Weighted Hardy-Type Inequalities for Differences and the Extension Problem for Spaces with Generalized Smoothness

It is well known that there are bounded domains Rⁿ whose boundaries are not smooth enough for there to exist a bounded linear extensionfor the Sobolev space into , but the embedding is nevertheless compact. For the Lipboundaries (0<<1) studied in [3, 4], there does not existin general an extension operator of into but there is a bounded linear extension of into and the smoothness retained by thisextension is enough to ensure that the embedding is compact. It is natural to ask if this is typicalfor bounded domains which are such that is compact, that is, that there exists a boundedextension into a space of functions in Rⁿ which enjoy adequatesmoothness. This is the question which originally motivatedthis paper. Specifically we study the ‘extension by zero’operator on a space of functions with given ‘generalized’smoothness defined on a domain with an irregular boundary, anddetermine the target space with respect to which it is bounded.     

Reducing Subspaces for a Class of Multiplication Operators

Let D be the open unit disk in the complex plane C. The Bergmanspace is the Hilbert space of analytic functions f in D such that where dA is the normalized area measure on D. If are two functions in , then the inner product of f and g is given by We study multiplication operators on induced by analytic functions. Thus for H (D), the space ofbounded analytic functions in D, we define by It is easy to check that M is a bounded linear operator on with ||M||=||||=sup{|(z)|:zD}.     

Multipliers on Banach spaces of functions on a locally compact Abelian group

Let E be a Banach space of functions on a locally compact Abeliangroup G satisfying certain conditions. It has been proved thatfor every bounded operator M on E commuting with translationsthere exists such that , where is a suitable subset of the group of the continuous morphismsfrom G into * and is a generalized Fourier transform of g defined on .     

Total Chromatic Number of Graphs of Order 2n + l having Maximum Degree 2n 1

Let G be a graph of order 2n + l having maximum degree 2n –1. We prove that the total chromatic number of G is 2n if andonly if e + ' n, where w is a vertex of minimum degree in G, is the complement of G —w, e is the size of , and ' is the edge independence number of .     

Some Compactness Results in Real Interpolation for Families of Banach Spaces

The paper shows that, if the operator T:A()B() is compact foralmost every , then is compact when or is the interpolation functor constructed for infinitefamilies of Banach spaces and S satisfies certain conditions.     

Ergodicity of a Class of Cocycles Over Irrational Rotations

It is proved that if is irrational and L²(S¹) with o(l/n)then for each mZ\{0} the corresponding skew product is ergodic. The rigidity of specialflows over irrational rotations with roof functions whose Fouriercoefficients are in o(l/n) is also shown.     

Asymptotic Expansions of Multiple Zeta Functions and Power Mean Values of Hurwitz Zeta Functions

Let (s, ) be the Hurwitz zeta function with parameter . Powermean values of the form are studied, where q and h are positive integers. These mean valuescan be written as linear combinations of , where _r(s₁,...,s_r;) is a generalization of Euler–Zagiermultiple zeta sums. The Mellin–Barnes integral formulais used to prove an asymptotic expansion of , with respect to q. Hence a general way of deducingasymptotic expansion formulas for is obtained. In particular, the asymptotic expansion of with respect to q is written down.     

Codeterminants for the Symplectic Schur Algebra

Standard codeterminants for Donkin's symplectic Schur algebra are defined. It is shown that they form a basis of . They are then used to give a purely combinatorial proof that is quasi-hereditary.     

One-Sided BMO Spaces

In this paper we introduce the one-sided sharp functions dennedby and where z⁺ = max(z, 0). We study the BMO spaces associatedto and and their relation with the good weights for theone-sided Hardy-Littlewood maximal functions. Finally, as anapplication of our results, we characterize the weights forone-sided fractional integrals and one-sided fractional maximaloperators.     

The Sector of Analyticity of the Ornstein-Uhlenbeck Semigroup on Lp Spaces with Respect to Invariant Measure

The sector of analyticity of the Ornstein–Uhlenbeck semigroupis computed on the space := L^p (R^N; µ) with respect to its invariant measure µ.If A= + Bx· denotes the generator of the Ornstein–Uhlenbecksemigroup, then the angle ₂ of the sector of analyticity in is /2 minus the spectral angleof BQ, Q being the matrix determining the Gaussian measure µ.The angle of analyticity in is then given by the formula      

Autoduality of the Compactified Jacobian

The following autoduality theorem is proved for an integralprojective curve C in any characteristic. Given an invertiblesheaf L of degree 1, form the corresponding Abel map A_L:C, which maps C into its compactifiedJacobian, and form its pullback map , which carries the connected component of 0 in the Picard schemeback to the Jacobian. If C has, at worst, points of multiplicity2, then is an isomorphism, and forming it commutes with specializing C. Much of the work in the paper is valid, more generally, fora family of curves with, at worst, points of embedding dimension2. In this case, the determinant of cohomology is used to constructa right inverse to . Then a scheme-theoretic version of the theorem of the cube is proved,generalizing Mumford's, and it is used to prove that is independent of the choice of L.Finally, the autoduality theorem is proved. The presentationscheme is used to achieve an induction on the difference betweenthe arithmetic and geometric genera; here, special propertiesof points of multiplicity 2 are used.     

