also obtained. Bruno Scárdua: Partially supported by ICTP-Trieste-Italy. Received: 27 September 2006     

A superalgebraic interpretation of the quantization maps of Weil algebras

Let G be a Lie group whose Lie algebra g is quadratic. In the paper ＂the non-commutative Weil algebra＂, Alekseev and Meinrenken constructed an explicit G-differential space homomorphism ￡, called the quantization map, between the Well algebra Wg = S（g^＊） χ∧A（g^＊） and Wg= U（g） χ Cl（g） （which they call the noncommutative Weil algebra） for g. They showed that ￡ induces an algebra isomorphism between the basic cohomology rings Hbas^＊（Wg） and Hbas^＊（Wg）. In this paper, we will interpret the quantization map .~ as the super Duflo map between the symmetric algebra S（Tg[1]） and the universal enveloping algebra U（Tg[1]） of a super Lie algebra T9[1] which is canonically associated with the quadratic Lie algebra g. The basic cohomology rings Hbas^＊（Wg） and Hbas^＊（Wg） correspond exactly to S（Tg[1]）^inv and U（Tg[1]）^inv, respectively. So what they proved is equivalent to the fact that the super Duflo map commutes with the adjoint action of the super Lie algebra, and that the super Duflo map is an algebra homomorphism when restricted to the space of invariants.     

A question in the theory of saturated formations of finite soluble groups

This paper examines the following question. If and are saturated formations then is defined to be the class of all soluble groups whose belong to . In general is a formation, but need not be a saturated formation. Here the smallest saturated formation containing is studied.     

Isometries and direct decompositions of pseudo MV-algebras

In the paper isometries in pseudo MV-algebras are investigated. It is shown that for every isometry f in a pseudo MV-algebra = (A, ⊕, ⁻, ^∼, 0, 1) there exists an internal direct decomposition of with commutative such that and for each x ∈ A. On the other hand, if is an internal direct decomposition of a pseudo MV-algebra = (A, ⊕, ⁻, ^∼, 0, 1) with commutative, then the mapping g: A → A defined by is an isometry in and .      

Lower bounds of some singular integrals and their applications

Let and . We are interested in the lower bounds of the integral:

The Hopf algebra Rep U_{q} \widehat{\frak g \frak l}_\infty

We define the Hopf algebra structure on the Grothendieck group of finite-dimensional polynomial representations of in the limitN→∞. The resulting Hopf algebra Rep is a tensor product of its Hopf subalgebras Rep_a ,a ∈ ℂ^×/q^2ℤ. Whenq is generic (resp.,q ² is a primitive root of unity of orderl), we construct an isomorphism between the Hopf algebra Rep _a and the algebra of regular functions on the prounipotent proalgebraic group (resp., ). Whenq is a root of unity, this isomorphism identifies the Hopf subalgebra of Rep _a spanned by the modules obtained by pullback with respect to the Frobenius homomorphism with the algebra generated by the coefficients of the determinant of an element of considered as anl×l matrix over the Taylor series. This gives us an explicit formula for the Frobenius pullbacks of the fundamental representations. In addition, we construct a natural action of the Hall algebra associated to the infinite linear quiver (resp., the cyclic quiver withl vertices) on Rep _a and describe the span of tensor products of evaluation representations taken at fixed points as a module over this Hall algebra.     

Baer cotorsion Pairs

LetR be a unital associative ring and two classes of leftR-modules. In [St3] the notion of a ( ) pair was introduced. In analogy to classical cotorsion pairs, a pair (V,W) of subclasses is called a ( ) pair if it is maximal with respect to the classes and the condition Ext _R ¹ (V, W)=0 for all . In this paper we study pairs whereR = ℤ and is the class of all torsion-free abelian groups andT is the class of all torsion abelian groups. A complete characterization is obtained assumingV=L. For example, it is shown that every pair is singly cognerated underV=L. The author was supported by a DFG grant.     

Selections of lattice-ordered Groups

It is well known that the quasitorsion class of archimedean -groups is the class of -groups G such that every closed convex -subgroup is a polar, and it is also well known that the class of -groups G such that every convex -subgroup is a polar is a torsion class. By defining a selection on -groups, these two results are generalized to show, whenever and are selections on -groups, the class of -groups G such that is a radical class. Three selections in particular — all convex -subgroups, all polars, and all closed convex -subgroups — and the radical classes determined by them are studied in some detail. Received March 7, 2006; accepted in final form August 29, 2006.     

On a subalgebra of the centre of a group ring II

Let p be a prime, G a finite group with p | |G| and F a field of characteristic p. By we denote the F-subspace of the centre of the group ring FG spanned by the p-regular conjugacy class sums. J. Murray proved that is an algebra, if G is a symmetric or alternating group. This can be used for the computation of the block idempotents of FG. We proved that is an algebra if the Sylow-p-subgroups of G are abelian. Recently, Y. Fan and B. Külshammer generalized this result to blocks with abelian defect groups. Here, we show that is an algebra if the Sylow-2-subgroups of G are dihedral. Therefore and are algebras for all primes p and all prime powers q. Furthermore we prove that is an algebra for the simple Suzuki-groups Sz(q), where q is a certain power of 2 and p is an arbitrary prime dividing |Sz(q)|. Received: 18 May 2007     

On meromorphic solutions of f 3 + g 3 = 1

We show that meromorphic solutions f, g of f ³ + g ³ = 1 in must be constant, if and have the same zeros (counting multiplicities). Received: 12 March 2007