On Ramanujan's Inequalities for exp(k)

Ramanujan claimed in his first letter to Hardy (16 January 1913)that where (k) lies between 2/21 and 8/45. This conjecture was provedin 1995 by Flajolet et al. The paper establishes the followingrefinement. where Both bounds for ^*(k) are sharp.     

Inverse Spectral Problems for Sturm-Liouville Equations with Eigenparameter Dependent Boundary Conditions

Inverse Sturm–Liouville problems with eigenparameter-dependentboundary conditions are considered. Theorems analogous to thoseof both Hochstadt and Gelfand and Levitan are proved. In particular, let ly = (1/r)(–(py')'+qy), , where det = > 0, c 0, det > 0, t 0 and (cs + dr –au – tb)² < 4(cr – ta)(ds – ub). Denoteby (l; ; ) the eigenvalue problem ly = y with boundary conditionsy(0)cos+y'(0)sin = 0 and (a+b)y(1) = (c+d)(py')(1). Define (; ; ) as above but with l replacedby . Let w_n denote the eigenfunctionof (l; ; ) having eigenvalue n and initial conditions w_n(0)= sin and pw'_n(0) = –cos and let _n = –aw_n(1)+cpw'_n(1).Define _n and _n similarly. As sample results, it is proved that if (l; ; ) and (; ; ) have the same spectrum, and (l;; ) and (; ; ) have the samespectrum or for all n, thenq/r = /.     

An Entire Function Defined by a Nonlinear Recurrence Relation

For the nonlinear recurrence relation it is proved that the limit exists and defines an entire function of ₂ = ₁(1-₁).     

On the Index of Vector Fields Tangent to Hypersurfaces with Non-Isolated Singularities

Let F be a germ of a holomorphic function at 0 in Cⁿ⁺¹, having0 as a critical point not necessarily isolated, and let be a germ of a holomorphic vectorfield at 0 in Cⁿ⁺¹ with an isolated zero at 0, and tangent toV := F^–1(0). Consider the O_V,0-complex obtained by contractingthe germs of Kähler differential forms of V at 0 (0.1) with the vector field X:=|_Von V: (0.2)     

The Moduli Spaces of Bielliptic Curves of Genus 4 with more Bielliptic Structures

Let C be an irreducible, smooth, projective curve of genus g 3 over the complex field C. The curve C is called biellipticif it admits a degree-two morphism : C E onto an ellipticcurve E such a morphism is called a bielliptic structure onC. If C is bielliptic and g6, then the bielliptic structureon C is unique, but if g=3,4,5, then this holds true only genericallyand there are curves carrying n>>1 bielliptic structures.The sharp bounds n 21,10,5 exist for g=3,4,5 respectively.Let M_g be the coarse moduli space of irreducible, smooth, projectivecurves of genus g=3,4,5. Denote by the locus of points in M_g $ representing curves carrying atleast n bielliptic structures. It is then natural to ask thefollowing questions. Clearly does hold? What are the irreducible components of ? Are the irreducible components of rational? How do the irreducible components of intersect each other? Let how many non-isomorphic elliptic quotients can it have? Completeanswers are given to the above questions in the case g=4.     

Link Polynomials of Higher Order

In this paper, we study certain polynomial invariants of links(singular or non-singular) that are related to the Homfly polynomialand Vassiliev's invariants. The Homfly polynomial H_L [3] (alsoknown as the Flypmoth polynomial) satisfies the well-known skeinrelation The Vassiliev invariants [1, 2] (of order 1) satisfy the relations and The invariants that we study satisfy the skein relations      

Exactness and Uniform Embeddability of Discrete Groups

A numerical quasi-isometry invariant R() of a finitely generatedgroup is defined whose values parametrize the difference between being uniformly embeddable in a Hilbert space and () being exact.     

Tangential Boundary Behaviour of Harmonic and Holomorphic Functions

Let K be a kernel on Rⁿ, that is, K is a non-negative, unboundedL¹ function that is radially symmetric and decreasing. We definethe convolution K * F by and note from L^p-capacity theory [11, Theorem 3] that, if F L^p, p > 1, then K * F exists as a finite Lebesgue integraloutside a set A Rⁿ with C_K,p(A) = 0. For a Borel set A, where We define the Poisson kernel for = {(x, y) : x Rⁿ, y > 0} by and set Thus u is the Poisson integral of the potential f = K * F, andwe write u=P_y*(K*F)=P_y*f=P[f]. We are concerned here with the limiting behaviour of such harmonicfunctions at boundary points of , and in particular with the tangential boundary behaviour ofthese functions, outside exceptional sets of capacity zero orHausdorff content zero.